2.1 Measures of Central Tendency-ungrouped

Measures of Central Tendencies

Numbers which, in some sense, give the central or middle values of the data

locates the center of the distribution of a set of data

the most typical value of a set of data

representative value of a given set of data

Mean

• arithmetic mean / average• the sum of the values divided by the number of values which were added.

Mean of ungrouped data

i1 2 3 nxx x x ... x

xn n

Where - sample mean xi - ith observation/item in the sample n - number of observations in the sample

x

i1 2 3 nxx x x ... x

xn n

Example 1: find the mean of the sample: 7, 11, 11, 8, 12, 7, 6, 6

7 11 11 8 12 7 6 6 68x 8.5

8 8

The sample mean is 8.5

Mean of ungrouped data

Example 2: find the mean of this sample:

18, 22, 25, 25, 26, 29, 45

Mean of ungrouped data

Weighted Mean

1 1 2 2 3 3 n nw

1 2 3 n

x (w ) x (w ) x (w ) ... x (w )x

w w w ... w

Where - weighted mean xi - ith observation/item in the sample wi – weight of the ith observation

wx

Examples of uses of weighted mean are in computing term GPA and in getting the mean responses for a Likert-type of questions.

It is used if the researcher wants

to know the feelings or

opinions of the respondents

regarding any topic or issues of

interest.

Likert-type questions

Choices are:5 – (SA) Strongly agree4 – (A) Agree3 – (N) Neutral2 – (D) Disagree1 – (SD) Strongly disagree

Check appropriate box 5 4 3 2 1

1 Student nurses serve as role models for their patients and the public.

2 Student nurses should set a good example by not smoking.

3 Patient's chances of quitting smoking are increased if a student nurses advises him or her to quit.

4 Smoking is harmful to your health.

5 Smoking other tobacco products is harmful to a person’s health.

Likert-type questions

54 3 2 1 Interpretati

on

1 7 11 2 0 0

2 9 10 1 0 0

3 2 8 8 2 0

4 16 4 0 0 0

5 17 3 0 0 0

x

Likert-type questions

4.25Strongly Agree4.40Strongly Agree

3.50 Agree

Likert-Type Mean Interpretation1.00 – 1.79 – Strongly Disagree1.80 – 2.59 – Disagree2.60 – 3.39 – Neutral3.40 – 4.19 – Agree4.20 – 5.00 – Strongly Agree

Strongly Agree4.80

4.85Strongly Agree

4.25 4.40 3.50 4.80 4.85Grand Mean 4.36

5

Strongly Agree4.36

Mean for Grouped Data

Where f – frequency of the class

X – Class Mark

n – sample size

nXf

x

Classes Freq. (f)

Class Mark(Xm)

fXm

12 - 22 23 - 33 34 - 44 45 - 55 56 – 66

47621

1728395061

Total 20

65932.95

20

Mean for Grouped Data

68

196

234100

61

659

nXf

x

Characteristics of the Mean

1. It can be calculated for any set of numerical data, so it always exist.

2. A set of numerical data has one and only one mean.

4. It is greatly affected by extreme or deviant values.

3. It is the most reliable since it takes into account every item in the set of data.

MedianThe median of a data is defined to be the ‘middle value’.

[(n 1) / 2]th term when n is odd

x (n / 2)th term + [(n/2)+1]th termwhen n is even

2

Note: it is important to arrange first the sample in ascending order before getting the median.

Thus, when n is odd, the median is the ‘center’ observation.

When n is even, it is the average of the two center observations.

Example 3: find the median of the this sample: 7, 11, 11, 8, 12, 7, 6, 6

The sample median is 7.5

(n / 2)th term + [(n/2)+1]th termx

2

Solution: Arrange the observations in ascending order.

6, 6, 7, 7, 8, 11, 11, 12

Since n = 8 (even), then

n/2 = 8/2 = 4 and (n/2) + 1=(8/2)+1=5 Thus,

4th term + 5th termx

2

7 + 8x 7.5

2

Median

Example 4: find the median of this sample: 18, 22, 25, 25, 26, 29, 45

The sample median is 25

Solution:The solution is already arranged in ascending order.

Since n = 7 (odd), then

(n + 1)/2 = (7 + 1)/2 = 4.

Thus, = 25

x [(n 1) / 2]th term

x 4th term

Median

Characteristics

1. The score or class in a distribution, below which 50% of the score fall and above which another 50% lie.

2. Not affected by extreme or deviant values.3. Appropriate to use when there are extreme or deviant values.

Illustration:18, 22, 25, 25, 26, 29, 45

Compare the mean (example 2) and the median (example 4)of the above sample.

Mean = 27.1 Median = 25

Which is the better measure of central tendency in thisexample? Why?

Median

1. It is used when we want to find the value which occurs most often.

2. It is a quick approximation of the average.

3. It is an inspection average.

4. It is the most unreliable among the three measures of central tendency because its value is undefined in some observations.

Mode

Mode

Mode: VS

2. The ages of 5 students are: 17, 18, 23, 20, & 19No Mode

3. The grades of 5 students are: 4.0, 3.5, 4.0, 3.5, & 1.0

Mode: 4.0 & 3.54. The weight of 5 persons in pounds are: 117, 218, 233, 120, & 117

Mode: 117

1. The following are the descriptive evaluation of 5 teachers.

Teacher Evaluation A VS B S C VS D VS E S

Examples

ComparisonFactor Mean Median Mode

Type of data

Quantitative

Quantitative

Quantitative and

Qualitative

Extreme scoreproblem

Yes No No

Always measurable

Yes Yes No

Number of score

1 1 0,1,2…

Characteristics

All scores included in computatio

n

Middle value

Popular value

(Grouped)Approximating the Median from the Freq. Distribution Table

Steps:1.) Construct the < cumulative frequency distribution.2.) Starting from the top, locate the class with the <cf greater than or equal to n/2 for the first time. median class

3.) Approximate using the formula:

+i

where = the lower class boundary of the median class i= class width n=total no. of observations= less than cumulative freq of the class preceding the median class = frequency of the median class

Approximating the Mode from the Freq. Distribution Table

Steps:1.) Locate the modal class. The modal class is the class with the highest frequency

2.) Approximate using the formula:

+i

where = the lower class boundary of the modal class i= class width = frequency of the modal class = frequency of the class preceding the modal class = frequency of the class following the modal class

Find the mean, median and mode.FDT of Final grades of 100 Math 103 students

Class Freq42-48 449-55 1256-62 2263-69 2770-76 1777-83 984-90 891-97 1

Measures of Location or Fractiles

-values below which a specified fraction or percentage of the observations in a given set must fall.

Percentiles

Percentiles are values that divide a set of observations in an array into 100 equal parts.

Percentiles Sort all observations in ascending order Compute the position L = (K/100) * N, where N

is the total number of observations. If L is a whole number, the K-th percentile is

the value midway between the L-th value and the next one.

If L is not a whole number, change it by rounding up to the nearest integer. The value at that position is the K-th percentile.

Decilesvalues that divide the array into 10 equal parts

D1 - the value below which 10% of the values fallD2 - the value below which 20% of the values fall…

Deciles Sort all observations in ascending order Compute the position L = (K/10) * N, where N

is the total number of observations. If L is a whole number, the K-th decile is the

value midway between the L-th value and the next one.

If L is not a whole number, change it by rounding up to the nearest integer. The value at that position is the K-th decile.

Quartilesvalues that divide the array into 4 equal parts.

Q1 – the value below which 25% of the values fallQ2 – the value below which 50% of the values fallQ3 - the value below which 75% of the values fall

To compute for Lower Quartiles Sort all observations in ascending order Compute the position L1 = 0.25 * N, where N

is the total number of observations. If L1 is a whole number, the lower quartile is

midway between the L1-th value and the next one.

If L1 is not a whole number, change it by rounding up to the nearest integer. The value at that position is the lower quartile.

To compute for Upper Quartile Sort all observations in ascending order Compute the position L3 = 0.75 * N, where N

is the total number of observations. If L3 is a whole number, the upper quartile is

midway between the L3-th value and the next one.

If L3 is not a whole number, change it by rounding up to the nearest integer. The value at that position is the upper quartile.

Example

The surveyed weights (in kilograms) of the students in Stat 231 were the following: 69, 70, 75, 66, 83, 88, 66, 63, 61, 68, 73, 57, 52, 58, and 77. Compute1.) P23 5.) Q32.) P85 6.) Q13.) D304.) D90

Approximating the ith Percentile from the FDT

where =the lower class boundary of the Pith classc = the class size of the Pith classn = the total number of observations in the distribution=the <cf of the class preceding the Pith class

+c