This document discusses various measures of central tendency used in statistics. It defines central tendency as a typical or average value for a probability distribution. The three most common measures are the mean, median, and mode. The mean is the average value calculated by adding all values and dividing by the total number. The median is the middle value when values are arranged in order. The mode is the most frequently occurring value. Other measures discussed include the geometric mean, harmonic mean, weighted mean, and truncated mean. Factors like the range, type of variable, and data distribution impact which measure is most appropriate.

1. GAYATRI R

2. MEASURES OF CENTRAL TENDENCY
O In statistics, a central tendency is a central or typical
value for a probability distribution. It may also be called a
center or location of the distribution. Colloquially,
measures of central tendency are often called averages.
The term central tendency dates from the late 1920s.
O Simpson and Kafka defined it as “ A measure of central
tendency is a typical value around which other figures
congregate”

3. FACTORS AFFECTING THE MEASURES OF
CENTRAL TENDENCY
ORange
OVariable
ODistribution of Data

4. RANGE
O Range is the difference between the highest &
lowest value in the distribution
O Depending upon the range / interval the usage of
the measure of central tendency differs

5. VARIABLE
o The changing quantity ie, the trait, factor or condition that can
exist in different amount or types is called variable. Mainly it is
classified into two types. They are
 Qualitative Variable
 Quantitative Variable
o Qualitative Variable –Qualitative/Categorical Variable which
cannot be used in Mathematical Operation
o Quantitative Variable – Quantitative Variable are numerical
values used in Mathematical Operation. They are of two types
namely
 Discrete Variable
 Continuous Variable

6. DISTRIBUTION OF DATA
O DISCRETE PROBABILITITY DISTRIBUTION:
If a random variable is a discrete variable, its probability
distribution is called a discrete probability distribution
An example will make this clear. Suppose you flip a coin two
times. This simple statistical experiment can have four possible
outcomes: HH, HT, TH, and TT.
Number of heads Probability
0 0.25
1 0.50
2 0.25

7. CONTD..
CONTINUOUS PROBABILITY DISTRIBUTION:
If a random variable is a continuous variable, its probability
distribution is called a continuous probability distribution. A
continuous probability distribution differs from a discrete
probability distribution in several ways like :
 The probability that a continuous random variable will
assume a particular value is zero.
 As a result, a continuous probability distribution cannot be
expressed in tabular form.
 Instead, an equation or formula is used to describe a
continuous probability distribution.

8. MEASURES OF CENTRAL TENDENCY
O Arithmetic mean
O Median
O Mode
O Geometric mean
O Harmonic mean
O Weighted mean
O Truncated mean
O Interquartile mean
O Midrange
O Mid hinge
O Trimean
O Winsorized mean

9. MEAN
 The mean (arithmetic mean or average) of a set of data is found by
adding up all the items and then dividing by the sum of the number
of items.
 The mean of a sample is denoted by (read “x bar”).
 The mean of a complete population is denoted by (the lower
case Greek letter mu).
 The mean of n data items x1, x2,…, xn, is given by the formula
or

10. Example:
Ten students were polled as to the number of siblings in their
individual families.
The raw data is the following set: {3, 2, 2, 1, 3, 6, 3, 3, 4, 2}.
Find the mean number of AGE for the ten students.

11. WEIGHTED MEAN
The weighted mean of n numbers x1, x2,…, xn, that are weighted by
the respective factors f1, f2,…, fn is given by the formula:
 
.
x f
w
f





12. Example:
Listed below are the grades of a students semester courses. Calculate
the Mean price.
Course Price (x) Quantity(f) x * f
Dark Chocolate 4 5 20
Milk Chocolate 3 3 9
Toffee 4 2 8
Candy 2 2 4

13. MEDIAN
• Another measure of central tendency, is the median.
• This measure divides a group of numbers into two parts,
with half the numbers below the median and half above it.
To find the median of a group of items:
1. Rank the items.
2. If the number of items is odd, the median is the middle item in the
list.
3. If the number of items is even, the median is the mean of the two
middle numbers.

14. Example:
Ten students in a math class were polled as to the number of
siblings in their individual families and the results were:
3, 2, 2, 1, 1, 6, 3, 3, 4, 2.
Find the median in number of siblings for the ten students.
Position of the median: 10/2 = 5
Between the 5th and 6th values
Data in order: 1, 1, 2, 2, 2, 3, 3, 3, 4, 6
Median = (2+3)/2= 2.5 siblings

15. Example
Nine students in a math class were polled as to the number of siblings
in their individual families and the results were:
3, 2, 2, 1, 6, 3, 3, 4, 2.
Find the median number of siblings for the ten students.
Position of the median: 9/2=4.5th value – 5th Value
In order: 1, 2, 2, 2, 3, 3, 3, 4, 6
Median = 3 siblings

16. Example
Median in a Frequency Distribution
Find the median for the distribution.
Position of the median is the sum of the frequencies divided by 2.
Position of the median =  (f)/2= 23/2=11.5th = 12th Term
Add the frequencies from either side until the sum is 12.
The 12th term is the median and its value is 4.
Value (x) 1 2 3 4 5
Frequency
(f)
4 3 2 6 8

17. MODE
 The mode of a data set is the value that occurs the most often
 If a distribution has two modes, then it is called bimodal.
 In a large distribution, this term is commonly applied even when
the two modes do not have exactly the same frequency
Example:
Ten students in a math class were polled as to the number of siblings
in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4,
2. Find the mode for the number of siblings
3, 2, 2, 1, 3, 6, 3, 3, 4, 2
The mode for the number of siblings is 3

18. Example
Mode in a Frequency Distribution
Find the mode for the distribution.
The mode in a frequency distribution is the value that has the largest
frequency.
The mode for this frequency distribution is 5 as it occurs eight times.
Value (x) 1 2 3 4 5
Frequency
(f)
4 3 2 6 8

19. GEOMETRIC MEAN
Geometric mean is the nth root of the product of the data
values, where there are n of these. This measure is valid only
for data that are measured absolutely on a strictly positive
scale.
GM = n√(a1 × a2 × ... × an)
Example: What is the Geometric Mean of 10, 51.2 and 8?
First we multiply them: 10 × 51.2 × 8 = 4096
Then (as there are three numbers) take the cube root:
3√4096 = 16

20. HARMONIC MEAN
Harmonic mean is the reciprocal of the arithmetic mean of the
reciprocals of the data values. This measure too is valid only for data
that are measured absolutely on a strictly positive scale.
Harmonic Mean = N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN)
Example:
To find the Harmonic Mean of 1,2,3,4,5.
Step 1:
Calculate the total number of values. N = 5
Step 2:
Now find Harmonic Mean using the above formula.
N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN) = 5/(1/1+1/2+1/3+1/4+1/5) =
5/(1+0.5+0.33+0.25+0.2) = 5/2.28
So, Harmonic Mean = 2.19

21. CONCLUSION
These are the Measures of Central tendency that are mostly
used in Social Science Research for data analysis