-
1
INTRODUCTION All researchers used statistics to help reach their
conclusions that would have been impossible to make with any
degrees of scientific validity without the benefit of statistics.
Researchers needed to use statistics as a tool to help them gain
perspective on the particular problems of interest to them. Why
learn statistics?
Statistics is an integral part of research activity Important
questions and issues are addressed in research and statistics can
be a valuable tool in
developing answers to these questions In conducting research,
statistical analysis will prove to be a useful aid in the
acquisition of knowledge Knowledge in statistics is important to
help one understand and interpret the reports. A knowledge of
statistical analysis helps to foster new and creative ways of
thinking about problems Statistical thinking can be a useful aid in
suggesting alternative answers to questions and posing new
ones Statistics helps to develop ones skills in critical
thinking, with both inductive and deductive inference Science is
best characterized as an interplay between theory and data and
statistics serves as a bridge
between theory and data
VARIABLES Most research is concerned with variables which is a
phenomenon that takes on different values of levels. In contrast, a
constant does not vary within given constraints. Researchers
distinguish between variables. One distinction is between an
independent variable and a dependent variable. Example: Suppose a
researcher is interested in the relationship between two variables:
the effect of information about the gender of a job applicant on
hiring decisions made by personnel managers. An experiment might be
designed in which 50 personnel managers are provided with
descriptions of a job applicant and asked whether they would hire
that applicant. The applicant is described to all 50 managers in
the same way on several pertinent dimensions. The only difference
is that 25 of the managers are told that the applicant is a woman,
and the other 25 managers are told that the applicant is a man.
Each manager then indicates his or her hiring decision. In this
experiment, the gender of the applicant is the independent variable
and the hiring decision is the dependent variable. The hiring
decision is termed the dependent variable because it is thought to
depend on the information about the gender of the applicant. The
gender of the applicant is termed the independent variable because
it is assumed to influence the dependent variable and does not
depend on the other variable (i.e. hiring decision). A useful tool
for identifying independent and dependent variables is the phrase
The effect of (independent variable) on (dependent variable). For
example: in a study on the effect of psychological stress on blood
pressure, the independent variable is the amount of psychological
stress an individual is feeling and the dependent variable is the
individual blood pressure. Similarly, if the effect of
child-rearing practices on intelligence is studied, the independent
variable is the type of child-rearing practice and the dependent
variable is the childs intelligence. The term independent variable
in general is any variable that is presumed to influence the
dependent variable. The distinction between independent and
dependent variables parallels cause-and-effect thinking with
independent variable being the cause and the dependent variable
being the effect. When reading studies or evaluating certain
statistics, it is useful to make distinctions between presumed
causes and the presumed effects.
-
2
STATISTICS DEFINED
Statistical investigations and analyses of data fall into two
broad categories:
THE NATURE OF DATA MEASUREMENT
A major feature of scientific research is measurement.
Measurement involves translating empirical relationships between
objects into numerical relationships. This frequently takes the
form of assigning numbers to respondents (or objects) in such a way
that the numbers have meaning and convey information about
differences between respondents (or objects). The four types or
levels of measurement used in sciences are (a) nominal, (b)
ordinal, (c) interval and (d) ratio. However, some scientific
researches, interval and ration were collapsed as one, thus, (a)
nominal, (b) ordinal or ranks and (c) interval or ratio.
STATISTICS
SPECIFIC NUMBERS: numerical measurement determined by a set
of
data
Twenty-three percent of people polled believed that learning
statistics is difficult
METHOD OF ANALYSIS: a collection of methods for planning
experiments, obtaining data, and then then organizing, summarizing,
presenting, analyzing, interpreting, and
drawingconclusions based on the data
STATISTICS
(Collection, Organization, Summary,
Presentation, Analysis and
Interpretation of Data)
DESCRIPTIVE
-deals with processing data without attempting to draw any
inferences/conclusions from them. It
refers to the representation of data in the form of tables,
graphs and to the description of some
characteristics of the data, such as averages and
deviations.
INFERENTIAL (INDUCTIVE)
-is a scientific discipline concerned with developing and using
mathematical tools to make forecasts and inferences. Basic to
the
development and understanding of inferential/inductive
statistics are the concepts
of probability theory.
-
3
4 LEVELS OF MEASUREMENT: NOMINAL, ORDINAL, INTERVAL AND
RATIO
Nominal measurement involves using numbers merely as labels. A
researcher might classify a group of people according to their
religion Catholic, Protestant, Jewish and all others and use
numbers 1, 2, 3, and 4 for these categories. Also gender is a
nominal measurement where male might take a value of 1 and 2 for a
female. In the nominal level, the numbers have no special quality
about them; they are used merely as labels. In research, the basic
statistics of interest for variables that involve nominal level are
frequencies, proportions, and percentages. Because nominal data
lack any ordering or numerical significance, they cannot be used
for calculations. Numbers are sometimes assigned to the different
categories (especially when data are computerized), but these
numbers have no real computational significance and any average
calculated with them is meaningless Ordinal measurement. A variable
is said to be measured on an ordinal level when the categories can
be ordered on some dimension. Suppose that a researcher is studying
the effects of stress during schooling on the grades as one index
of academic performance. The researcher takes several students who
differ in letter grades and assigns the number 1 to a grade of A,
the number 2 to a grade of B, the number 3 to the next letter
grade, and the number 4 to the last and lowest letter grade. In
this case, letter grade is measured on an ordinal level, which
allows the students to be ordered from best to worst. Another
example is when the researcher wants to know how often do teenagers
aged 18 and above watch R-18 slasher/horror movies. The researcher
will take respondents who differs in the extent of watching and
assigns the number 4 to always, the number 3 to oftentimes, the
number 2 to sometimes, the number 1 to seldom and the number 0 to
never. Thus, with ordinal level, the researcher classifies
individual into different categories but are ordered along a
dimension of interest. Ordinal data provide information about
relative comparisons, but not the magnitudes of the differences.
They should not be used for calculations. Interval measurement.
Interval measures have all the properties of ordinal measures but
allows us to do more than order objects on a dimension. They also
provide information about the magnitude of the differences between
objects. For example, interval measures not only would tell us that
one student is better in math than another, but also would convey a
sense of how much better one student is than another. Technically
speaking, interval measures have property that numerically equal
distance on the scale represents equal distances on the dimension
being measured. Also, in an interval level, measurements do not
start from 0 starting point like the cases of number grades and
calendar year. For example, a researcher might study the
relationship between IQ scores and EQ intelligences. The difference
between an IQ score of 50 and 100 is the same as the difference
between an IQ score of 100 and 150. In both instances, the
difference of 50 points corresponds to the same absolute amount of
scores. Interval measures provide information about the magnitude
of differences because of this useful property. However, since
interval measurements have no 0 starting point,
DATA
QUALITATIVE
NOMINAL ORDINAL
QUANTITATIVE
INTERVAL RATIO
-
4 we cannot say that a person whose IQ score of 100 is twice as
intelligent, than a person whose IQ score is only 50. Likewise a
temperature reading of 30oC does not mean 3 times hotter than a
temperature reading of 10oC (this is also true for degrees
Fahrenheit) unless the unit is degrees Kelvin. Ratio measurement.
Ratio measures have all the properties of interval measures but
provide even more information. Specifically, ratio measures have 0
starting point that map onto underlying dimension in such a way
that ratios between the numbers represent ratios of the dimension
being measured. For example, if we use inches to measure the
underlying dimension of height, in the case that a child who is 50
inches tall is twice the height of a child who is 25 inches tall.
Similarly, a student who got a score of 75 points in a 100-point
test has thrice the score of a student who got a score of 25
points. Moreover, a runner who runs a 1-km distance in a time of 10
minutes is twice as faster than a runner who runs the same distance
in a time of 20 minutes. Three Different Ways of Measuring the
Heights of Four Building
We have measured height on a n interval scale. Note that on this
scale, even though building B has a score of 4(that is 4 feet above
the criterion) and building D has a score of 2 (2 feet above the
criterion), it is not the case that building B is twice as tall as
building D. We cannot make a ratio statement because all measures
were taken relative to an arbitrary criterion (100 feet). Finally,
we can measure each building from the ground (0 as a starting
point) which is a true zero point rather than a n arbitrary
criterion. Building D is 102 feet high, building B is 104 feet
high, building C is 180 feet high and building A is 204 feet high
(figure d). We can now state with confidence that building A is
twice as tall as building D.
The figure on the left shows graphically the heights of four
buildings and indicates how tall each one is. The first way of
measuring the heights of these buildings is to assign the number 1
to the shortest building, the number 2 to the next shortest
building, the number 3 to the nest and the number 4 to the tallest
building (figure b). This assignment represents ordinal
measurement. It allows us to order the buildings on the dimension
of height but it does not tell us anything about the magnitude of
the heights. A second method is to measure by how many feet each
building exceeds the 100-feet criterion. In this case, building D
is 2 feet taller than the criterion, building B is 4 feet taller
than the criterion, building C is 80 feet taller than the criterion
and building A is 104 feet taller than the criterion (figure c). In
contrast to ordinal level, now not only can we order the buildings
on a dimension of height, but also we have information about the
relative magnitudes of the heights. Building B is 2 feet taller
than building D, building C is 76 feet taller than building B, and
so on.
-
5 Summary: The four levels of measurements can be thought of as
a hierarchy. At the lowest level, nominal measurement allows us
only to categorize phenomena into different groups. The second
level, ordinal measurement, not only allows us to classify
phenomena into different groups but also indicates the relative
ordering of the groups on a dimension of interest. The third level,
interval measurement, possesses the same properties as ordinal but,
in addition, is sensitive to the magnitude of the differences in
the groups on the dimension. However, ratio statements are not
possible at this level since the measurement is based on some
criterion which is arbitrary. The fourth and final level, ratio
measurement, have all the properties of nominal, ordinal and
interval measurements and also permit ratio judgments to be made (0
as a starting point).
THE MEASUREMENT HIERARCHY
The four types of measurement can be thought of a hierarchy. At
the lowest level, nominal measurement allows us only to categorize
or classify phenomena into different groups. The second level,
ordinal measurement, not only allow us to categorize or classify
phenomena into different groups but also indicates the relative
ordering of the groups on a dimension of interest. Interval
measurement, the third level, possesses the same properties as
ordinal but in addition, is sensitive to the magnitude of the
differences in the groups on the dimension. However, ratio
statements are not possible at this level. It is only at the final
level, ratio measurement, that such statements are possible. Ratio
measures have all the properties of nominal, ordinal and interval
measures and also permit ratio judgments to be made. Variables
measured on the ordinal, interval, or ratio level are known as
quantitative variables, whereas variables measured in nominal level
are called qualitative variables.
Exercise: The following data describe the different data
associated with a state senator. For each data entry, indicate the
corresponding level of measurement. (1) The senators name is Carah
Bao. (2) The senator is 58 years old. (3) The years in which the
senator was elected to the senate are 1963, 1969, 1981, and 1994.
(4) Her total taxable income last year was $78,317.19. (5) The
senator sponsored a bill to protect water rights. Out of 1100
voters in her district, 400 hundred said they strongly favoured the
bill, 300 said they favoured the bill, 200 said they were neutral,
150 said they did not favour the bill and 50 said they strongly did
not favour the bill. (6) The senator is married now. (7) However,
the senator has married three times. (8) A leading news magazine
claims the senator is ranked seventh for her voting record on bills
regarding public education
-
6 Answers:
(1) Name is nominal (2) Years of age is ratio (3) Years when the
senator was elected are interval (4) Income is a ratio (5) Degree
of agreement (strongly favoured, favoured, neutral, not favoured,
strongly not favoured) is an
ordinal (ranks) (6) Marital status is nominal (7) Number of
times the senator married constitutes counting which is ratio. (0,
1, 2, 3, . . .) (8) Rank is a nominal data
Applicants for different positions of ABC Company
1 2 3 4 5 6 7 8 9 10 Age
(years) Civil
Status Nationality Religion No. of
dependents
Degree earned
Sex Job applying
for
IQ Score Years of relative
experience (months)
24 Single Thai Christian 2 BSMath M Statistician 110 6 23 Single
Thai Buddhist 0 BSMath F Statistician 128 10 28 Married Thai
Buddhist 3 BSAcc M Accountant 115 16 27 Married Filipino Baptist 3
BSME M Engg Head 133 10 29 Married Filipino Catholic 4 BSME F Engg
Head 110 3 28 Married American Protestant 1 BSAcc F Accountant 95 0
32 Widow American Baptist 0 BSMath F Researcher 115 12 35 Married
Chinese Protestant 0 BSEE M Researcher 95 8 25 Single Chinese
Catholic 0 BSCoE F Systems
Analyst 130 20
27 Single Filipino Baptist 1 BSCS M Systems Analyst
105 10
29 Single Thai Buddhist 1 BSAcc F Accountant 125 14 24 Single
Chinese Catholic 0 BSME M Researcher 120 6
Answers:
1. Age Interval/Ratio
2. Civil Status Nominal
3. Nationality Nominal
4. Religion Nominal
5. Number of Dependents Interval/Ratio
6. Degree Earned Nominal
7. Sex Interval/Ratio
8. Job Applying for Nominal
9. IQ Score Interval/Ratio
10. Years of Relative Experience Interval/Ratio
-
7
ACTIVITY No. 1 (Level of Measurements)
Identify whether the following observations are nominal,
ordinal, interval/ratio. Write N for nominal, O for
ordinal, IR for interval/ratio.
_____1. Weight in pounds of new born babies
_____2. Speed of a car in miles per hour
_____3. Degree of agreement or disagreement of respondents about
the appropriateness of a television
program for children below 10 years old (Strongly agree, Agree,
Disagree or Strongly Disagree).
_____4. Length of Milkfish in a fish pond.
_____5. Eye color
_____6. Skin tone
_____7. IQ level as low, average or high
_____8. Sound intensity of the noise made by students in a
cafeteria
_____9. Educational attainment
_____10. Number of children in a family
_____11. Socioeconomic status of residence in Khon Khaen City
(Low, Average, High)
_____12. Population of Thailand in the year 2010
_____13. Monthly salary of employees in the College of Asian
Scholars
_____14. Religious affiliation
_____15. Gender of applicants
_____16. Anxiety level whether low, moderate, high or very
high
_____17. Academic performance in math (poor, fair, good, very
good)
_____18. Weight in pounds of babies born in the month of
December 2008
_____19. Number of coffee-break hours per day spent by
executives
_____20. Length in hours of the study time spent per day by
students
_____21. Military ranks
_____22. Home address of students
_____24. The year when you were born
_____25. Softdrinks preference of Thai people
_____26. Number of foreigners migrating to Thailand every
year.
_____27. Length of hair of females.
_____28. The boiling point of water is 1000C.
_____29. His cellphone number is 0929-9999875.
_____30. Johns height is 168 cm.
_____31. The number of children with missing/decayed teeth in a
community is 200.
_____32. The following data are the densities of sample
substances taken from River Kwai (in gm/cc): 23.6,
19.8, 15.0, 7.8, 1.6 and 2.4
_____33. The average speed of motorboats crossing in a river
everyday is 5 meters per second.
_____34. Anxiety level of 8 selected female students in
University of Baguio
Maria Low Luisa Average Marissa Low Martha High
Lana High Maridel Low Kelly Average Sandy Low
_____35. Religion of 5 job applicants at ABC Company
Applicant A Roman Catholic Applicant D Baptist
Applicant B English Catholic Applicant E Protestant
Applicant C Seventh Day Adventist
-
8 _____36. Average monthly income in pesos of 5 families in
Irisan, Baguio City
Family A - 23,000.00 Family D - 18,000.00
Family B - 12,000.00 Family E - 55,000.00
Family C - 14,500.00
_____37. Contents of cola softdrink in ounces (oz)
Bottle A 2.3 oz Bottle C 2.6 oz Bottle E 2.3
Bottle B 2.5 oz Bottle D 2.2 oz
_____38. The age in months of babies admitted at NDC Hospital
for treatment of bronchopneumonia are as
follows:
14, 6, 29, 43, 40, 32, 60, 58
_____39. Weights in pounds of the students in Statistics
Luis 120 Lucia 200 Gerry 166
Manuel 125 Felna 145
_____40. Scores of students in Statistics Exam: 34, 56, 45, 78,
67, 98, 78, 66, 57, 75, 34, 43, 24, 77, 80
_____41. The average score of students in an English quiz is
45.8
_____42. The total area of farm lands in a certain town is
120,000 square meters.
_____43. The volume of a softdrink bottle is 1.5 liters.
_____44. The speed of a car travelling along a highway is 60
miles per hour.
_____45. The length of a snake caught in a forest is 4
meters.
-
9
Population (N) and Sample (n) One of the goals of a statistical
investigation is to explore the characteristics of a large group of
items on the basis of a few. Sometimes it is physically,
economically, or for some other reason almost impossible to examine
each item in a group under study. In such situation the only
recourse is to examine a sub-collection of items from this group.
In statistics we commonly use the terms population and sample.
DEFINITIONS:
Data are collections of observations (such as measurements,
genders, survey responses). A population is the complete collection
of all individuals (scores, people, measurements, and so on) to
be studied. The collection is complete in the sense that it
includes all of the individuals to be studied.
Example: Suppose an ornithologist is interested in investigating
migration patterns of birds in the Northern Hemisphere. Then all
the birds in the Northern Hemisphere will represent the population
of interest to him. His choice of the population restricts him, for
it does not include birds that are native to Australia and do not
migrate to the Northern Hemisphere. Example: Every ten years the
Bureau of Census conducts a survey of the entire population of a
country accounting for every person regarding sex, age, and other
characteristics. In this case the entire population of the country
is the population in the statistical sense. A population can be
finite or infinite and is made up of study units
Example: If we are conducting a telephone interview to study all
adults (our target population) in a particular city, we do not have
access to those persons who do not have a telephone. Example: We
may wish to study in a particular community the effect of a drug A
among all men with cholesterol levels above a specified value;
however short of sampling all men in the community, only those men
who for some reason visit a doctors office, clinic, or hospital are
available for a blood sample to be taken.
Unfortunately the target population is not always readily
accessible, and we can study only that part of it that is
available. There are many ways to collect information about the
study population. One way is to conduct a sample. A sample is a
subcollection of members selected from a population.
Population
Study Unit
Target Population
The whole group of study units which we are interested in
applying our inferences or conclusions
Study Population
The group of study units to which we can legitimately apply our
inferences or conclusions
-
10 Example: A fisheries researcher is interested in the behavior
pattern of Hermit crab along the coast of the Gulf of Siam. It
would be inconceivable and impossible to investigate every crab
individually. The only way to make any kind of educated guess about
their behavior would be by examining a small sub-collection, that
is, a sample. Example: Suppose a machine has produced 10,000
electric bulbs and we are interested in getting some idea about how
long the bulbs will last. It would not be practical to test all the
bulbs, because the bulbs that are tested will never reach the
market. So we might pick 50 of these bulbs to test. Our interest is
in learning about the 10,000 bulbs and we study 50. The 10,000
bulbs constitute the population and the 50 bulbs a sample.
Relationship between population and sample
Sample data must be collected in an appropriate way, such as
through a process of random selection. If sample data are not
collected in an appropriate way, the data may be so completely
useless that no amount of statistical torturing can salvage them.
The terms population and sample are relative. A collection that
constitute a population in one context may well be a sample in
another context. For instance, if we wish to learn how people in
Khon Khaen City feel about a certain national issue, then all the
residents of Khon Khaen City would constitute the population of
interest. However, assuming that Khon Khaen City represents a cross
section of Thailand population, if we use the response from these
residents to understand the feelings about the issue among all the
Thai people, then the residents of Khon Khaen City would represent
a sample.
RANDOM SAMPLING TECHNIQUES SAMPLE SIZE An important
consideration in conducting research is the size of your sample. It
must be large enough so that erratic behavior of very small samples
will not produce misleading results. Repetition of a research or an
experiment is called replication. A large sample is not necessarily
a good sample. Although it is important to have a sample that is
sufficiently large, it is more important to have a sample in which
the elements have been chosen in an appropriate way, such as random
selection. Use a sample size large enough so that we can see the
true nature of any effects or phenomena, and obtain the sample
using an appropriate method, such as one based on randomness.
-
11 RANDOMIZATION One of the worst mistakes is to collect data in
a way that is inappropriate. We cannot overstress this very
important point: Data carelessly collected may be so completely
useless that no amount of statistical torturing can salvage
them.
COMMON METHODS OF SAMPLING In a random sample members of the
population are selected in such a way that each has an equal chance
of being selected. Sampling is a process or procedure which
involves taking a part of a population, making observation on this
representatives and the generalizing the findings to the bigger
population. (Ary, Jacob and Razavieh, 1981). Probability Sampling
is a random sampling technique that each element in a population
has an equal chance of being selected. Non-probability Sampling is
a non-random sampling technique that each element in a population
has no equal chance of being selected.
SAMPLING
TECHNIQUE
PROBABILITY SAMPLING
SIMPLE RANDOM
SAMPLING
FISH-BOWL TECHNIQUE
LOTTERY TECHNIQUE
TABLE OF RANDOM NUMBERS
SYSTEMATIC SAMPLING
STRATIFIED SAMPLING
CLUSTER SAMPLING
NON-PROBABILITY
SAMPLING
ACCIDENTAL / CONVENIENCE
SAMPLING
PURPOSIVE SAMPLING
QUOTA SAMPLING
SNOW-BALL SAMPLING
-
12
SAMPLING STRATEGIES APPROPRIATE TO PARTICULAR FEATURES OF THE
POPULATION
Personal Attributes Geographical Spread Sampling Strategies
Homogeneous Concentrated Simple Random or Systematic
Dispersed 1.) Cluster Sampling 2.) Simple Random or
Systematic
Heterogeneous
Concentrated 1.) Stratified Sampling 2.) Simple Random or
Systematic
Dispersed
1.) Stratified 2.) Cluster 3.) Simple Random or Systematic
Determination of sample size (n) provided that the Population
size (N) is known
Slovins Formula Lynch et. al Formula
21 Ne
Nn
N = Population Size n = sample size e = margin of error (0.10,
0.05, or 0.01)
)1(
)1(22
2
ppZNd
ppNZn
Z = value of the normal variable for a reliability level Z =
1.645 (90% reliability in obtaining the sample size)) Z = 1.96 (95%
reliability in obtaining the sample size) Z = 2.575 (99%
reliability in obtaining the sample size) p = 0.50 (proportion of
getting a good sample) (1 p) = 0.50 (proportion of getting a poor
sample) d = 0.01, 0.025, 0.05, or 0.10 (choice of sampling error) N
= population size n = sample size
Example:
Find a minimum sample n if a population size N is 5000 with a
margin of error due to sampling of 5%.
Given : N = 5000 e = 5% = 0.05 Slovins Formula: Find a minimum
sample n if a population size N is 5000 with a margin of error due
to sampling of 5% and a 95% reliability in obtaining the sample
size. Given: N = 5000 d = 5% = 0.05
z = 1.96 (95% reliability) Modified Lynch et. Al Formula:
35775.3564604.13
4802
9604.05.12
4802
)96.1)(25.0()05.0)(5000(
)96.1)(5000)(25.0(
)25.0(
)25.0(22
2
22
2
zNd
Nzn
37037.3705.13
5000
5.121
5000
)05.0)(5000(1
5000
1 22
Ne
Nn
-
13 Stratified Sampling: The following are the population from 5
different communities. Use Modified Lynch et al. to find the sample
size for each community with a margin of error due to sampling of
5% and a 99% reliability in obtaining the sample size.
Community Population Size (N)
A 800 B 400 C 500 D 600 E 700
Total N = 3000
Community Population Size (N)
Ratio i = nN 5433000 = 0.181
Sample size per community
A 800 0.181 800 144.8 = 145 B 400 0.181 400 72.4 = 72 C 500
0.181 500 90.5 = 91 D 600 0.181 600 108.6 = 109 E 700 0.181 700
126.7 = 127
Total N = 3000 n = 544 Note: The minimum sample size n was 543,
however in the computation the value of n is 544 which is accepted
as long as it is not less than 543.
Community Population Size (N)
A 145 B 72 C 91 D 109 E 127
Total n = 544 A researcher wants to know the study habits of the
students in a particular school. Determine the size of the sample
units from each level using 2% margin of error with 95% reliability
in obtaining the sample size.
Gender Year Level Total Freshman Sophomore Junior Senior
Male 750 600 550 500 2400 Female 580 650 450 670 2350 Total 1330
1250 1000 1170 N = 4750
Ratio i = n N = 1595 4750 = 0.3358 (up to 4 decimal places for
accuracy)
54304.54315765625.9
96875.4972
65765625.15.7
96875.4972
)575.2)(25.0()05.0)(3000(
)575.2)(3000)(25.0(
)25.0(
)25.0(22
2
22
2
zNd
Nzn
159585.15948604.2
9.4561
9604.09.1
9.4561
)96.1)(25.0()02.0)(4750(
)96.1)(4750)(25.0(
)25.0(
)25.0(22
2
22
2
zNd
Nzn
-
14 Male Freshman 0.3358 750 = 251.85 = 252 Female Freshman
0.3358 580 = 194.76 = 195 Male Sophomore 0.3358 600 = 201.48 = 201
Female Sophomore 0.3358 650 = 218.27 = 218 Male Junior 0.3358 550 =
184.69 = 185 Female Junior 0.3358 450 = 151. 11 = 151 Male Senior
0.3358 500 = 167. 9 = 168 Female Senior 0.3358 670 = 224.99 = 225
-------- -------- 806 789 Sample size n = 806 + 789 = 1595
Gender Year Level Total Freshman Sophomore Junior Senior
Male 252 201 185 168 806 Female 195 218 151 225 789 Total 447
419 336 393 1595
NON-PROBABILITY SAMPLING
1.) Accidental/Convenience Sampling Simply use results that are
readily available or accessible. Usually the first person who comes
along who typifies a unit of analysis serves as the respondent of
the study.
2.) Purposive Sampling Implemented with the researcher defining
a criterion or set of criteria for
determining the respondents of the study. It is the researchers
judgment that becomes the basis for selecting an element or group
that will serve as the unit of analysis. It is useful in
qualitative or exploratory studies. The objective is not to have
many respondents but to make sure that the person who would be
interviewed will provide a wealth of information. The aim is not to
quantify but to characterize an event being studied.
3.) Quota Sampling - Similar to stratified sampling except that
the selection of the elements per stratum is
done through the application of random sampling strategy. Quota
sampling entails grouping elements according to certain
characteristics and ensuring that each group is represented. Quota
sampling is helpful if the sampling frame is not available per
group or stratum. It refines the application of convenience
sampling since there is conscious intent on the part of the
researcher to view the probable differences of every stratum or
group with regard to the critical variables of the study.
4.) Snowball or Referral Sampling Involves having a respondent
refers other people who are in a position
to answer some of the questions of the researcher. This is a
particularly helpful in the study of highly sensitive topics where
the identity of respondents is difficult to divulge or may even be
unknown to many. In other words, if the sampling frame cannot be
provided and the topic has security implications, a researcher
could obtain referrals from the first respondent to the other
respondents who may be willing to talk.
A sampling error is the difference between a sample result and
the true population result; such an error results from chance
sample fluctuations. A nonsampling error occurs when the sample
data are incorrectly collected, recorded, or analyzed (such as by
selecting a biased sample, using a defective measurement
instrument, or copying the data incorrectly).
-
15
ACTIVITY No. 2 Determining Sample Size and Stratified
Sampling
Use Slovins and Lynch et al formulas in determining the sample
size of the following problems and use stratified sampling if
necessary.
1. A researcher uses a 5% margin of error in computing for his
sample size. If the population size is 15,000 what is the sample
size with 95% reliability?
a. Slovin Formula b. Lynch et. al Formula
2. The following is a table about a population in a certain
community:
Gender Age in Years Row Total
11 20 21 30 31 40 41 50 Male 240 400 350 260 1250
Female 250 300 400 250 1200 Column Total 490 700 750 510 N =
2450
a. What would be the required sample size with 95% reliability
at 5% margin of error? (Use Lynch et.
Al formula) b. Use stratified sampling to find the minimum
sample size in each stratum.
Gender Age in Years Row Total
11 20 21 30 31 40 41 50 Male
Female
Column Total
n =
-
16
METHODS OF PRESENTATION OF DATA Statistical data collected
should be arranged in such a manner that will allow a reader to
distinguish their essential features. Depending on a type of
information and the objectives of the person presenting the
information, data may be presented using one or a combination of
three forms: TEXTUAL, TABULAR, and GRAPHICAL. TEXTUAL FORM The
textual or paragraph form is utilized when the data to be presented
are purely qualitative or when very few numbers are involved. This
method is, generally, not desirable when too many figures are
involved as the reader may fail to grasp the significance of
certain quantitative relationships, but it becomes an effective
device when the objective is to call the readers attention to some
data that require special emphasis.
Example: From a newspaper report, it was gathered that China has
a population of 707 million, India has 505 million, US has 207
million, USSR (before the break-up) has 245 million, and Indonesia
has 125 million. That more than half of the worlds people, about
2.1 billion live in Asia, 456 million in Europe, 354 million in
North America, 195 million in South America, and 20 million in
Oceana. Shanghai has 10,820,000; Tokyo has 8,841,000; New York has
7,895,000; and Moscow has 7,050,000. TABULAR FORM A more effective
device of presenting data because the data are presented in more
concise and systematic manner. People who want to make some
comparisons and draw relationships usually find tabular arrangement
more convenient and understandable than the textual presentation.
The data are presented through tables consisting of vertical
columns and horizontal rows with headings describing these rows and
columns. Example:
Continent/Region Population Country Population Cities Population
Asia 2,100,000,000 China
India Indonesia
707,000,000 505,000,000 125,000,000
Shanghai Tokyo
10,820,000
8,841,000 North America 354,000,000 USA 207,000,000 New York
7,895,000 Europe 465,000,000 USSR 245,000,000 Moscow 7,050,000
South America 195,000,000 Oceana 20,000,000
GRAPHICAL OR PICTORIAL FORM Among the different methods of
presenting data, the graph or chart is perhaps the most effective
device for attracting peoples attention. Readers who look for
comparisons and trends may skip statistical tables but may pause to
examine graphs. Graph has a great advantage over tables because
graph conveys quantitative values and compares more readily than
tables.
-
17
MEASURES OF CENTER (CENTRAL TENDENCY)
Definitions:
A measure of central tendency is a single value that is used to
identify the center of the data A representative or average value
that indicates where the middle of the data set is located.
o It is thought of as a typical value of the distribution. o
Precise yet simple o Most representative value of the data
There are several different ways to determine the center, so we
have different definitions of measures of center, including the
mean, median, and mode.
Mean The arithmetic mean of a set of values is the number
obtained by adding the values and dividing the total by the number
of values. The (arithmetic) mean is generally the most important of
all numerical descriptive measurements, and it is what most people
call an average.
Median The median of a data set is the middle value when the
original data values are
arranged in order of increasing (or decreasing) magnitude.
Mode The mode of a data set is the value that occurs most
frequently. When two values occur with the same greatest frequency,
each one is a mode and the data set is bimodal. When more than two
values occur with the same greatest frequency, each is a mode and
the data set is said to be multimodal. When no value is repeated,
we say that there is no mode and the data set is said to be
nonmodal.
Procedures for Finding
Measures of Center USES OF MEAN, MEDIAN AND MODE 1. When a
quantitative data is measured on a level that at least approximates
interval characteristics and the distribution of observations is
not too skewed, all three measures of center are meaningful. 2.
When a distribution is skewed, both the mean and the median should
be reported. 3. When a quantitative (or some qualitative) data is
measured on an ordinal level that departs markedly from interval
characteristics, the mean is not an appropriate index of center but
the mode or median must be used instead. Other uses of the Mean,
Median and Mode 4. When a qualitative data is measured (that is,
nominal measures), the mean or median are meaningless because these
concepts require ordering objects along a dimension. In this case,
the mode (that is, the most frequency occurring category) is the
only applicable descriptor of center. 5. When a quantitative data
that contain some outliers (extreme values that fall outside the
overall pattern), trimmed mean will be used. Because the mean is
very sensitive to extreme values, we say that it is not a resistant
measure of center. The trimmed mean is more resistant.
-
18 RESISTANT MEASURE: A resistant measure is one that is not
influenced by extremely high or low data values (outliers). A
measure of center that is more resistant than the mean but still
sensitive to specific data values is the trimmed mean. To compute
the 5% trimmed mean, order the data from the smallest to largest,
delete the bottom 5% of the data, and then delete the top 5% of the
data. Finally compute the mean of the remaining 90% of the data.
THE MODE: The mode )(x is the most frequent, most typical, or most
common value in a distribution.
For example, there are more Catholics in the Philippines than
people of any other Christina religion; and so we refer to this
religion as the mode. Similarly, if at a given university, Nursing
is the most popular course, this too would represent the mode. The
mode is the only measure of center available for nominal-level
variables and it can be used to describe the most common score in
any distribution regardless of the level of measurements. To find
the mode, find the score or category that occurs most often in a
distribution. It can be easily found by inspection, rather than by
computation. Example: Scores: 1, 2, 3, 1, 1, 6, 5, 4, 1, 4, 4, 3
The mode is 1 because it is the number that occurs more than any
other scores in the set (it occurs four times). Note: The mode is
not the frequency of the most frequent score (f = 4), but the value
of the most frequent score ( x = 1) Example: Scores: 6, 6, 7, 2, 6,
1, 2, 3, 2, 4 Some frequency distributions contain two or more
modes. In the following set of data above, the scores 2 and 6 both
occur most often. Graphically, such distributions have two points
of maximum frequency. These distributions are referred to as being
bimodal in contrast to the more common unimodal variety, which has
only a single point of maximum frequency. THE MEDIAN )~(x : When
ordinal or interval data are arranged in order of size, it becomes
possible to locate
the median, the middlemost point in a distribution. Thus, the
median is a measure of center that cuts the distribution into two
equal parts. If the number of cases in a distribution is odd, the
median falls exactly in the middle of the distribution but if the
number of cases in a distribution is even, the median is always
that point above which 50% of the cases fall and below which 50% of
the cases fall. It means that we add the two middlemost values and
divided by 2. The data should be in order from low to high (or high
to low) in order to locate the median. Example: Scores: 1, 2, 3, 1,
1, 6, 5, 4, 1, 4, 4
Array: 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 6 Number of cases: n = 11
(odd) Position of median = (n + 1)/2 (for odd) = (11 + 1)/2 = 6th
position either from left or right (top or bottom) The median x~ =
3 is the 6th score in the distribution counting from either
end.
-
19 Example: Scores: 6, 6, 7, 2, 6, 1, 2, 3, 2, 4
Array: 1, 2, 2, 2, 3, 4, 6, 6, 6, 7 Number of cases: n = 10
(even) Position of median = n/2 (for even)
= 10/2 = 5th position (from left and right) or (top and bottom)
) 3 = 5th position from left 4 = 5th position from right Median x~
= (3 + 4)/2 = 7/2 = 3.5 THE MEAN )(x : By far the most commonly
used measure of center, the arithmetic mean, is obtained by
adding
up a set of scores and dividing by the number of scores. Thus,
mean is defined formally as the sum of a set of scores divided by
the total number of scores in the set.
By formula:
Population Mean Sample Mean
Example:
Respondent X (IQ) Computation
1. Albert 125 2. Beth 92
3. Connie 72
4. Drake 126
5. Elmer 120
6. Fritz 99
7. Gertz 130
8. Henry 100
X = 864
Unlike the mode, the mean is not always the score that occurs
most often. Unlike the median, the mean is not necessarily the
middlemost point in a distribution. The mean is the point in a
distribution around which the scores above it balance with those
scores below it. Thus, the mean is a balance point that the sum of
the deviations that fall above the mean is equal in absolute value
to the sum of the deviations that fall below the mean. The Weighted
Mean Researchers sometimes find it useful to obtain a mean of means
that is, to calculate a total mean for a number of different
groups. Suppose the students in three different sections of
Sociology class receive the following mean scores on their final
examinations for the course: Section 1 Section 2 Section 3 Mean: 85
72 79 Number of cases: (n) 28 28 28
N
X
n
XX
1088
864
n
XX
-
20 Because exactly the same numbers of students were enrolled in
each section of the course, it is quite simple to calculate a total
mean score: When groups differ in size, you must weight each group
mean it its size (n). The weighted mean may be calculated by first
multiplying each group mean by its respective number of cases (n)
before summing the products, and then dividing by the total number
in all groups: Where: = mean of a particular group
= number of cases in a particular group
= number in all cases combined (n1 + n2 + n3 + + nk)
= weighted mean
Section 1 Section 2 Section 3 Mean: 85 72 79 Number of cases:
(n) 28 40 32 Thus, the mean final grade for all sections combined
was 77.88 Weighted mean can also apply in relation to Likert Scale
(1 = Strongly Disagree, 2 = Disagree, 3 = Agree, 4 = Strongly
Agree) Example: Suppose a survey was conducted regarding their
extent of watching horror films, the following data were gathered:
Question: Always
(5) Oftentimes
(4) Sometimes
(3) Seldom
(2) Never
(1) To what extent do you watch horror film movies? n = 28 n =
39 n = 15 n = 26 n = 12 Range Verbal interpretation Thus, the
weighted mean of 3.375 suggests that on average the 4.21 5.00
Always respondents sometimes watch horror films. 3.41 4.20
Oftentimes 2.61 3.40 Sometimes 1.81 2.60 Seldom 1.00 1.80 Never
67.783
797285
3
321
XXX
total
groupgroup
wN
XnX
groupX
groupn
totalN
wX
88.77100
7788
100
252828802380
324028
)79(32)72(40)85(28
wX
375.3120
405
1226153928
)1)(12()2)(26()3)(15()4)(39()5)(28(
total
groupgroup
wN
XnX
-
21
Comparing the Mode, Median and Mean The time comes when a
researcher chooses a measure of center for a particular research
situation. Will he/she employ the mode, the median, or the mean?
The decision involves several factors, including the following: 1.
Level of measurement 2. Shape or form of the distribution of data
3. Research objective
OBTAINING THE MODE, MEDIAN AND MEAN FROM A SIMPLE FRREQUENCY
DISTRIBUTION Example: Suppose a researcher conducted personal
interviews with 20 lower-income respondents in order to determine
their ideal conceptions of family size. Each respondent was asked:
Suppose you could decide exactly how large your family should be.
Including all children and adults, how many people would you like
to see in your family? Raw Data: 2, 3, 3, 2, 2, 1, 4, 4, 6, 5, 7,
8, 9, 3, 7, 3, 7, 6, 8, 7 These data can be rearranged as a simple
frequency distribution as follows X f 1 1 2 3 3 4 Mode 4 2 5 1 6 2
7 4 Mode 8 2 9 1 ----------- n = 20 The median is the middlemost
score in the ordered list of scores. If there is an odd number of
cases, the median is the score in the exact middle of the list; if
there is an even number of cases, the median is halfway between the
two middlemost scores.
n = 20 n/2 = 20/2 = 10th score
4 = 10th score from top 5 = 10th score from bottom
x~ = (4+ 5)/2 = 4.5
-
22 Determine the sum of the scores = 97 X f fX Calculate the
mean 1 1 (1)(1) = 1 2 3 (3)(2) = 6 3 4 (4)(3) = 12 4 2 (2)(4) = 8 5
1 (1)(5) = 5 6 2 (2)(6) = 12 7 4 (4)(7) = 28 8 2 (2)(8) = 16 9 1
(1)(9) = 9 ------------------------------------ n = 20 fX = 97
Summary: Modes )(x = 3 and 7
Median )~(x = 4.5
Mean )(x = 4.85
There is a wide range of family size preferences, from living
alone (1) to having a big family (9). Using either the mean = 4.85
or the median = 4.5, we might conclude that the average respondents
ideal family contained between four and five members. Knowing that
the distribution is bimodal, however, we see that there were
actually two ideal preferences for family size in this group of
respondents one for a small family (Mode = 3) and the other for a
large family (Mode = 7). Example: Given the impact of television on
childrens attitudes and behaviour, an important concern of
behavioural scientists is the amount of time children of various
ages spend watching television. The following data are the weekly
viewing times (in hours) of 12-year-olds. Describe or interpret the
data set using the measures of center.
18 17 22 20 25 20 16 19 18 22 26 23 23 23 24 24 22 21 19 20
20
Solution: Arrange the data either in ascending or descending
order
X f fX 16 1 (1)(16) = 16 17 1 (1)(17) = 17 18 2 (2)(18) = 36 19
2 (2)(19) = 38 20 4 (4)(20) = 80 21 1 (1)(21) = 21 22 3 (3)(22) =
66 23 3 (3)(23) = 69 24 2 (2)(24) = 48 25 1 (1)(25) = 25 26 1
(1)(26) = 26 ______________________________ n = 21 fX = 442
85.420
97
n
X
n
fXX
-
23 Mode: The highest frequency (f) is 4 which corresponds to 20.
Thus the modal time is 20 hours Median: Since n = 21 which is odd
Position of median = (n + 1)/2 (for odd) = (21 + 1)/2 = 22/2 = 11th
position either from left or right (top or bottom) Median time = 21
hours Mean:
05.2121
442
n
X
n
fXx
Interpretation: The weekly viewing times (in hours) of
12-year-olds ranges from 16 to 26 hours. Most of the 12 year-old
children spent watching television for 20 hours in a week (mode).
Half of the children spent watching television in a week for 21 or
more hours (median). On average, a 12-year old child spent about 21
hours watching television in a week.
=====================================================================================
Activity No. 3
Measures of Center
Problem: Tuitions at private colleges and universities vary
quite a bit. Below are lists of tuitions per unit of
basic subjects at accredited colleges and universities in
Thailand. A sample of 30 colleges and universities
showed annual tuition per unit (Baht) as follows:
270 290 345 295 300 245 240 325 300 295 310 265 275 285 330
295 270 285 270 265 275 320 310 335 345 335 265 280 245 260
Describe or interpret the data set using measures of center.
-
24
MEASURES OF DISPERSION/VARIABILITY/VARIATION In summarizing a
given set of data, sometimes, the measures of center (central
tendency) alone are not sufficient to give useful information. They
have to be supplemented by other measures of description, and such
description is the MEASURES OF VARIABILITY. A measure of
variability indicates the extent to which values in a distribution
are spread around the central tendency. A measure of variation is a
single value that is used to describe the spread of the
distribution. A measure of central tendency alone does not uniquely
describe a distribution
INTERPRETING AND UNDERSTANDING STANDARD DEVIATION We understand
that the standard deviation measures the variation of values about
the mean. Values close together will yield a small standard
deviation, whereas values spread farther apart will yield a larger
standard deviation. Because variation is such an important concept
and because the standard deviation is such an important tool in
measuring variation, there are ways of developing a sense for
values of standard deviations. CONCEPTS: Variation refers to the
amount that values vary among themselves Values that are relatively
close together have lower measures of variation, and values that
are spread farther apart have measures of variation that are
larger
(1) Range (R)
Difference between the highest and the lowest observed values in
a distribution. A very rough measure of spread Provides useful but
limited information since it depends only on the extreme values
(2) Sample Variance (s2)
Important measure of variation Shows variation about the
mean
Measures of Variability or Dispersion
Measures of Absolute Variation
Range Variance Standard Deviation
Measures of Relative Dispersion
Quantiles
Median Quartiles Deciles Percentiles
Coefficient of Variation
-
25
RAW DATA (UNGOUPED DATA)
Population variance Sample variance Formula 1:
Formula 2: ( )
( )
(3) Sample Standard Deviation (SD) Most important measure of
variation Square root of Variance Has the same units as the
original data
RAW DATA (UNGOUPED DATA)
Population Standard Deviation Sample Standard Deviation
s= X2-(X)2
n(n-1)
Remarks: If there is a large amount of variation, then on
average, the data values will be far from the
mean. Hence, the SD will be large. If there is only a small
amount of variation, then on average, the data values will be close
to the
mean. Hence, the SD will be small.
Comparing Standard Deviations Example: Team A - Heights of five
marathon players in inches
Mean = 65 S = 0
65 65 65 65 65
N
X
2
2
1
2
2
n
XXS
N
X
2
1
2
n
XXS
-
26
( )
( )
Height (X) ( ) 65 (65 65)2 = 0 65 (65 65)2 = 0 65 (65 65)2 = 0
65 (65 65)2 = 0 65 (65 65)2 = 0
Height (X) X2 65 652 = 4225 65 652 = 4225 65 652 = 4225 65 652 =
4225 65 652 = 4225
X = 325 ( )
X = 325 X2 = 21125
( )
( ) ( ) ( )
( )
Example: Team B - Heights of five marathon players in inches
Mean = 65 S = 4.0
62 67 66 70 60
( )
( )
Height (X) 2XX 62 (62 65)2 = 9 67 (67 65)2 = 4 66 (66 65)2 = 1
70 (70 65)2 = 25 60 (60 65)2 = 25
Height (X) X2 62 622 = 3844 67 672 = 4489 66 662 = 4356 70 702 =
4900 60 602 = 3600
X = 325 ( )
X = 325 X2 = 21189
( )
( ) ( ) ( )
( )
1
2
2
n
XXS
1
2
2
n
XXS
1
2
2
n
XXS
04
0
15
02
S
1
2
2
n
XXS
164
64
15
642
S
-
27
OBTAINING THE SAMPLE VARIANCE AND STANDARD DEVIATION FROM A
SIMPLE FREQUENCY DISTRIBUTION
Example: Suppose a researcher conducted personal interviews with
20 lower-income respondents in order to determine their ideal
conceptions of family size. Each respondent was asked: Suppose you
could decide exactly how large your family should be. Including all
children and adults, how many people would you like to see in your
family? Raw Data: 2, 3, 3, 2, 2, 1, 4, 4, 6, 5, 7, 8, 9, 3, 7, 3,
7, 6, 8, 7 These data can be rearranged as a simple frequency
distribution as follows: X f fX fX2= (fX)(X)
1 1 (1)(1) = 1 (1)(1) = 1 2 3 (3)(2) = 6 (6)(2) = 12 3 4 (4)(3)
= 12 (12)(3) = 36 4 2 (2)(4) = 8 (8)(4) = 32 5 1 (1)(5) = 5 (5)(5)
= 25 6 2 (2)(6) = 12 (12)(6) = 72 7 4 (4)(7) = 28 (28)(7) = 196 8 2
(2)(8) = 16 (16)(8) = 128 9 1 (1)(9) = 9 (9)(9) = 81 n = 20 fX = 97
fX2 = 583 Solving for the sample variance: Solving for the sample
standard deviation
43.292.52 ss
WHEN THE VARIOUS MEASURES OF VARIABILITY ARE USED
1. Range. This is the least reliable of the measures and is used
only when one is in a hurry to get a measure of variability. It may
be used with ordinal, interval, or ratio data.
2. Standard Deviation and the Variance. The standard deviation
is used whenever a distribution approximates a normal distribution.
It is the basis for much of the statistics. As the most reliable
measure of variability it is used with interval and ratio data. Use
standard deviation or variance when two means are equal.
)1(
22
2
nn
fxfxnS
)1(
22
nn
fxfxnS
92.5
380
2251
380
940911660
)19(20
97)583(20
)1(
222
2
nn
fxfxnS
-
28
THE SAMPLE STANDARD DEVIATION (s) and SAMPLE VARIANCE (s2)
RELATIONSHIP BETWEEN THE STANDARD DEVIATION AND VARIANCE
Variance = (Standard deviation)2 s2 = (s)2
Standard deviation =
STANDARD DEVIATION: A MEASURE OF DISTANCE
Theres an important difference between the standard deviation
and its co-measure, the mean. The mean is a measure of position but
the standard deviation is a measure of distance (on either side of
the mean of the distribution)
(1) Majority within one standard deviation for most frequency
distribution, a majority (as often as 68%) of all observations are
within one standard deviation on either side of the mean.
(2) Minority deviate outside two standard deviation for most
frequency distribution, a small minority (often as small as 5%) of
all distributions deviate more than two standard deviations on
either side of the mean.
(3) Usual or normal within two standard deviation for most
frequency distribution the usual or normal values (as often as 95%)
of all observations are within two standard deviations on either
side of the mean.
MAJORITY OF THE DISTRIBUTION In a normal distribution, majority
of the scores/values lie within one standard deviation from the
left and right of the mean. This is based on the principle that
majority (64.26%) of sample values lie with 1 standard deviation of
the mean. Majority of Scores/Values = (mean) 1(standard deviation)
= Lower Range: Upper Range:
-
29 USUAL OR NORMAL VALUES RANGE RULE OF THUMB (Rough estimates
of the minimum and maximum usual sample values) The Range Rule of
Thumb is based on the principle that for many data sets, the vast
majority (95.44%) of sample values lie within 2 standard deviations
of the mean. For interpretation: If the standard deviation s is
known/given, use it to find rough estimates of the minimum and
maximum usual sample values as follows: Lower Range: Minimum usual
value = (mean) 2(standard deviation) Upper Range: Maximum usual
value = (mean) + 2(standard deviation)
SKEWNESS Definition: A distribution of data is skewed
(asymmetric) if it is not symmetric and if it extends more to one
side than the other. (A distribution of data is symmetric if the
left half of its histogram is roughly a mirror image of its right
half)
Definition: Skewness is a degree of asymmetry (or departure from
symmetry) of a distribution.
Lopsided to the right = Skewed to the left = Negatively Skewed
Lopsided to the left = Skewed to the right = Positively Skewed Data
not lopsided = Symmetric = Zero Skewness
For skewed distributions, the mean tends to lie on the same side
of the mode as the longer tail. Thus a measure of the asymmetry is
supplied by the difference: mean minus mode. This can be made
dimensionless if we divide it by a measure of dispersion, such as
standard deviation. To avoid using the mode, we can employ the
empirical formula (mean mode) = 3(mean median). Thus the
coefficient of skewness (I) is given by the formula:
( )
Where : I = index of skewness The equation above is called
Pearsons second coefficient of skewness. Intervals I 1.00 data can
be considered to be significantly skewed to the right I 1.00 data
can be considered to be significantly skewed to the left Example:
Find the Pearsons second coefficient of skewness for the Ages of
Oscar-winning Best Actors and Actresses (Mathematics Teacher
magazine)
-
30 Actors: 32 37 36 32 51 53 33 61 35 45 55 69 76 37 42 40 32 60
38 56 48 48 40 43 62 43 42 44 41 56 39 46 31 47 45 60 Actresses: 50
44 35 80 26 28 41 21 61 38 49 33 74 30 33 41 31 35 41 42 37 26 34
34 35 61 60 34 24 30 37 31 27 39 34 Summary: Actor Actress Mean
45.97 38.94
Median 43.5 35 Mode 32 34 Standard Deviation 11.08 13.55
Actor Pearsons second coefficient of skewness: I = 3(45.97 43.5)
11.08 = 0.6687725663 0.67 Interpretation: approximates a normal
distribution Actress Pearsons second coefficient of skewness: I =
3(38.94 35) 13.55 = 0.872324723 0.87 Interpretation: approximates a
normal distribution
-
31 Level of acceptability of a four-year Fish Technology course
along the area of Marketability as perceived by the Community,
Local Government and the Academe To compute for the standard
deviation based on the number of items (Community)
Majority Range: Usual or Normal Range: Lower Range: 4.12 0.31 =
3.81 Lower Range: 4.12 2(0.31) = 3.50 Upper Range: 4.12 + 0.31 =
4.43 Upper Range: 4.12 + 2(0.31) = 4.74 Interpretation: The 100
community respondents who perceived the level of acceptability of a
four-year Fish Technology course along the area of Marketability,
has an overall mean rating of 4.12 with a standard deviation of
0.31. Based on these two results, it implies that majority of these
community respondents who perceived the level of acceptability,
their mean response ranges from 3.81 (acceptable) to 4.43 (very
acceptable). Likewise, it is expected that it is usual or normal
for these community respondents that their perceived mean ratings
ranges from 3.50 (acceptable) to 4.74 (very acceptable).
Indicators Mean Response Community (N = 100)
LGU (N = 50)
Academe (N = 100)
1 4.00 4.24 4.04 2 4.33 4.38 4.36 3 4.65 4.06 4.60 4 3.74 3.60
4.18 5 4.18 4.06 4.16 6 3.81 4.28 3.93 7 4.11 3.82 4.01 Overall
Mean
4.12
4.06
4.18 Standard Deviation
Indicators
( )( )
Using Variance Formula:
( )
( ) ( ) ( )
( )( )
Standard Deviation:
Community (N = 100)
X X2 1 4.00 (4.00)2 = 16.0000 2 4.33 (4.33)2 = 18.7489 3 4.65
(4.65)2 = 21.6225 4 3.74 (3.74)2 = 13.9876 5 4.18 (4.18)2 = 17.4724
6 3.81 (3.81)2 = 14.5161 7 4.11 (4.11)2 = 16.8921
n = 7 (number of items)
X = 28.82
X2 = 119.2396
-
32
Activity No. 4 Exploratory Data Analysis
Male
X X2 3
10 5 4 2 6 7 8 4 3 5 4
12 4 8 5 5 9 7 5
10 6
10 3
X = X2 =
The following are the number of cigarettes smoke on an average
day according to gender on Status of Cigarette Smoking and Drinking
Liquor among ESL Teachers in Baguio City Korean Schools. Answer the
following:
(1) What is the mean and median number of cigarettes for male
group?
(2) What is the standard deviation for the data set? (3) What is
the coefficient of skewness for the data set? (4) The majority of
the male group smoke cigarettes
between what two values? (5) The male group usually smokes
cigarettes between
what two values?
-
33
HYPOTHESIS TESTING
One of the principal objectives of research is comparison: How
does one group differ from another? This typical question can be
handled by the primary tools of classical statistical inference
estimation and hypothesis testing. The unknown characteristic, or
parameter, of a population is usually estimated from a statistic
computed from sample data. Ordinarily, a researcher is interested
in estimating the mean and the standard deviation of some
characteristic of the population. The purpose of statistical
inference is to reach conclusions from sample data and to support
the conclusions with probability statements. With such information,
a researcher will be able to decide whether an observed effect is
real or is due to chance. Testing the significance of the
difference between two means, two standard deviations, two
proportions/percentages is an important area of inferential
statistics. Comparison of two or more variables
often arises in research or experiments and to be able to make
valid conclusions, one has to apply an
appropriate test statistic.
Fundamentals of Hypothesis Testing HYPOTHESIS
A hypothesis is a conjecture or statement that aims to explain
certain phenomena. To seek for the answers to
queries, a researcher tries to find and present evidences then
tests the resulting hypothesis using statistical
tools and analysis. In statistical analysis, assumptions are
given in the form of a null hypothesis, the truth of
which will either be rejected or failed to be rejected
(accepted) within a certain critical interval.
Components of a Formal Hypothesis Test
(a) Null Hypothesis (denoted by Ho) is a statement about a value
of a population parameter (such as the
mean), and it must contain the condition of equality and must be
written with the symbol =, , or . (b) Alternative Hypothesis /
Research Hypothesis (denoted by H1) is the statement that must be
true if the
null hypothesis is false and it must be written with the symbol
, < or >. NULL AND ALTERNATIVE HYPOTHESES
You might legitimately ask, What does it really mean when
researchers test hypothesis or perform tests of significance? The
concept is actually simple and direct. We are trying to find out if
two (or more) things are the same or if they are different. What
actually are null and alternative hypotheses? The null hypothesis
is that there is no difference between or among population means,
variances or proportions. For now, remember that the key part of
the definition is no difference. The hypothesis that is subjected
to testing to determine whether its truth can be rejected or failed
to be
rejected (accepted) is the null hypothesis (H0). This hypothesis
states that there is no significant relationship or
no significant difference between two or more variables, or that
one variable does not affect another variable.
In statistical research, the hypotheses should be written in
null form.
Example: Suppose you want to know whether method A is more
effective than method B in teaching high
school mathematics. The null hypothesis for this study will be
one of the following:
Ho: There is no significant difference between effectiveness of
method A and method B in
teaching high school mathematics. (AMETHOD = BMETHOD)
-
34 Ho: Method A is as effective as method B in teaching high
school mathematics (AMETHOD = BMETHOD)
The other type of hypothesis is the alternative hypothesis (H1
or HA) that challenges the null hypothesis. The
alternative hypothesis is what is known as the research
hypothesis. This hypothesis specifies that there is a
significant relationship or significant difference between two
or more variables or that one variable affects
another variable. Sometimes the alternative hypothesis is
referred to as the research hypothesis. The
alternative hypothesis or research hypothesis is what the
researcher expects to find. This is why the research,
and hence the statistical analysis, is being done.
In the example above, the alternative hypothesis can be one of
the following:
Non-Directional (Area inTwo-tails)
H1: There is a significant difference between the effectiveness
of method A and method B in
teaching high school mathematics. (AMETHOD BMETHOD)
Directional (Area in Right-Tail)
H1: Method A is more effective than method B in teaching high
school mathematics. (AMETHOD > BMETHOD)
Directional (Area in Left-Tail)
H1: Method A is less effective than method B in teaching high
school mathematics. (AMETHOD < BMETHOD)
Examples:
Null Hypothesis (Ho) Non-Directional Alternative Hypothesis
(H1)
(Research Hypothesis)
Directional Alternative Hypothesis (H1)
(Research Hypothesis) Europeans are no more or less obedient to
authority than Americans
Europeans differ from Americans with respect to obedience to
authority
Americans are more obedient to authority than Europeans
Christians have the same suicide rate as Non-Christians
Christians do not have the same suicide rate as
Non-Christians
Christians have more suicide rates than Non-Christians
The mean age of gamblers in the Asia is 30 years old
The mean age of gamblers in the Asians not 30 years old
The mean age of gamblers in the Asia is below years old
The mean monthly salary of statistics professors is at least
60,000.
The mean monthly salary of statistics professors is different
from 60,000.
The mean monthly salary of statistics professors is more than
60,000.
One-half of all internet users make on-line purchases
All internet users making on-line purchases is not one-half
Fewer than one-half of all Internet users make on-line
purchases
The proportion of defective
computers is equal to 0.05.
The proportion of defective
computers is different from 0.05.
The proportion of defective
computers is less than 0.05.
Womens heights have a standard
deviation that is equal to 2.8 inches
which is the standard deviation for
mens heights.
Womens heights have a standard
deviation that is different from 2.8
inches which is the standard
deviation for mens heights.
Womens heights have a
standard deviation less than 2.8
inches which is the standard
deviation for mens heights.
Test Statistic a statistic used to determine the relative
position of the mean, variance or proportion in the hypothesized
probability distribution of sample means. Test Statistic is a value
computed from the sample data that is used in making the decision
about the rejection of the null hypothesis. The test statistic
converts the sample statistic (such as the sample mean) to a score
(such as the z score) with the assumption that the
-
35 null hypothesis is true. The test statistic can therefore be
used to gauge whether the discrepancy between the sample and the
claim is significant. Critical Region The region on the far end of
the distribution. If only one end of the distribution, commonly
termed the tail, is involved, the region is referred to as
one-tailed test; if both ends are involved, the region is known as
two-tailed test. When the computed test statistic (z, t, F, 2,
etc.) falls in the critical region, reject the null hypothesis. The
critical region is sometimes called the rejection region. The
probability that a test statistic falls in the critical region is
denoted by . The critical region is the set of all values of the
test statistic that cause us to reject the null hypothesis.
Nonrejection Region the region of the sampling distribution not
included in ; that is, the region located under the middle portion
of the curve. Whenever the test statistic falls in this region, the
evidence does not permit the researcher to reject the null
hypothesis. The implication is that the results falling in this
region are not unexpected. The nonrejection region is denoted by (1
- ). Critical Value The number that divides the distribution
(normal or skewed) into the region where the null hypothesis will
be rejected and the region where the null hypothesis will fail to
be rejected. A critical value is any value that separates the
critical region (where we reject the null hypothesis) from the
values of the test statistic that do not lead to rejection of the
null hypothesis. The critical values depend on the nature of the
null hypothesis, the relevant sampling distribution, and the
significance level .
Test of Significance a procedure used to establish the validity
of a claim by determining whether the test statistic falls in the
critical region. If it does, the results are referred to as
significant. This test is sometimes called the hypothesis test.
The significance level (denoted by ) is the probability that the
test statistic will fall in the critical region
when the null hypothesis is actually true. If the test statistic
falls in the critical region, we will reject the null
hypothesis, so is the probability of making the mistake of
rejecting the null hypothesis when it is true. The
common level of significances are 10%, 5% and 1% but the most
preferred in educational/
psychological/sociological research is 5%.
To test the null hypothesis of no significance in the difference
between the two variables, one must set the
level of significance first. This is the probability of
committing a type I error (). A type I error is the probability
of rejecting the null hypothesis when in fact it is a true
hypothesis. The probability of accepting a null hypothesis
when in fact it is a false hypothesis is called a type II error
().
-
36
DIRECTIONAL (One-Tailed) AND NON-DIRECTIONAL (Two-tailed)
TESTS
In testing statistical hypotheses, you must always ask a key
question: Am I interested in the deviation of one population mean
from another population mean in one or both directions? The answer
is usually implicit in the way Ho and H1 are stated. If you are
interested in determining whether the mean of one data is
significantly different from the mean of the other data, you should
perform a two-tailed test, because the difference could either be
negative or positive. If you are interested in whether the mean of
one data is significantly larger or smaller than the other mean
data, you should perform a one-tailed test. A one-tailed test is
indicated for questions like: Is a new drug superior to a standard
drug? Does the air pollution level exceed safe limits? Has the
death rate been reduced for those who quit smoking? A two-tailed
test is indicated for questions like: Is there a difference between
cholesterol levels of men and women? Does the mean age of a group
of volunteers differ from that of the general population? Notice
the difference in the way these questions are worded. In a
potential one-tailed test, you will see words like exceed, reduced,
higher, lower, more, less, and better.
A test is called directional (area in one-tail) if the region of
rejection lies on one extreme side of the
distribution (either left or right) and non-directional (area in
two-tails) if the region of rejection is located on
both ends of the distribution.
Non-directional (Two-tailed) test:
The critical region is in the two extreme
regions (tails) under the curve.
-
37
Conclusions in Hypothesis Testing The original claim sometimes
becomes the null hypothesis and at other times becomes the
alternative hypothesis. The standard procedure of hypothesis
testing requires that always test the null hypothesis and that
initial conclusion will always be one of the following:
1.) Reject the null hypothesis 2.) Fail to reject the null
hypothesis
ACCEPT vs. FAIL TO REJECT Some texts say accept the null
hypothesis instead of fail to reject the null hypothesis. Whether
to use the term accept or fail to reject, we should recognize that
we are not proving the null hypothesis but merely saying that the
sample evidence is not strong enough to warrant rejection of the
null hypothesis. The term accept is somewhat misleading, because it
seems to imply incorrectly that the null hypothesis has been
proved. The phrase fail to reject says more correctly that the
available evidence is not strong enough to warrant rejection of the
null hypothesis.
TESTING HYPOTHESIS
The following are suggested steps when testing the truth of a
hypothesis
1. Formulate the null hypothesis (Ho) and the alternative
hypothesis (H1)
2. Set the desired level of significance ()
3. Determine the appropriate test statistic to be used in
testing the null hypothesis
4. Compute for the value of the statistic to be used
5. Find the critical value (tabular value) from a table
6. Compare the computed value to the tabular value and state the
decision rule:
If the absolute computed value is greater than the tabulated
value (tabled value), reject the null
hypothesis.
7. Make a conclusion and interpret the result in a non-technical
manner.
Directional (Right-tailed) test:
The critical region is in the extreme right
region (tail) under the curve.
Directional (Left-tailed test):
The critical region is in the extreme left
region (tail) under the curve.
-
38
Activity No. 5
Null and Alternative Hypotheses
The following are claims about a phenomenon. Identify whether
each hypothesis stated as null or alternative. If it is an
alternative, further identify whether the alternative hypothesis is
one-tailed (directional) or two-tailed (non-directional) test?
1. The mean amount of Coke in cans is at least 12 ounces.
2. Salaries among women business analysts have a standard
deviation greater than 126,000.00
3. More than 50% of gun owners favour stricter gun laws.
4. Nasal congestion occurs at a higher rate among drug users
than those who do not use drug.
5. Proportion of drinkers among convicted arsonists is greater
than the proportion of drinkers convicted
of fraud.
6. Ages of faculty cars vary less than the ages of student
cars.
7. The treatment and placebo groups have the same mean.
8. Men and women have different mean height.
9. Obsessive-compulsive patients and healthy persons have the
same mean brain volume.
10. There is no difference between the mean for
obsessive-compulsive patients and the mean for healthy
persons.
11. The mean amount of carbon monoxide in filtered cigarettes is
equal to the mean amount of carbon
monoxide for non-filtered cigarettes.
12. Dyspepsia occurs at a higher rate among drug users than
those who do not use drug.
13. There is a difference between the pre-training and
post-training mean weights.
14. Women with a college degree have incomes with a higher mean
than women with a high school
diploma.
15. Waiting times for the single line have lower standard
deviation than the waiting times for any one of
several lines.
16. Dozenol tablets are more soluble after being stored for one
year than before storage.
17. Percentage of women ticketed for speeding is less than the
percentage of men.
18. The average number of sold paracetamol tablets is more than
100 per day.
19. There is a significance difference in the scores of the
engineering and computer science students in a
mathematics quiz administered by their professor.
20. There is no significant difference between the mean heights
of the two groups of trees planted with
two different types of soil.
1. 8. 15. 2. 9. 16. 3. 10. 17. 4. 11. 18. 5. 12. 19. 6. 13. 20
7. 14.
-
39
PARAMETRIC VERSUS NON-PARAMETRIC STATISTICS
Parametric statistics require quantitative dependent variables
and are usually applied when these variables are measured on either
interval or ratio characteristics. Statistical techniques that
involve analysis of means, variances and sums of squares are under
parametric statistics. Parametric statistics require assumptions
about the distribution of scores within the population of interest.
The nonparametric statistics focus on differences between
distributions of scores and that can be used to analyze
quantitative variables that are measured on an ordinal or even
nominal level. Nonparametric statistics do not require many of the
assumptions about distributional properties of scores that
parametric statistics rely on.
BETWEEN-VERSUS WITHIN-SUBJECTS DESIGNS
Experiment 1 Consider an experiment where the investigator is
interested in the relationship between two variables: type of drug
and learning. The investigator wants to know whether two drugs A
and B, differentially affect performance on a learning task. Fifty
participants are randomly assigned to one of two conditions. In the
first condition, 25 participants are administered drug A and then
read a list of 15 words. They are asked to recall as many words as
possible. A learning score is derived by counting the number of
words correctly recalled (scores can range from 0 to 15). In the
second condition, a different 25 participants read the same list of
15 words and respond to the same recall task after being
administered drug B. The relative effects of the drugs on learning
are determined by comparing the responses of the two groups. In
this experiment, the investigator is studying the relationship
between two variables: (1) type of drug and (2) learning as
measured on a recall task. Type of drug is the independent variable
and the learning measure is the dependent variable. The independent
variable is set up so that participants who received drug A did not
received drug B and those who received drug B did not received drug
A, that is the two groups included different individuals. A
variable of this type is known as a between-subjects variable
because the values of the variables are split up between
participants instead of occurring completely within the same
individuals. Research designs that involve between-subjects
independent variables are referred to as between-subjects designs
or independent groups designs. Experiment 2 Consider a similar
experiment that is conducted in a slightly different approach. A
group of 25 participants are administered drug A and then given a
learning task. One month later, the same 25 participants return to
the experiment and are given the learning task after being
administered drug B. The performance of these participants under
the influence of drug B is then compared with their earlier
performance under the influence of drug A. In this experiment, the
25 participants or subjects who received drug A also received drug
B, that is, the same individuals participated in both conditions. A
variable of this type is known as a within-subjects variable.
Research designs that involve within-subjects independent variable
are referred to as within-subjects designs or correlated groups
designs or repeated measures designs.
SELECTION OF STATISTICAL TEST The importance of the selection of
a statistical test rests on distinguishing between qualitative and
quantitative variables and between within-subjects and
between-subjects designs. The requires steps are: (1) identify the
independent and dependent variables, (2) classify each as being
qualitative or quantitative, (3) classify the independent variable
as being between-subjects or within-subjects in nature and (4) note
the number of levels that each variable has
-
40
INFERENCES ABOUT TWO MEANS
Two samples are independent if the sample values selected from
one population are not related to or somehow paired with the sample
values selected from the other population. If the values in one
sample are related to the values in the other sample, the samples
are dependent. Such samples are often referred to as matched pairs,
or paired samples.
START
PAIRED t-TEST
t TEST
Pool the sample
variances
CASE 2
t TEST
Do not pool the
sample variances
CASE 3
Dependent
(Matched)
Samples
NO
Independent
Samples?
YES
Equal
Population
Variances? NOYES
INFERENCES ABOUT TWO MEANS:
Independent Samples Assumptions
1.) The two samples are independent. 2.) The two samples are
simple random samples selected from normally distributed
populations.
When these conditions are satisfied, use one of the three
different procedures corresponding to the following cases: Case 1:
The values of both population variances are known (In reality, this
case seldom occurs)
Case 2: The two populations have equal variances (That is 22
2
1 )
Case 3: The two populations have unequal variances (That is
22
2
1 )
Case 1: Both Population Variances Are Known In reality Case 1
almost never occurs. Finding population variances typically
requires that we know all of the values of both populations, and we
can therefore find the values of their population means so there is
no need to make inferences about their means.
-
41 Remember: ( )
Null Hypothesis (Ho): There is no significant difference between
two population means 1 = 2 Alternative Hypothesis (H1): There is a
significant difference between two population means 1 2
(Non-directional or two-tailed test) 1 < 2 (Directional:
Right-tailed Test)
1 > 2 (Directional: Left-Tailed Test)
Notation for parameters and statistics when considering two
populations
Choosing Between Cases 2 and 3: Preliminary F test approach:
Apply the F test to test the null hypothesis that 12 = 22. Use the
conclusion of the test as follows:
Use case 2 if F 2.50 and conclude that the two groups have equal
variances Use case 3 if F > 2.50 and conclude that the two
groups have different or unequal variances CASE 2: Equal Population
Variances: Pool the Two Sample Variances Hypothesis Test: t-Test
for two population means (assume equal variances)
21
212
21
2
2
1
2
21
2
2
1
2
2121 )()()()(
nn
nns
xx
n
s
n
s
xx
n
s
n
s
xxt
p
pppp
where pooled variance: )2(
)1()1(
21
2
22
2
112
nn
snsnsp
degrees of freedom: df = n1 + n2 2
-
42
-
43 Example: Independent simple random samples of 35 faculty
members in private institutions and 30 faculty members in public
institutions yielded the data on annual income in thousands of
dollars in the following table. At the 5% significance level, do
the data provide sufficient evidence to conclude that mean salaries
for faculty in private and public institutions differ?
Private Institutions Public Institutions
19.881 x
s1 = 26.21 n1 = 35
18.732 x
s2 = 23.95 n2 = 30
Solution:
Step 1: State the null and alternative hypotheses Ho: Mean
salaries for faculty in private and public institutions does not
differ (1 = 2) H1: Mean salaries for faculty in private and public
institutions differ (1 2) where 1 and 2 are the mean salaries of
all faculty in private and public