This document provides information about various measures of central tendency including the mean, median, and mode. It defines each measure and provides examples of how to calculate them using both raw data and grouped data. The mean is the average value and can be influenced by outliers. The median is the middle value and is not affected by outliers. The mode is the most frequent value. Each measure is suited for certain types of data distributions and the document discusses when each should be used.

1. 21COM2T412: Business Statistics
Dr. Neerupa Chauhan
Asst. Professor
Kristu Jayanti College, Autonomous
(Reaccredited A++ Grade by NAAC with CGPA 3.78/4)
Bengaluru – 560077, India

2. Summary of Measures
Central Tendency
Mean
Median
Mode
Quartile
Summary Measures
Variation
Variance
Standard Deviation
Coefficient of
Variation
Range

3. Measures of Central Tendency
• A measure of central tendency is a
descriptive statistic that describes the
average, or typical value of a set of
scores.
• There are three common measures of
central tendency:
– the mean
– the median
– the mode

4. The mean is:
the arithmetic average of all the scores
(X)/N
The mean of a population is
represented by the Greek letter ; the
mean of a sample is represented by X
The Mean

5. Calculating the Mean for
Grouped Data
N
X
f
X


where: f X = a score multiplied by its frequency
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5 Mean = 6
Mean affected by extreme values

6. You should use the mean when
the data are interval or ratio scaled
Many people will use the mean with ordinally
scaled data too
and the data are not skewed
The mean is preferred because it is
sensitive to every score
If you change one score in the data set, the
mean will change
When To Use the Mean

7. Find the mean of the following data:
Mean =
[3(10)+10(9)+9(8)+8(7)+10(6)+
2(5)]/42 = 7.57
Score Number of
students
10 3
9 10
8 9
7 8
6 10
5 2
Calculating the Mean for Grouped Data

8. The Median
• The median is simply another name for
the 50th percentile
– It is the score in the middle; half of the
scores are larger than the median and half
of the scores are smaller than the median
– Not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5

9. How To Calculate the Median
• Conceptually, it is easy to calculate the
median
– There are many minor problems that can
occur; it is best to let a computer do it
• Sort the data from highest to lowest
• Find the score in the middle
– middle = (n + 1) / 2
– If n, the number of scores, is even the median
is the average of the middle two scores

10. Calculating the Median for
Grouped Data
h
f
cf
N
l
Median 



2
/
• To use this formula first determine median class.
Median class is that class whose less than type cumulative
frequency is just more than N / 2 ;
• l = lower limit of median class ;
• cf = less than type cumulative frequency of premedian
class;
• f = frequency of median class
• h = class width.

11. When To Use the Median
The median is often used when the
distribution of scores is either positively
or negatively skewed
The few really large scores (positively
skewed) or really small scores (negatively
skewed) will not overly influence the
median

12. Median Example
• What is the median of the following
scores:
10 8 14 15 7 3 3 8 12 10 9
• Sort the scores:
15 14 12 10 10 9 8 8 7 3 3
• Determine the middle score:
middle = (n + 1) / 2 = (11 + 1) / 2 = 6
• Middle score = median = 9

13. Median Example
• What is the median of the following
scores:
24 18 19 42 16 12
• Sort the scores:
42 24 19 18 16 12
• Determine the middle score:
middle = (n + 1) / 2 = (6 + 1) / 2 = 3.5
• Median = average of 3rd and 4th scores:
(19 + 18) / 2 = 18.5

14. The Mode
The mode is the score that occurs most
frequently in a set of data
Not Affected by Extreme Values
There May Not be a Mode
There May be Several Modes
Used for Either Numerical or Categorical Data
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode

15. Calculating the Mode for
Grouped Data
h
f
f
f
f
f
l
Mode
m
m











2
1
1
2
To use this formula first determine modal class.
Modal class is that class which has maximum
frequency ;
l = lower limit of modal class;
fm = maximum frequency;
f1 = frequency of pre modal class ;
f2 = frequency of post modal class
h = class width.

16. When To Use the Mode
The mode is not a very useful measure
of central tendency
It is insensitive to large changes in the data
set
That is, two data sets that are very different
from each other can have the same mode
The mode is primarily used with nominally
scaled data
It is the only measure of central tendency that is
appropriate for nominally scaled data

17. Wages ( C.I.) 40-60 60-80 80-100 100-120 120-140 140-160
No.of workers
(freq)
50 80 30 20 50 20
X 1 2 3 4 5 Total
No. of Families
(freq)
20 50 20 5 5 100
Problem 1 : Wages (in Rs) paid to workers of an organization are given
below. Calculate Mean, Median and Mode.
Problem 2 : Weekly demand for marine fish (in kg) (x) for 100 families is
given below. Calculate Mean, Median and Mode.
Calculate Mean, Median &
Mode

18. Thank you