This document discusses measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure. The mean is the average value and is calculated by summing all values and dividing by the total number of values. The median is the middle value when data is arranged in order. The mode is the most frequently occurring value in a data set. An example calculating the mean, median, and mode for a data set of lemon weights is provided and solved step-by-step.

1. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
skhot1976@gmail.com
Paper XIV
Unit. 2. Biostatistics
2.5 Measures of central tendency
Mean, Median, Mode
(for Ungrouped data)
Dr. Sudhakar Sambhaji Khot
M.Sc., Ph.D., SET
Assistant Professor in Botany
Y. C. Warana Mahavidyalaya, Warananagar

2. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT 2
B.Sc. Part- III Botany
Paper- XIV DSE –F26
Bioinformatics, Biostatistics and Economic Botany
Unit 2: Biostatistics (11)
2.1 Introduction, definition, terminology.
2.2 Collection and presentation of data:
Types of data, techniques of data collection- Census method,
sampling method- simple random, stratified and systematic sampling.
Classification, tabulation, graphical representation- Histogram and polygon.
2.3 Measures of central tendency and Dispersion:
Arithmetic mean, Mode, Median,
Range, Deviation, Mean deviation, Standard Deviation, Coefficient of Variation.
2.4 Statistical methods for testing the hypothesis’
i) Students’ T-test
ii) Chi-square test

3. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT
2.3 Measures of central tendency and Dispersion :
Arithmetic Mean, Median, Mode
‘Central Tendency: a single value that express and represent the entire set of data.
Lays in between the range of lowest and highest value of a data.
Common measures of central tendency are mean, median and mode.
Merits:
 Well defined
 Easy to calculate
 All items are considered
 Used for other statistical calculations
Arithmetic mean ( μ or x̄ ): Average
Demerits:
 Affected by extreme high/ low value
of item
 May give absurd values
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑴𝒆𝒂𝒏
μ =
𝒙
𝑵
=
𝒙𝟏
+𝒙𝟐
+𝒙𝟑
+⋯𝒙𝑵
𝑵
For ungrouped data
𝑺𝒂𝒎𝒑𝒍𝒆 𝑴𝒆𝒂𝒏
x̄ =
𝒙
𝒏
=
𝒙𝟏
+𝒙𝟐
+𝒙𝟑
+⋯𝒙𝒏
𝒏

4. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT
2.3 Measures of central tendency and Dispersion :
Arithmetic Mean, Median, Mode
Merits:
 Well defined
 Easy to calculate
 All items are considered
 Used for other statistical calculations
Arithmetic mean ( μ or x̄ ): Average
Demerits:
 Affected by extreme high/ low value
of item
 May give absurd values
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑴𝒆𝒂𝒏
μ =
𝒙
𝑵
=
𝒙𝟏
+𝒙𝟐
+𝒙𝟑
+⋯𝒙𝑵
𝑵
For ungrouped data
𝑺𝒂𝒎𝒑𝒍𝒆 𝑴𝒆𝒂𝒏
x̄ =
𝒙
𝒏
=
𝒙𝟏
+𝒙𝟐
+𝒙𝟑
+⋯𝒙𝒏
𝒏
Example: Educational Qualifications in Resume
Qualification Board/ University Yr of
Passing
% of marks
obtained
S.S.C. Maharashtra State Board 1991 62.85 %
B.Sc. Shivaji Univ., Kolhapur 1997 65.12 %
M.Sc. Shivaji Univ., Kolhapur 1999 67.17 %
D.Y.Ed. Bhavnagar Univ., Bhavnagar (Guj) 2002 55.86 %
Ph.D. Bhavnagar Univ., Bhavnagar (Guj) 2003
SET SET - MAH State 2016 66.86 %

5. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT
2.3 Measures of central tendency and Dispersion :
Arithmetic mean, Median, Mode
Merits:
 Simple to calculate
 Can be calculated without knowing
the values of other items
 Not affected by extreme values
 Can be calculated graphically
Median (Md): middle value of a data arranged in ascending / descending order of magnitude
It is positional average.
Divides data in two equal parts
Demerits:
 Not based on all observations
 Not used for further statistical
calculations
For ungrouped data:
𝑴𝒆𝒅𝒊𝒂𝒏 𝑴𝑫 =
𝒏 + 𝟏
𝟐
𝒕𝒉
𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏
Example: Bargaining for the Price value
Rs. 1200/-
Vendor
I’ll pay 1000/- only
Customer
Lets finalize at 1100/-
Vendor Ok, Deal
Customer
How much for this?
Customer

6. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT
2.3 Measures of central tendency and Dispersion :
Arithmetic mean, Mode, Median
Merits:
 Easy to found out
 No need of calculations
 Not affected by extreme values
 Can be calculated graphically
Mode (Mo): value of variable which occurs most frequently in a distribution
It is positional average.
Data may be unimodal, bimodal, polymodal or amodal
Demerits:
 Not clearly defined
 Not based on all observations
 Not reliable
 Not used for further statistical calculations
For ungrouped data: Mo= most frequent value
Example: Dish for dinner
10 friends went for dinner……
Each one had different preference….
Decided to have same dish for everyone
Asked the ‘waiter’ for special dish of the restaurant…
He named four menus said ‘Dish A, B, C and D’…
All friends Voted ….
one selected ‘Dish A’
Six selected ‘Dish B’
Three selected ‘Dish C’
So, they ordered ‘Dish B’.

7. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT 7
Example 1: The weight of 11 lemons in gms is as below. Calculate the
mean, median and mode.
S.No. 1 2 3 4 5 6 7 8 9 10 11
Weight 56 60 48 65 52 58 65 65 70 68 53
S.No. Ascending order of weight
1 48
2 52
3 53
4 56
5 58
6 60
7 65
8 65
9 65
10 68
11 70
Total: 660
𝑴𝒆𝒂𝒏 =
𝟔𝟔𝟎
𝟏𝟏
Mean = 60 gms
Solution: steps
1. Rearrange data in
ascending
/descending
2. Calculate central
tendency as per
formula
𝑴𝒆𝒂𝒏 x̄ =
𝒙
𝒏
=
𝒙𝟏
+𝒙𝟐
+𝒙𝟑
+⋯𝒙𝒏
𝒏

8. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT 8
Solution:
𝑴𝒆𝒅𝒊𝒂𝒏 (𝑴𝑫) =
𝒏 + 𝟏
𝟐
𝒕𝒉
𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏
Here, n=11
Therefore,
𝑴𝒆𝒅𝒊𝒂𝒏 =
𝟏𝟏 + 𝟏
𝟐
𝒕𝒉
𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏
𝑴𝒆𝒅𝒊𝒂𝒏 =
𝟏𝟐
𝟐
𝒕𝒉
𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏
Median=6th Observation in rearranged data = 60
gms
∴ 𝑴𝒆𝒅𝒊𝒂𝒏 (𝑴𝑫) = 𝟔𝟎 𝒈𝒎𝒔

9. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT 9
Solution: 𝑴𝒐𝒅𝒆 = 𝑴𝒐𝒔𝒕 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒕 𝒗𝒂𝒍𝒖𝒆
Here,
item 65 is repeated for maximum times = 3 times.
So, 65 is most frequent value
Therefore,
Mode = 65 gms

10. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany
Dr. S. S. KHOT 10
Example 1: The weight of 11 lemons in gms is as below. Calculate the
mean, median and mode.
S.No. 1 2 3 4 5 6 7 8 9 10 11
Weight 56 60 48 65 52 58 65 65 70 68 53
𝑴𝒆𝒂𝒏 =
𝒙
𝑵
=
𝑿𝟏 + 𝑿𝟐 + 𝑿𝟑 + ⋯ 𝑿𝒏
𝑵
Mean = 60 gms
𝑴𝒆𝒅𝒊𝒂𝒏 (𝑴𝑫) =
𝒏 + 𝟏
𝟐
𝒕𝒉
𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏 ∴ 𝑴𝒆𝒅𝒊𝒂𝒏 (𝑴𝑫) = 𝟔𝟎 𝒈𝒎𝒔
𝑴𝒐𝒅𝒆 = 𝑴𝒐𝒔𝒕 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒕 𝒗𝒂𝒍𝒖𝒆 Mode = 65 gms
Therefore, for given data the central tendency is:

11. skhot1976@gmail.com B.Sc.-III Paper- XIV (DSE –F26) Bioinformatics, Biostatistics and Economic Botany