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QQS1013

ELEMENTARY STATISTIC

CHAPTER 2DESCRIPTIVE STATISTICS

2.1 Introduction

2.2 Organizing and Graphing Qualitative Data

2.3 Organizing and Graphing Quantitative Data

2.4 Central Tendency Measurement

2.5 Dispersion Measurement

2.6 Mean, Variance and Standard Deviation for

Grouped Data

2.7 Measure of Skewness

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OBJECTIVES

After completing this chapter, students should be able to:

Create and interpret graphical displays involve qualitative

and quantitative data.

Describe the difference between grouped and ungrouped

frequency distribution, frequency and relative frequency,

relative frequency and cumulative relative frequency.

Identify and describe the parts of a frequency distribution:

class boundaries, class width, and class midpoint.

Identify the shapes of distributions.

Compute, describe, compare and interpret the three

measures of central tendency: mean, median, and mode for

ungrouped and grouped data. Compute, describe, compare and interpret the two measures

of dispersion: range, and standard deviation (variance) for

ungrouped and grouped data.

Compute, describe, and interpret the two measures of

position: quartiles and interquartile range for ungrouped and

grouped data.

Compute, describe and interpret the measures of skewness:

Pearson Coefficient of Skewness.

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2.1 Introduction

Raw data - Data recorded in the sequence in which there are

collected and before they are processed or ranked.

Array data - Raw data that is arranged in ascending or descendingorder.

Example 1

Here is a list of question asked in a large statistics class and the raw

data given by one of the students:

1. What is your sex (m=male, f=female)?Answer (raw data): m

2. How many hours did you sleep last night?Answer: 5 hours

3. Randomly pick a letterS or Q.Answer: S

4. What is your height in inches?Answer: 67 inches

5. Whats the fastest youve ever driven a car (mph)?Answer: 110 mph

Example 2

Quantitative raw data

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Qualitative raw data

These data also called ungrouped data

2.2 Organizing and Graphing Qualitative Data

2.2.1 Frequency Distributions/ Table

2.2.2 Relative Frequency and Percentage Distribution

2.2.3 Graphical Presentation of Qualitative Data

2.2.1 Frequency Distributions / Table

A frequency distribution for qualitative data lists all categories and

the number of elements that belong to each of the categories. It exhibits the frequencies are distributed over various categories

Also called as a frequency distribution table or simply a frequency

table.

The number of students who belong to a certain category is called

thefrequencyof that category.

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2.2.2 Relative Frequency and Percentage Distribution

A relative frequency distribution is a listing of all categories along

with their relative frequencies (given as proportions or percentages).

It is commonplace to give the frequency and relative frequency

distribution together.

Calculating relative frequency and percentage of a category

Relative Frequency of a category

= Frequency of that categorySum of all frequencies

Percentage = (Relative Frequency)* 100

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Example 3

A sample of UUM staff-owned vehicles produced by Proton was

identified and the make of each noted. The resulting sample follows (W =

Wira, Is = Iswara, Wj = Waja, St = Satria, P = Perdana, Sv = Savvy):

W W P Is Is P Is W St Wj

Is W W Wj Is W W Is W WjWj Is Wj Sv W W W Wj St W

Wj Sv W Is P Sv Wj Wj W W

St W W W W St St P Wj Sv

Construct a frequency distribution table for these data with their relative

frequency and percentage.

Solution:

Category FrequencyRelative

FrequencyPercentage (%)

Wira 19 19/50 = 0.380.38*100

= 38

Iswara 8 0.16 16

Perdana 4 0.08 8

Waja 10 0.20 20

Satria 5 0.10 10

Savvy 4 0.08 8

Total 50 1.00 100

2.2.3 Graphical Presentation of Qualitative Data

1. Bar Graphs

A graph made of bars whose heights represent the frequencies of

respective categories.

Such a graph is most helpful when you have many categories to

represent.

Notice that agap is inserted between each of the bars.

It has=> simple/ vertical bar chart

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=> horizontal bar chart

=> component bar chart

=> multiple bar chart

Simple/ Vertical Bar Chart

To construct a vertical bar chart, mark the various categories on the

horizontal axis and mark the frequencies on the vertical axis

Refer to Figure 2.1 and Figure 2.2,

Figure 2.1 Figure 2.2

Horizontal Bar Chart

To construct a horizontal bar chart, mark the various categories on

the vertical axis and mark the frequencies on the horizontal axis.

Example 4: Refer Example 3,

Figure 2.30 5 10 15 20

Wira

Iswara

Perdana

Waja

Satria

Savvy

Frequency

TypesofVehicle

UUM Staff-owned Vehicles Produced By

Proton

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Another example of horizontal bar chart: Figure 2.4

Figure 2.4: Number of students at Diversity College who areimmigrants, by last country of permanent residence

Component Bar Chart

To construct a component bar chart, all categories is in one bar and

every bar is divided into components.

The height of components should be tally with representativefrequencies.

Example 5

Suppose we want to illustrate the information below, representing

the number of people participating in the activities offered by an

outdoor pursuits centre during Jun of three consecutive years.

2004 2005 2006Climbing 21 34 36Caving 10 12 21Walking 75 85 100

Sailing 36 36 40Total 142 167 191

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Solution:

Figure 2.5

Mulztiple Bar Chart

To construct a multiple bar chart, each bars that representative any

categories are gathered in groups.

The height of the bar represented the frequencies of categories.

Useful for making comparisons (two or more values).

Example 6: Refer example 5,

Figure 2.6

0

20

40

60

80

100

120

140

160

180200

2004 2005 2006

Numberofparticipants

Year

Activities Breakdown (Jun)

Sailing

Walking

Caving

Climbing

0

20

40

60

80

100

120

2004 2005 2006

Numberofparticipants

Year

Activities Breakdown (Jun)

Climbing

Caving

Walking

Sailing

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Another example of horizontal bar chart: Figure 2.7

Figure 2.7: Preferred snack choices of students at UUM

The bar graphs for relative frequency and percentage distributions

can be drawn simply by marking the relative frequencies or

percentages, instead of the class frequencies.

2. Pie Chart

A circle divided into portions that represent the relative frequencies

or percentages of a population or a sample belonging to different

categories.

An alternative to the bar chart and useful for summarizing a single

categorical variable if there are not too many categories.

The chart makes it easy to compare relative sizes of each

class/category.

The whole pie represents the total sample or population. The pie is

divided into different portions that represent the different categories.

To construct a pie chart, we multiply 360o by the relative frequency

for each category to obtain the degree measure or size of the angle

for the corresponding categories.

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Example 7 (Table 2.6 and Figure 2.8):

Table 2.6 Figure 2.8

Example 8 (Table 2.7 and Figure 2.9):

MovieGenres

Frequency RelativeFrequency

Angle Size

Comedy

ActionRomanceDrama

Horror

Foreign

ScienceFiction

54

362828

22

16

16

0.27

0.180.140.14

0.11

0.08

0.08

360*0.27=97.2o

360*0.18=64.8o

360*0.14=50.4

o

360*0.14=50.4o

360*0.11=39.6o

360*0.08=28.8o

360*0.08=28.8o

200 1.00 360o

Figure 2.9Figure 2.9

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3. Line Graph/Time Series Graph

A graph represents data that occur over a specific period time of

time.

Line graphs are more popular than all other graphs combined

because their visual characteristics reveal data trends clearly and

these graphs are easy to create.

When analyzing the graph, look for a trend or pattern that occurs

over the time period.

Example is the line ascending (indicating an increase over time) or

descending (indicating a decrease over time).

Another thing to look for is theslope, orsteepness, of the line. A line

that is steep over a specific time period indicates a rapid increase or

decrease over that period.

Two data sets can be compared on the same graph (called a

compound time series graph) if two lines are used.

Data collected on the same element for the same variable at different

points in time or for different periods of time are called time series

data.

A line graph is a visual comparison of how two variablesshown on

the x- and y-axesare related or vary with each other. It shows

related information by drawing a continuous line between all the

points on a grid.

Line graphs compare two variables: one is plotted along the x-axis

(horizontal) and the other along the y-axis (vertical).

The y-axis in a line graph usually indicates quantity (e.g., RM,

numbers of sales litres) or percentage, while the horizontal x-axis

often measures units of time. As a result, the line graph is often

viewed as a time series graph

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Example 9

A transit manager wishes to use the following data for a presentation

showing how Port Authority Transit ridership has changed over the

years. Draw a time series graph for the data and summarize thefindings.

YearRidership

(in millions)1990

1991

1992

1993

1994

88.0

85.0

75.7

76.6

75.4

Solution:

The graph shows a decline in ridership through 1992 and then leveling offfor the years 1993 and 1994.

75

77

79

81

83

85

87

89

1990 1991 1992 1993 1994

Ridership(in

millions)

Year

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Exercise 1

1. The following data show the method of payment by 16 customers in asupermarket checkout line. Here, C = cash, CK = check, CC = credit card, D =

debit and O = other.

C CK CK C CC D O C

CK CC D CC C CK CK CC

a. Construct a frequency distribution table.b. Calculate the relative frequencies and percentages for all categories.c. Draw a pie chart for the percentage distribution.

2. The frequency distribution table represents the sale of certain product in ZeeZeeCompany. Each of the products was given the frequency of the sales in certain

period. Find the relative frequency and the percentage of each product. Then,

construct a pie chart using the obtained information.

Type ofProduct

Frequency RelativeFrequency

Percentage Angle Size

A

B

C

D

E

13

12

5

9

11

3. Draw a time series graph to represent the data for the number of worldwide airlinefatalities for the given years.

Year 1990 1991 1992 1993 1994 1995 1996No. of

fatalities440 510 990 801 732 557 1132

4. A questionnaire about how people get news resulted in the following information

from 25 respondents (N = newspaper, T = television, R = radio, M = magazine).

N N R T T

R N T M R

M M N R N

T R M N M

T R R N N

a. Construct a frequency distribution for the data.b. Construct a bar graph for the data.

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5. The given information shows the export and import trade in million RM for fourmonths of sales in certain year. Using the provided information, present this

data in component bar graph.

Month Export Import

SeptemberOctober

November

December

2830

32

24

2028

17

14

6. The following information represents the maximum rain fall in millimeter (mm)in each state in Malaysia. You are supposed to help a meteorologist in your

place to make an analysis. Based on your knowledge, present this information

using the most appropriate chart and give your comment.

State Quantity (mm)

Perlis

Kedah

Pulau Pinang

Perak

Selangor

Wilayah Persekutuan

Kuala Lumpur

Negeri Sembilan

Melaka

Johor

Pahang

Terengganu

Kelantan

Sarawak

Sabah

435

512

163

721

664

1003

390

223

876

1050

1255

986

878

456

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2.3 Organizing and Graphing Quantitative Data

2.3.1 Stem and Leaf Display

2.3.2 Frequency Distribution

2.3.3 Relative Frequency and Percentage

Distributions.

2.3.4 Graphing Grouped Data

2.3.5 Shapes of Histogram

2.3.6 Cumulative Frequency Distributions.

2.3.1 Stem-and-Leaf Display

In stem and leaf display of quantitative data, each value is

divided into two portions a stem and a leaf. Then the leaves

for each stem are shown separately in a display.

Gives the information of data pattern.

Can detect which value frequently repeated.

Example 10

25 12 9 10 5 12 23 736 13 11 12 31 28 37 61441 38 44 13 22 18 19

Solution:

0 9 5 7 6

1 2 0 2 3 1 2 4 3 8 9

2 5 3 8 2

3 6 1 7 8

4 1 4

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2.3.2 Frequency Distributions

Afrequency distribution for quantitative data lists all the classes and

the number of values that belong to each class.

Data presented in form of frequency distribution are called grouped

data.

The class boundary is given by the midpoint of the upper limit of

one class and the lower limit of the next class. Also calledreal class

limit.

To find the midpoint of the upper limit of the first class and the

lower limit of the second class, we divide the sum of these two limits

by 2.

e.g.:

400 401400.5

2

class boundary

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Class Width (class size)

Class width = Upper boundaryLower boundary

e.g. :Width of the first class = 600.5400.5 = 200

Class Midpoint or Mark

Lower limit + Upper limitclass midpoint or mark =

2

e.g:

401 600Midpoint of the 1st class = 500.5

2

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Constructing Frequency Distribution Tables

1. To decide the number of classes, we used Sturges formula,

which is

c = 1 + 3.3 log n

where c is the no. of classes

n is the no. of observations in the data set.

2. Class width,

Largest value - Smallest valueNumber of classes

Range

i

ic

This class width is rounded to a convenient number.

3. Lower Limit of the First Class or the Starting Point

Use the smallest value in the data set.

Example 11

The following data give the total home runs hit by all players of each of

the 30 Major League Baseball teams during 2004 season

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Solution:

i) Number of classes, c = 1 + 3.3 log 30= 1 + 3.3(1.48)= 5.89 6 class

ii) Class width,

242 135

6

17.8

18

i

iii) Starting Point = 135

Table 2.10 Frequency Distribution for Data of Table 2.9

Total Home Runs Tally f135152

153170171188

189206

207224

225242

|||| ||||

||||||

|||| |

|||

||||

10

25

6

3

4

30f

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2.3.3 Relative Frequency and Percentage Distributions

Frequency of that classRelative frequency of a class =

Sum of all frequencies

=

Percentage = (Relative frequency) 100

f

f

Example 12 (Refer example 11)

Table 2.11: Relative Frequency and Percentage Distributions

Total HomeRuns

Class Boundaries RelativeFrequency

%

135152153170

171188

189206

207224

225242

134.5 less than 152.5152.5 less than 170.5

170.5 less than 188.5

188.5 less than 206.5

206.5 less than 224.5

224.5 less than 242.5

0.33330.0667

0.1667

0.2

0.1

0.1333

33.336.67

16.67

20

10

13.33

Sum 1.0 100%

2.3.4 Graphing Grouped Data

1. Histograms

A histogram is a graph in which the class boundaries are

marked on the horizontal axis and either the frequencies,

relative frequencies, or percentages are marked on the vertical

axis. The frequencies, relative frequencies or percentages are

represented by the heights of the bars.

In histogram, the bars are drawn adjacent to each other and

there is a space between y axis and the first bar.

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0

2

4

6

8

10

12

1

Frequency

Total home runs

Example 13 (Refer example 11)

Figure 2.10: Frequency histogram for Table 2.10

2. Polygon

A graph formed by joining the midpoints of the tops of

successive bars in a histogram with straight lines is called apolygon.

Example 13

Figure 2.11: Frequency polygon for Table 2.10

0

2

4

6

8

10

12

1

Frequency

Total home runs

134.5 152.5 170.5 188.5 206.5 224.5 242.5

134.5 152.5 170.5 188.5 206.5 224.5 242.5

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For a very large data set, as the number of classes is increased (and

the width of classes is decreased), the frequency polygon eventually

becomes a smooth curve called a frequency distribution curve or

simply afrequency curve.

Figure 2.12: Frequency distribution curve

2.3.5 Shape of Histogram

Same as polygon.

For a very large data set, as the number of classes is increased

(and the width of classes is decreased), the frequency polygon

eventually becomes a smooth curve called a frequency

distribution curveor simply afrequency curve.

The most common of shapes are:

(i) Symmetric

Figure 2.13 & 2.14: Symmetric histograms

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(ii) Right skewed and (iii) Left skewed

Figure 2.15 & 2.16: Right skewed and Left skewed

Describing data using graphs helps us insight into the main

characteristics of the data.

When interpreting a graph, we should be very cautious. We should

observe carefully whether the frequency axis has been truncated or

whether any axis has been unnecessarily shortened or stretched.

2.3.6 Cumulative Frequency Distributions

Acumulative frequency distribution gives the total number of

values that fall below the upper boundary of each class.

Example 14: Using the frequency distribution of table 2.11,

Total HomeRuns

Class Boundaries Cumulative Frequency

135152

153170

171188

189206

207224

225242

134.5 less than 152.5

152.5 less than 170.5

170.5 less than 188.5

188.5 less than 206.5

206.5 less than 224.5

224.5 less than 242.5

10

10+2=12

10+2+5=17

10+2+5+6=23

10+2+5+6+3=26

10+2+5+6+3+4=30

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Ogive

An ogive is a curve drawn for the cumulative frequency distribution

by joining with straight lines the dots marked above the upper

boundaries of classes at heights equal to the cumulative frequencies

of respective classes.

Two type of ogive:

(i) ogive less than

(ii) ogive greater than

First, build a table of cumulative frequency.

Example 15 (Ogive Less Than)

Earnings (RM) CumulativeFrequency

(F)

Less than 29.5Less than 39.5Less than 49.5

Less than 59.5Less than 69.5Less than 79.5Less than 89.5

05

11

17202330

Figure 2.17

5663

37

30 3940 4950 5960 - 69

70

7980 - 89

30

Number ofstudents (f)

Total

Earnings(RM)

CumulativeFrequency

0

5

10

1520

25

30

35

29.5 39.5 49.5 59.5 69.5 79.5 89.5

Earnings

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Example 16 (Ogive Greater Than)

Figure 2.18

Figure 2.18

566337

30 3940 4950 5960 - 6970 7980 - 89

30

Number of

students (f)

Total

Earnings

(RM)

302519131070

More than 29.5More than 39.5More than 49.5More than 59.5More than 69.5More than 79.5More than 89.5

CumulativeFre uenc F

Earnings

RM

0

5

10

15

20

25

30

35

29.5 39.5 49.5 59.5 69.5 79.5 89.5

EarningsCumulativeFrequency

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2.3.7 Box-Plot

Describe the analyze data graphically using 5 measurement:

smallest value, first quartile (K1), second quartile (median or

K2), third quartile (K3) and largest value.

2.4 Measures of Central Tendency

2.4.1 Ungrouped Data(1) Mean

(2) Weighted mean

(3) Median

(4) Mode

2.4.2 Grouped Data(1) Mean

(2) Median

(3) Mode

Smallest

value

Largest

value

K1 Median K3

Largestvalue

K1 Median K3

Largestvalue

K1 Median K3

Smallestvalue

Smallest

value

For symmetry data

For left skewed data

For right skewed data

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2.4.3 Relationship among mean, median & mode

2.4.1 Ungrouped Data

1. Mean

Mean for population data:x

N

Mean for sample data:x

xn

where: x = the sum af all valuesN = the population size

n = the sample size, = the population mean

x = the sample mean

Example 17

The following data give the prices (rounded to thousand RM) of five

homes sold recently in Sekayang.

158 189 265 127 191

Find the mean sale price for these homes.

Solution:

158 189 265 127 191

5

930

5

186

x

x

n

Thus, these five homes were sold for an average price of RM186thousand @ RM186 000.

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The mean has the advantage that its calculation includes each valueof the data set.

2. Weighted Mean

Used when have different needs.

Weight mean :

w

wxx

w

where w is a weight.

Example 18

Consider the data of electricity components purchasing from a factory inthe table below:

Type Number of component (w) Cost/unit (x)

1

23

4

5

1200

5002500

1000

800

RM3.00

RM3.40RM2.80

RM2.90

RM3.25

Total 6000

Solution:

1200(3) 500(3.4) 2500(2.8) 1000(2.9) 800(3.25)

1200 500 2500 1000 800

17800

6000

2.967

w

wx

x w

=

=

=

Mean cost of a unit of the component is RM2.97

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3. Median

Median is the value of the middle term in a data set that has been

ranked in increasing order.

Procedure for finding the Median

Step 1: Rank the data set in increasing order.

Step 2: Determine the depth (position or location) of the median.

1

2

n Depth of Median =

Step 3: Determine the value of the Median.

Example 19

Find the median for the following data:

10 5 19 8 3

Solution:

(1) Rank the data in increasing order3 5 8 10 19

(2) Determine the depth of the Median1

2

5 1

2

3

n

Depth of Median =

=

=

(3) Determine the value of the median

Therefore the median is located in third position of the data set.

3 5 8 10 19

Hence, the Median for above data = 8

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Example 20

Find the median for the following data:

10 5 19 8 3 15

Solution:

(1) Rank the data in increasing order

3 5 8 10 15 19

(2) Determine the depth of the Median

1

2

6 1

2

3.5

n

Depth of Median =

=

=

(3) Determine the value of the Median

Therefore the median is located in the middle of 3rd

position and 4th

position of the data set.

8 109

2

Median

Hence, the Median for the above data = 9

The median gives the center of a histogram, with half of the data

values to the left of (or, less than) the median and half to the right of

(or, more than) the median.

The advantage of using the median is that it is not influenced by

outliers.

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4. Mode

Mode is the value that occurs with the highest frequency in adata set.

Example 21

1. What is the mode for given data?

77 69 74 81 71 68 74 73

2. What is the mode for given data?

77 69 68 74 81 71 68 74 73

Solution:

1. Mode = 74 (this number occurs twice): Unimodal

2. Mode = 68 and 74: Bimodal

A major shortcoming of the mode is that a data set may have

none or may have more than one mode.

One advantage of the mode is that it can be calculated for both

kinds of data, quantitative and qualitative.

2.4.2 Grouped Data

1. Mean

Mean for population data:

fx =

N

Mean for sample data:

fxx =

n

Where x the midpoint andf is the frequency of a class.

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Example 22

The following table gives the frequency distribution of the number of

orders received each day during the past 50 days at the office of a mail-

order company. Calculate the mean.

Solution:

Because the data set includes only 50 days, it represents a sample. The

value of fx is calculated in the following table:

Numberof order

f x fx

10121315

1618

1921

412

20

14

1114

17

20

44168

340

280

n = 50 fx= 832

The value of mean sample is:

fx 832x = = =16.64

n 50

Thus, this mail-order company received an average of 16.64 orders per

day during these 50 days.

Numberof order

f

1012

1315

1618

1921

4

12

20

14

n = 50

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2. Median

Step 1: Construct the cumulative frequency distribution.

Step 2: Decide the class that contain the median.

Class Median is the first class with the value of cumulative

frequency is at least n/2.

Step 3: Find the median by using the following formula:

Where:n = the total frequencyF = the total frequency before class mediani = the class width

= the lower boundary of the class median= the frequency of the class median

Example 23

Based on the grouped data below, find the median:

Time to travel to work Frequency

110

11202130

31404150

8

1412

97

Median mm

n- F

2= L + i f

mL

mf

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Solution:

1st Step: Construct the cumulative frequency distribution

Time to travelto work

Frequency CumulativeFrequency

110

11202130

3140

4150

8

1412

9

7

8

2234

43

50

Class median is the 3rd

class

So, F= 22, = 12, = 21.5 and i = 10

Therefore,

Thus, 25 persons take less than 24 minutes to travel to work and another

25 persons take more than 24 minutes to travel to work.

252

50

2

n

mf

mL

2

25 2221 5 10

12

24

Median

=

=

mm

n- F

= L if

-.

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3. Mode

Mode is the value that has the highest frequency in a data set.

For grouped data, class mode (or, modal class) is the class with

the highest frequency.

To find mode for grouped data, use the following formula:

Where:

is the lower boundary of class mode

is the difference between the frequency of class mode and

the frequency of the class before the class mode

is the difference between the frequency of class mode and

the frequency of the class after the class mode

i is the class width

Example 24

Based on the grouped data below, find the mode

Time to travel to work Frequency

110

1120

21303140

4150

8

14

129

7

Mode 1mo

1 2

= L + i

+

moL

1

2

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2.4.3 Relationship among mean, median & mode

As discussed in previous topic, histogram or a frequency

distribution curve can assume either skewed shape or

symmetrical shape.

Knowing the value of mean, median and mode can give us

some idea about the shape of frequency curve.

(1) For a symmetrical histogram and frequency curve with one

peak, the value of the mean, median and mode are identical

and they lie at the center of the distribution.(Figure 2.20)(2) For a histogram and a frequency curve skewed to the right, the

value of the mean is the largest that of the mode is the smallest

and the value of the median lies between these two.

Figure 2.20: Mean, median, andmode for a symmetric histogram

and frequency distribution curve

Figure 2.21: Mean, median, and mode fora histogram and frequency distributioncurve skewed to

the right

(3) For a histogram and afrequency curve skewed to

the left, the value of the

mean is the smallest and

that of the mode is the

largest and the value of the

median lies between thesetwo.

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Figure 2.22: Mean, median, and mode for a histogram and

frequency distribution curve skewed to the left

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2.5 Dispersion Measurement

The measures of central tendency such as mean, median and

mode do not reveal the whole picture of the distribution of adata set.

Two data sets with the same mean may have a completely

different spreads.

The variation among the values of observations for one data

set may be much larger or smaller than for the other data set.

2.5.1 Ungrouped data

(1) Range

(2) Standard Deviation

2.5.2 Grouped data

(1) Range

(2) Standard deviation

2.5.3 Relative Dispersion Measurement

2.5.1 Ungrouped Data

1. Range

RANGE = Largest valueSmallest value

Example 25:

Find the range of production for this data set,

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Solution:

Range = Largest valueSmallest value

= 267 27749 651

= 217 626

Disadvantages:o being influenced by outliers.o Based on two values only. All other values in a data set are

ignored.

2. Variance and Standard Deviation

Standard deviation is the most used measure of dispersion.

A Standard Deviation value tells how closely the values of a data

set clustered around the mean.

Lower value of standard deviation indicates that the data set value

are spread over relatively smaller range around the mean.

Larger value of data set indicates that the data set value are spread

over relatively larger around the mean (far from mean).

Standard deviation is obtained the positive root of the variance:

Variance Standard Deviation

Population

N

N

xx

2

2

2

22

Sample

1

2

2

2

n

n

x

x

s

22ss

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Example 26

Let x denote the total production (in unit) of company

Company ProductionA

B

C

D

E

62

93

126

75

34

Find the variance and standard deviation,

Solution:

Company Production (x) x2

A

B

C

D

E

62

93

126

75

34

3844

8649

15 876

5625

1156

1156 351502 x

2

5

5 1

1182 50

39035150-=

=

2

2

2

xx -

ns =n -1

.

Since s2

= 1182.50;

Therefore,

1182 50

34 3875

s .

.

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The properties of variance and standard deviation:

(1) The standard deviation is a measure of variation of all values

from the mean.

(2) The value of the variance and the standard deviation are nevernegative. Also, larger values of variance or standard deviation

indicate greater amounts of variation.

(3) The value ofs can increase dramatically with the inclusion of

one or more outliers.

(4) The measurement units of variance are always the square ofthe measurement units of the original data while the units of

standard deviation are the same as the units of the original

data values.

2.5.2 Grouped Data

1. Range

Class Frequency

4150

5160

61707180

8190

91 - 100

1

3

713

10

6

Total 40

Upper bound of last class = 100.5

Lower bound of first class = 40.5Range = 100.540.5 = 60

Range = Upper bound of last classLower bound of first class

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2. Variance and Standard Deviation

Variance Standard Deviation

Population

2

2

2

fx

fx NN

22

Sample

2

2

2

1

fxfx

nsn

22ss

Example 27

Find the variance and standard deviation for the following data:

Solution:

No. of order f x fx fx2

101213151618

1921

41220

14

111417

20

44168340

280

48423525780

5600

Total n = 50 857 14216

No. of order f

1012

1315

1618

1921

4

12

20

14

Total n = 50

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1. Quartiles

Quartiles are three summary measures that divide ranked dataset into four equal parts.

The 1st quartilesdenoted as Q1

1

4

1Depth of Q =

n

The 2nd quartilesmedian of a data set or Q2

The 3rd quartilesdenoted as Q3

3 1

4

3Depth of Q =

(n )

Example 29

1. Table below lists the total revenue for the 11 top tourism company inMalaysia

109.7 79.9 21.2 76.4 80.2 82.1 79.4 89.3 98.0 103.586.8

Solution:

Step 1: Arrange the data in increasing order

76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7

121.2

Step 2: Determine the depth for Q1 and Q3

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1 11 13

4 4

1Depth of Q = = =

n

3 11 13 1 94 4

3Depth of Q = = =

(n )

Step 3: Determine the Q1 and Q3

76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7121.2

Q1 = 79.9

Q3 = 103.5

2. Table below lists the total revenue for the 12 top tourism company inMalaysia

109.7 79.9 74.1 121.2 76.4 80.2 82.1 79.4 89.3

98.0 103.5 86.8

Solution:

Step 1: Arrange the data in increasing order

74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5109.7 121.2

Step 2: Determine the depth for Q1

and Q3

1 12 13 25

4 4

1Depth of Q = = =

n.

3 12 13 19 75

4 4

3Depth of Q = = =

(n ).

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Step 3: Determine the Q1 and Q3

74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5109.7 121.2

Q1 = 79.4 + 0.25 (79.979.4) = 79.525

Q3 = 98.0 + 0.75 (103.598.0) = 102.125

2. Interquartile Range

The difference betweenthe third quartile and the first quartile

for a data set.

IQR = Q3Q1

Example 30

By referringto example 29, calculate the IQR.

Solution:

IQR = Q3Q1 = 102.12579.525 = 22.6

2.6.2 Grouped Data

1.Quartiles

From Median, we can get Q1 and Q3 equation as follows:

1

1

1 Q

Q

n- F

4Q L + if

;

3

3

3 Q

Q

3 n- F

4Q L + if

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Example 31

Refer to example 23, find Q1 and Q3

Solution:

1st Step: Construct the cumulative frequency distribution

Time to travelto work

Frequency Cumulative Frequency

110

11

202130

3140

4150

8

1412

9

7

8

2234

43

50

2nd Step: Determine the Q1 and Q3

1

n 50Class Q 12 5

4 4

.

Class Q1 is the 2nd

class

Therefore,

1

1

1

4

12 5 810 5 10

14

13 7143

Q

Q

n- F

Q L if

. -.

.

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3

3 503nClass Q 37 5

4 4.

Class Q3 is the 4th

class

Therefore,

3

3

3

4

37 5 3430 5 10

9

34 3889

Q

Q

n- F

Q L if

. -.

.

2.Interquartile Range

IQR = Q3Q1

Example 32:

Referto example 31, calculate the IQR.

Solution:

IQR = Q3Q1 = 34.388913.7143 = 20.6746

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2.7 Measure of Skewness

To determine the skewness of data (symmetry, left skewed,

right skewed) Also called Skewness Coefficient orPearson Coefficient of

Skewness

IfSk+ve right skewed

IfSk-ve left skewed

IfSk= 0

IfSk takes a value in between (-0.9999, -0.0001) or (0.0001,

0.9999)

approximately symmetry.

Example 33

The duration of cancer patient warded in Hospital Seberang Jaya recorded

in a frequency distribution. From the record, the mean is 28 days, median

is 25 days and mode is 23 days. Given the standard deviation is 4.2 days.

a. What is the type of distribution?b. Find the skewness coefficient

Solution:

This distribution is right skewed because the mean is the largest value

28 2311905

4 2

3 3 28 2521429

4 2

Mean - Mode

OR

Mean - Median

k

k

S .s .

S .s .

So, from the Skvalue this distribution is right skewed.

s

ModeMeanS

or

s

ModeMeanS

k

k

)(3

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Exercise 2:

1. A survey research company asks 100 people how many times they have been tothe dentist in the last five years. Their grouped responses appear below.

Number of Visits Number of Responses

04 16

59 25

1014 48

1519 11

What are the mean and variance of the data?

2. A researcher asked 25 consumers: How much would you pay for a televisionadapter that provides Internet access? Their grouped responses are as follows:

Amount ($) Number of Responses

099 2

100199 2

200249 3

250299 3

300349 6

350399 3

400499 4

500999 2

Calculate the mean, variance, and standard deviation.

3. The following data give the pairs of shoes sold per day by a particular shoe storein the last 20 days.

85 90 89 70 79 80 83 83 75 76

89 86 71 76 77 89 70 65 90 86

Calculate thea. mean and interpret the value.b. median and interpret the value.c. mode and interpret the value.d. standard deviation.

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4. The followings data shows the information of serving time (in minutes) for 40

customers in a post office:

2.0 4.5 2.5 2.9 4.2 2.9 3.5 2.8

3.2 2.9 4.0 3.0 3.8 2.5 2.3 3.5

2.1 3.1 3.6 4.3 4.7 2.6 4.1 3.14.6 2.8 5.1 2.7 2.6 4.4 3.5 3.0

2.7 3.9 2.9 2.9 2.5 3.7 3.3 2.4

a. Construct a frequency distribution table with 0.5 of class width.

b. Construct a histogram.

c. Calculate the mode and median of the data.

d. Find the mean of serving time.

e. Determine the skewness of the data.

f. Find the first and third quartile value of the data.

g. Determine the value of interquartile range.

5. In a survey for a class of final semester student, a group of data was obtained for

the number of text books owned.

Number ofstudents

Number of textbook owned

12

9

11

15

108

5

5

3

2

10

Find the average number of text book for the class. Use the weighted mean.

6. The following data represent the ages of 15 people buying lift tickets at a skiarea.

15 25 26 17 38 16 60 21

30 53 28 40 20 35 31

Calculate the quartile and interquartile range.

7. A student scores 60 on a mathematics test that has a mean of 54 and a standarddeviation of 3, and she scores 80 on a history test with a mean of 75 and a

standard deviation of 2. On which test did she perform better?

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8. The following table gives the distribution of the shares price for ABC Companywhich was listed in BSKL in 2005.

Price (RM) Frequency

1214

1517

1820

2123

2426

27 - 29

5

14

25

7

6

3

Find the mean, median and mode for this data.