Summary measures

SUMMARY MEASURES

Measures of Central Tendency


Most sets of data has a distinct tendency to group or cluster around a central point.

Thus, for any particular set of data, a single typical value can be used to describe the entire data set. Such a value is referred to as measure of central tendency or location.

Objectives of Averaging

To get single value that describes the characteristic of the entire group.

To facilitate comparison.


Arithmetic Mean Median, Quartiles, Percentiles and Deciles Mode

Arithmetic Mean Ungrouped (Raw) Data

Arithmetic Mean (A.M.) of a set of n values, say,

, is defined as:ni xxxx .....,......, 21

nn

xxx

nsobservatioofNumber

nsobservatioofSumx

n

x ..........

n

1ii21

Example:

The data (next slide) gives value of equity holdings of 10 of the India’s billionaires.

Name Equity Holdings (Millions of Rs.)

Kiran Mazumdar-Shaw 2717

The Nilekani family 2796

The Punj family 3098

K.K. Birla 3534

The Murthi family 4310

Keshub Mahindra 4506

The Kirloskar family 4745

Ajay G. Piramal 4923

S.P. Hinduja 5071

Uday Kotak 5034

Solution

4073.4

10

40734

10

5034.........27962717

x

x

“The sum of deviations of all the observations from A.M. is equal to zero.”

The productivity of employees in banks, as measured by “business per employee” for three banks, for the year 2005 – 2006, are given as follows:

Bank No. of Employees

Business per Employee

Total Business

Bank of Baroda

38737 396 15339852

Bank of India

41808 381 15928848

Corporation Bank

10754 527 5667358

Sum 91299 1304 36936058

Solution:

404.6

91299

36936058

banks threein the Employees ofNumber Total

Banks threein the Business Total employeeper Average

A.M. (Grouped Data)

When the data is grouped, the following type of frequency table is prepared

Class Interval Mid-point of Class interval Frequency

--- ----- ----

--- ----- -----

ix if

k

ii

k

iii

f

xfx

1

1

ixClass Interval

Frequency Mid - Values

2000-3000 2 2500 5000

3000-4000 2 3500 7000

4000-5000 4 4500 18000

5000-6000 2 5500 11000

Sum 10 41000

ixif

if

Data relating to equity holdings of the group of 10 billionaires:

Therefore, mean of the above data is 4100

Combined A.M. of Two Sets of Data

Let there be two sets of data with

Number of observations =

A.Ms. =

Then

21 and nn

21 and xx

21

2211

nn

xnxnx

Example

The average turnover of 200 small and medium enterprises (SMEs) financed by ‘X’ bank in a state is Rs. 50 crores, and the average turnover of 300 SMEs financed by ‘Y’ bank in the same state is Rs. 60 crores. Find the combined mean for the small and Medium enterprises financed by both the banks.

Solution:

56

500

6030050200

500

SMEsofNumber

SMEsallofturnoverTotalFinancedSMEsofMeanCombined

Weighted Arithmetic Mean

Formula

i

iiww

w

wxxor

Weighted Arithmetic Mean

Example: A college may decide that for admission to its XI class, it will attach the following weights to the class X marks obtained in subjects as follows:

Mathematics 3 Science 2 English 1

Solution:

If a student has 60% marks in English, 90% marks in Mathematics, and 80% in Science, his ‘average’ score would be

%7.81

321

601802903

MEDIAN (Ungrouped Data)

The median is the value in the middle when data is arranged in ascending order.

MEDIANArrange the data in ascending order (smallest value to largest value)(a)For an odd number of observations the median is the middle value.(b) For an even number of observations, the median is the average of two middle values.

MEDIANArrange the data in ascending order (smallest value to largest value)(a)For an odd number of observations the median is the middle value.(b) For an even number of observations, the median is the average of two middle values.

Table: 2

Monthly starting salaries for a sample of 12 Business School Graduates.

Graduate Monthly Starting Salary ($)

Graduate Monthly Starting Salary ($)

1 2850 7 2890

2 2950 8 3130

3 3050 9 2940

4 2880 10 3325

5 2755 11 2920

6 2710 12 2880

We first arrange the data in ascending order.

2710; 2755; 2850; 2880; 2880; 2890; 2920; 2940; 2950; 3050; 3130; 3325

Because n = 12 is even, we identify the middle two values: 2890 and 2920. The median is the average of these values.

Median =

Middle two values

2

2905 2920 2890

Median (Grouped Data)

The median for the grouped data can be calculated from the following formula:

f

ipcfNLMedian

- 2/

ixClass Interval

Frequency Cumulative Frequency

2000-3000 2 2

3000-4000 2 4

4000-5000 4 8

5000-6000 2 10

Sum 10

if

Data relating to equity holdings of the group of 10 billionaires:

Solution

f

ipcfNLMedian

- 2/

4250

4

1000454000

52/102/

Median

N

Percentiles:

The pth percentile is a value such that at least p percent of the observations are less than or equal to this value and at least (100-p) percent of the observations are greater than or equal to this value.

Example: The 70th percentile score indicates that 70% of students scored lower than this individual and approx. 30% of the students scored higher than this individual.

Calculating the pth percentile

Step 1: Arrange the data in ascending order.

Step 2: Compute an index i.

Where p is the percentile of the interest and n is the number observations.

Step 3: (a) If I is not an integer, round up. The next integer greater than i denotes the position of the pth percentile.

(b) If i is an integer, the pth percentile is the average of the values in positions I and i+1.

np

i 100

Determine the 85th percentile for the starting salary data:

Step 1: Arrange the data in ascending order.

Step 2:

Step 3: Because i is not an integer, round up. The position of the 85th percentile is the data value in the 11th position.

Data value at 11th position = 3130

100

np

i

2710; 2755; 2850; 2880; 2880; 2890; 2920; 2940; 2950; 3050; 3130; 3325

2.10 12 100

85

Calculation of the 50th percentile for the starting salary data.

Applying step 2:

Because I is an integer, step 3(b) states that the 50th percentile is the average of the sixth and seventh data values; thus the 50th percentile is (2890 + 2920)/ 2 = 2905.

Note: 50th percentile is also the median.

6 12 100

50

i

Quartiles:

It is often desirable to divide the data into four parts, with each part containing approximately one-fourth, or 25% of the observations.

.percentile75or quartile, third Q

median) (also percentile50or quartile, second Q

percentile25or quartile,first Q

th 3

th 2

th 1

25% 25% 25% 25%

1Q 2Q 3Q

Computation of first and third quartiles

Since i is an integer. Therefore,

100

Q 1

np

i

For

3 12

100

25

30002/)30502950(

28652/)28802850(

3

1

Q

Q

2710; 2755; 2850; 2880; 2880; 2890; 2920; 2940; 2950; 3050; 3130; 3325

29052 Q 30003 Q28651 Q

Median

Mode (Ungrouped Data):

The mode is the most frequently occurring value in a set of data.

Example: The annual salaries of quality-control managers in selected states are shown below. What is the modal annual salary?

State Salary State Salary

Arizona $35,000 Massachusetts $40,000

California 49,100 New Jersy 65,000

Colorado 60,000 Ohio 50,000

Florida 60,000 Tennessee 60,000

Idaho 40,000 Texas 71,400

Lllinois 58,000 West Virginia 60,000

Louisiana 60,000 Wyoming 55,000

Maryland 60,000

A persual of the salaries reveals that the annual salary of $60,000 appears more often (six times) Than any other salary. The mode is, therefore, $60,000.

Mode (Grouped Data):

Class Interval

Frequency

2000-3000 2

3000-4000 2

4000-5000 4

5000-6000 2

Sum 10

Solution:

For grouped data mode can be calculated as:

21

1

iLM o

4500

4

1000 24000

Relationship between mean, median and mode

In a symmetrical distribution, the values of mean, median, and mode are equal.

In other words, when all these three values are not equal to each other, the distribution is not symmetrical.

Mean=median=mode(a) Symmetrical

Mode Median Mean(b) Skewed to the Right

Mean Median Mode(c) Skewed to the Left

A distribution that is not symmetrical, but rather has most of its values either to the right or to the left of the mode, is said to be skewed.

Mean – Mode = 3(Mean - Median)

Or Mode = 3 Median – 2 Mean

In case of right or positively skewed distribution. The order of magnitude of these measures will be

Mean > Median > Mode

Left or negatively skewed

Mean < Median < Mode

Five-Number Summary

Smallest value First quartile Median Third quartile Largest value

Technology

Summary measures