This document discusses measures of central tendency including mean, median, and mode. It provides definitions and formulas for calculating each measure from both grouped and ungrouped data. For mean, it is the sum of all values divided by the number of values. Median is the middle value when values are arranged in order. Mode is the most frequently occurring value. The document also discusses requirements for a good measure of central tendency and provides examples calculating the measures from sample medical data sets to illustrate their use and assess skewness.

1. Jagdish Powar
Statistician cum tutor
Communicate Medicine
SMBT, IMSRC, Nashik.

2. Competency & Learning objectives
Competency Learning Objectives
CM6.4
Enumerate, discuss and demonstrate
Common sampling techniques, simple
statistical methods, frequency
distribution, measures of central
tendency and dispersion
The student should be able to
 Define measures of central
tendency-Mean, Mode, Median
 Calculate the different measures
of central tendency from given
dataset,
 Comment on skewness of given
dataset
2PSM, SMBT, Nashik

3. What is measure of central tendency?
• A measure of central tendency is a summary
statistic that represents the center point or value
of a dataset. These measures simply summarize a
set of data in a single value. These are
representative of data and around which large
portion of observations gathered.
• In statistics, the three most common measures of
central tendency are the mean, median and
mode.
3PSM, SMBT, Nashik

4. Requirement of a good measure of central
tendency
1. It should be rigidly defined.
2. It should be based on all observations (values).
3. It should be easy to calculate and understand.
4. It should have sampling stability i.e. it should not
be affected by sampling fluctuations.
5. It should be capable for the further mathematical
calculations.
6. It should not be affected by extreme values.
4PSM, SMBT, Nashik

5. Mean is very simple and commonly used measure of central
tendency. Mean is sum of all observations divided by
number of observations.
For ungrouped data:-
Mean ( 𝑥) =
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
𝑛𝑜.𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
=
𝛴𝑥
𝑛
For grouped data:-
Mean ( 𝑥) =
𝛴𝑓∗𝑥
𝑁
N = 𝛴𝑓 x= midpoints of class interval
f= frequency
5PSM, SMBT, Nashik

6. Central values when observations are arranged in ascending or descending
order.
• For ungrouped data:-
Arrange the observations in ascending or descending order
n = Odd value Median=
𝑛+1
2
th value
n = Even value Median =
𝑛
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+
𝑛
2
+1 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
• For grouped data:-
Median = L +
(
𝑁
2
−𝐶𝐹)
𝑓
∗ 𝑖
L- lower class limit of median class
CF- lcf of pre-median class
f- Frequency of median class
i- width of median class
The class with LCF is just greater than N/2 is known median class
6PSM, SMBT, Nashik

7. MODE
The observation which occurs most frequently in series is known as
mode.
 For ungrouped data
Mode= frequently occurred observation.
 For grouped data
Mode = L +
(𝑓𝑚−𝑓1)
(2∗𝑓𝑚−𝑓1−𝑓2)
∗ 𝑖
L- lower class limit of modal class
fm- frequency of modal class
f1- frequency of pre-modal class
f2- frequency of post-modal class
i - Width of modal class.
The class with maximum frequency is model class
7PSM, SMBT, Nashik

8. SKEWNESS
Positive Skewed (Right tailed): – Mean > Median > Mode
Negative Skewed (Left tailed): – Mean < Median < Mode
Symmetric (Normal): – Mean = Median = Mode
Relation between mean, median and mode
Mean – Mode = 3(Mean – Median)
Mode= 3*Median - 2* Mode
8PSM, SMBT, Nashik

9. 1)The following number gives the incubation period of 10
consecutive patients of hepatitis. Calculate mean, mode,
median and comment on its skewness
24, 22, 35, 15, 26, 19, 24, 18, 22, 25
Sol-
Mean:-
Mean ( 𝑥) =
𝛴𝑥
𝑛
=
24+22+⋯……………..+22+25
10
=
230
10
=23
Median:-
Ascending order
15 18 19 22 22 24 24 25 26 35
n = 10, Even value
Median =
𝑛
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+
𝑛
2
+1 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
=
5 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+6 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
=
22+24
2
= 23
9PSM, SMBT, Nashik

10. Mode
Mode= frequently occurred observation
The observations 22 & 24 occurred most frequently in
given data set
Mode=22, 24 (Bi-model distribution)
Skewness:-
Here Mean = Median= 23
Symmetric or Normal distribution
10PSM, SMBT, Nashik

11. 2) Following table gives the age distribution of patients from RHTC, SMBT-
Ekdara
Calculate Mean, median, mode and comment on skewness.
Sol-
Mean:-
Mean ( 𝑥) =
𝛴𝑓∗𝑥
𝑁
N = 𝛴𝑓, x= midpoints of class interval, f= frequency
Mean ( 𝑥) =
𝛴𝑓∗𝑥
𝑁
=
6400
160
=40
Mean age= 40 years
Age (Years) 10-20 20-30 30-40 40-50 50-60 60-70
No. of patients 15 25 30 55 25 10
Class
Interval
f
x
(midpoint)
f*x
10.-20 15 15 225
20-30 25 25 625
30-40 30 35 1050
40-50 55 45 2475
50-60 25 55 1375
60-70 10 65 650
160 6400
11PSM, SMBT, Nashik

12. Median:-
CI f LCF
10.-20 15 15
20-30 25 40
30-40 30 70
40-50 55 125
50-60 25 150
60-70 10 160
160
Median class
N/2=160/2 =80
LCF=125 > N/2(80) hence median class is 40-50
Median = L +
(
𝑁
2
−𝐶𝐹)
𝑓
∗ 𝑖
L- lower class limit of median class=40
CF- lcf of pre-median class=70
f- Frequency of median class =55
i- width of median class=10
Median = 40 +
(80−70)
55
∗ 10
=40+1.81=41.81
Median age= 41.81 years
12PSM, SMBT, Nashik

13. Mode
CI f
10.-20 15
20-30 25
30-40 30
40-50 55
50-60 25
60-70 10
The class interval 40-50 is with maximum
frequency hence model class is 40-50.
Mode = L +
(𝑓𝑚−𝑓1)
(2∗𝑓𝑚−𝑓1−𝑓2)
∗ 𝑖
L- lower class limit of modal class=40
fm- frequency of modal class=55
f1- frequency of pre-modal class=30
f2- frequency of post-modal class=25
i - Width of modal class=10
Mode =40 +
(55−30)
(2∗55−30−25)
∗ 10
=40+4.55=44.55
Mode age=44.55 years
Mean (40)< Median(41.81) < Mode(44.55)
Negative skewed distribution.
13PSM, SMBT, Nashik

14. PSM, SMBT, Nashik 14