Measures of Central Tendency

Measures of central tendency

• A measure of central tendency is a typical value around which other figures concentrate.

• An average is a single value within the range of the data that is used to represent all the values in the series.

Objectives of an average

• It reduces the mass information into a single value.

• Average provides facility to make comparisons.

• Average is very useful in statistical analysis.• Average is also used in sampling. We can get a

clear picture of a group or population by the help of a sample.

Types of Averages

• Arithmetic Mean• Median• Mode

Arithmetic Mean

• It is defined as the sum of the observations divided by the number of observations.

• It is denoted by x

1. Individual Series

(i) Direct Method

where, x denotes the value of the

variable n denotes the number of

observations.

(ii) Short Cut Method

where, A is assumed mean n denotes number of

terms.

xn

x1 Axdd

nAx ;

1

Ques 1. The number of new orders received by a company over the last 25 working days were recorded as follows:

3, 0, 1, 4, 4, 4, 2, 5, 3, 6, 4, 5, 1, 4, 3, 2, 0, 2, 0, 5, 4, 2, 3, 3, 1.

Calculate the average number of orders received over all similar working days.

Ques 2. The following data gives the equity holdings of 10 of the India’s billionaires.

Calculate the average equity holding.

Name Equity Holdings(millions of Rs)

Kiran Mazumdar-Shaw 2717

The Nilekani family 2796

The punj family 3098

Karsanbhai K. Patel 3144

Shashi Ruia 3527

K.K.Birla 3534

B.Ramalinga Raju 3862

Habil F. Khorakiwala 4187

The Murthy family 4310

Keshub Mahindra 4506

2. Discrete Series(i) Direct Method

where, x denotes the values of

the variable f denotes the frequency N = ∑f ; denotes the no.

of observations.

(ii) Short Cut Method

where, d = x - A A is assumed meanN = ∑f ; denotes the no. of

terms

xfN

x1

dfN

Ax1

Ques 1. The HR manager at a city hospital began a study of the overtime hours of the registered nurses. The following data was recorded during a month.

Calculate the arithmetic mean of overtime hours during the month.

Overtime hours

No. of nurses

5 3

6 4

7 2

9 4

10 2

12 5

13 3

15 2

Ques 2. The following are the figures of profits earned by 1,400 companies during 2008-2009.

Calculate the average profit for all the companies.

Profits(Rs. Lakhs)

No. of companies

300 500

500 300

700 280

900 120

1100 100

1300 80

1500 20

3. Continuous Series(i) Direct Method:

where, x denotes the mid point of various class interval f denotes the frequency of each class N = ∑f ; denotes the total frequency

xfN

x1

(ii) Short Cut Method:

where, d = x – A x denotes the mid point of various class interval A is assumed mean f denotes the frequency of each class N = ∑f ; denotes the total frequency

dfN

Ax1

(iii) Step Deviation Method:

where, u = (x - A)/h h denotes the width of class interval A is assumed mean N = ∑f ; denotes the no. of terms

hufN

Ax1

Ques 1. Calculate mean for the following frequency table: Weekly rent No. of persons

paying rent

200-400 6

400-600 9

600-800 11

800-1000 14

1000-1200 20

1200-1400 15

1400-1600 10

1600-1800 8

1800-2000 7

Ques 2. In an examination of 675 candidates the examiner supplied the following information:

Marks obtained

No. of candidates

0 – 10 7

10 – 20 32

20 – 30 56

30 – 40 106

40 – 50 175

50 – 60 164

60 – 70 86

70 – 80 44

Ques. Six types of workers are employed in each of two workshops but at different rates of wages as follows:

In which of the two workshops is the average rate of wages per worker higher and by how much?

Workshop ADaily wagesper worker

No. of workers

Workshop BDaily wages

No. of workersper worker

92.50 2 93.00 11

93.50 14 93.25 50

94.00 20 93.50 8

93.00 7 94.25 10

94.25 6 94.50 10

94.50 1 95.00 2

Ques. Calculate mean from the following data pertaining to the profits(in crore Rs.) of 125 companies: Profits(Rs. crore)(less than)

No. of companies

10 4

20 16

30 40

40 76

50 96

60 112

70 120

80 125

Merits & Demerits• Merits

– All values are used– It has unique value & easy to calculate – The sum of the deviations from the mean is 0

• Demerits– The mean is affected by extreme values

E.g., average salary at a company 12,000; 12,000; 12,000; 12,000; 12,000; 12,000;

12,000; 12,000; 12,000; 12,000; 20,000; 390,000 Mean = $44,167– It is not suitable for open end classes

MEDIAN

The median is the measure of central tendency which appears in the middle of an ordered sequence of values. That is, half of the observations in a set of data are lower than it and half of the observations are greater than it.

1. Individual Series

• Arrange the data in increasing or decreasing order.

• Median is given by:

Median= termthn

2

)1(

Questions:1.From the following data of wages of 7 workers, compute the median wage: 2600, 2650, 2580, 2690, 2660, 2606, 2640

2. Discrete Series

• Calculate the cumulative frequency.• Find (N+1)/2th term.• Select the cumulative frequency in which that

value lies.• The value of the variable corresponding to

that cumulative frequency is the median.

2. Calculate the median for following data:

X f

45 10

55 20

65 20

75 15

85 15

95 20

3. Continuous Series• Calculate the cumulative frequency.• The value corresponding to N/2th term gives the median class.• Then median is calculated by the formula:

ervalclassofwidthh

classmedianthe

preceedingclasstheoffrequencycumulativecf

classmediantheoffrequencyf

classmediantheofitlowerl

where

hcfN

flMedian

int

lim

,

2

1

3. Calculate the median for following data:

Class interval Frequency

10 - 15 11

15 – 20 20

20 – 25 35

25 – 30 20

30 – 35 8

35 - 40 6

4. Calculate the median from the following data pertaining to the profits (in crore Rs.) of 125 companies:

Profits (less than) No. of companies

10 4

20 16

30 40

40 76

50 96

60 112

70 120

80 125

• Merits – Median is unique– Median is less affected by extreme values as compared to

mean– It can be used for open–end distribution– Graphical presentation of median is possible– Median is used for studying qualitative attributes

• Demerits – For median, it is necessary to arrange the data– It is not capable for further algebraic treatment– It does not use each and every observation of the data set

Mode• The most frequent score in the distribution.

• A distribution where a single score is most frequent has one mode and is called unimodal.

• A distribution that consists of only one of each score has n modes.

• When there are ties for the most frequent score, the distribution is bimodal if two scores tie or multimodal if more than two scores tie.

Mode (con’t)

2

2

3

3

4

4

4

4

4

10

Number of Words Recalled in Performance Study

The mode is 4.

Mode (con’t)

72 72 73 76 78

81 83 85 85 86

87 88 90 91 92

This distribution is bimodal.

1. Individual Series: In Individual series we guess mode by inspection. We observe that term in the

series which occurs maximum number of times. This term is called mode.

2. Discrete Series: In discrete series mode is that value of variable whose frequency is maximum.

3. Continuous Series:

The class interval that is corresponding to the maximum frequency is the modal class. Then mode is calculated by the formula:

classalofwidthh

classalthesuceedingclassoffrequencyf

classalpreceedingclassoffrequencyf

classaloffrequencyf

classalofitlowerl

where

hfff

fflz

mod

mod

mod

mod

modlim

,

2

2

0

1

201

01

Ques 1. Find the mode of following data:

Class Frequency

0 – 5 29

5 – 10 195

10 – 15 241

15 – 20 117

20 – 25 52

25 – 30 10

30 – 35 6

35 – 40 3

40 – 45 2

Ques 2. Find the mode of following data:

Class Frequency

0 – 10 5

10 – 20 15

20 – 30 20

30 – 40 20

40 – 50 32

50 – 60 14

60 – 70 14

70 – 80 5

Merits of Mode:

• It is easy to understand. Sometimes it is found only by inspection.• It is least affected by extreme values.• It can also be located graphically.• It is also useful when items related to fashion are considered.

Demerits of Mode:

• It is not based on all the observations.• Equal intervals are needed for the calculation of mode.

Empirical relation between mean, median and mode

The relationship between arithmetic mean, median and mode is given by following formula:

Mode = 3 Median – 2 Mean

Ques 1. Find out an estimate of mode, when mean and median are 12 and 16.5

Ques 2. If mean = 26, mode = 28.5. Find an estimate of median.