MEASURES OF CENTRAL

TENDENCY

UNIT 3

1

OBJECTIVES :By the end of this chapter, you should be able to

: Define the Central Tendency measurementCalculate the measure of central tendency for

ungrouped data using the mean, median, mode quartiles, percentiles and deciles.

Calculate the measure of central tendency for grouped data using the mean, median, mode quartiles, percentiles and deciles.

Determine the empirical relationship between mean, median and mode

2

IntroductionThe measure of central tendency is usually called the average. Central tendency is a single value situated at the center of a data and can be taken as a summary value for that data set.

Three types of averages are often used as measures of central tendency. They are mean, median and mode.

3

A) Mean It is the average of a group of data.That is computed by taking the sum of all data values and then dividing it by the number of data.

For example, if the statistics test scores of five students are 70,80,60,90, and 50, then the mean score of the statistics test for five students is ( 70 + 80 + 60 + 90 + 50 )/ 5 = 70.

 

Introduction

4

B) MedianIs value that list in the center of the data.To determine the median, one has to

arrange the data in ascending or descending order and then select the central value of the data as the median.

Using the data supplied earlier and arranging the data in ascending order we obtain 50, 60, 70, 80, 90. The center value is 70 and we conclude that the median is 70.

Introduction

5

C) Mode Value of data with the highest frequency in a data set.

For example : If the data set is 50, 60, 70, 70, 70, 80, 90, 90, then the mode is 70 because it occurs most Frequently (three times) in the data set.

Introduction

6

Example 1 : The data below shows the monthly wages in RM of

ten factory workers. Calculate the value of mean500, 450, 400, 380,360,420,470, 390, 460, 500

7

Mean (for ungrouped data)

n data, ofNumber x , data of valuesall of SumxMean,

nx xMean,

solution :

104330

= 433

8

Median (for ungrouped data)

21Median

n

Example 1 : The data below shows the monthly wages in

RM of ten factory workers. Calculate the value of Median500, 450, 400, 380,360,420,470, 390, 460,

500 solutio

n : 360, 380, 390, 400, 420, 450, 460,470, 500, 500

median) of 435(value2

8702

450420Median

(Location) 5.52

11

21nMedian

9

Mode (for ungrouped data)

Mode : Value of data with the highest frequency in a data set.

Example 1 : The data below shows the monthly wages in

RM of ten factory workers. Calculate the value of Mode500, 450, 400, 380,360,420,470, 390, 460,

500 solutio

n : 360, 380, 390, 400, 420, 450, 460,470, 500, 500The value of mode is 500

Mean (for grouped data)3 method to find the value of mean

i)Basic Meanii)Estimate Meaniii)Coding meanThese 3 methods will give same answer even though they are different from each other

10

1) BASIC MEAN

11

frequency of Totalmidpoint) x frequency ( of SumxMean

,

ffx

xMean,

Class interval f0-4 25-9 4

10-14 615-19 820-24 425-29 230-34 4

   

2) ESTIMATE MEAN

12

ffd

AxMean,

Where :A = Estimate (any value of midpoint)f = Frequencyd = x – A ( differentiate between midpoint and estimate mean, A)

Class interval

f

0-4 25-9 4

10-14 615-19 820-24 425-29 230-34 4

     

3) CODING MEAN

13

C x ffu

AxMean,

Where :A = Estimate (any value of midpoint), C = class intervalf = Frequencyu =

C

Ax

MEDIANStep 1 :Add a column for the cumulative frequency and Location of data in frequency distribution table.Step 2 : Determine the position of median in the class intervals using the Median Class

Step 3 : The value of median is calculate as follows

14

2

f Class Median

Cx fm

fm2

f

Lm ~x Median,

Where :Lm = Lower boundary of the median class

= Sum of frequency = Cumulative frequency before the median classfm = Frequency of the median classC = Size of the Class Interval

f

f m

15

Years of experien

ce

Number of employee,

f1-45-8

9-1213-1617-2021-2425-28

 

Example 21 : (Page 38) The table below shows the years of working experience for 120 employee of Jimmy’s Company. Calculate the value of median using the formula.