This document discusses the calculation of arithmetic mean from data in various formats. It defines arithmetic mean as the sum of all values divided by the number of observations. It then provides examples of calculating the arithmetic mean using direct and shortcut methods for individual series, discrete series with frequencies, continuous series with class intervals, and series with open-ended classes. The different methods are demonstrated using example data sets. References on biostatistics are also included.
2. Arithmetic Mean
• It represents the entire data by one value which is obtained by adding together all the
values and dividing this total value by the number of observations
1. Calculation of Arithmetic Mean of an individual Series
• This can be calculated by direct method or short-cut method. The result will be same for
both methods.
Direct Method: 𝑿̄̅ =
∑𝑿̄̅
𝒏
X̄= arithmetic mean
∑X= sum of all values of the variable x i.e. X1, X2, X3, …… Xn
n= No. of observations
Example 1.
Calculate the arithmetic mean of the following set of observations: 7, 6, 8, 10, 13, 14
X̄= ∑x/n
=58/6= 9.67
3. Short Cut Method: X = 𝑨 +
∑𝒅
𝒏
X̄= arithmetic mean
A= Assumed mean
d= deviation of items from the assumed mean (x-A)
∑d= sum of all deviations
n= No. of observations
Example 2: Calculate the arithmetic mean of the data given:
X̄= A+ ∑d/n
=24+13/10
=25.3
No. of
spikelets per
spike
Deviations
from the
assumed
mean
Assumed
mean=24
18 -6
19 -5
20 -4
21 -3
22 -2
28 4
29 5
30 6
31 7
35 11
n=10 ∑d=-20+33=13
4. 2. Calculation of Arithmetic Mean in Discrete Series
The values of the variables are multiplied by their respective frequencies. The number of
observations is the total number of frequencies.
Direct Method: 𝑿̄̅ =
∑𝒇𝑿̄̅
𝒇
X̄= arithmetic mean
∑fX= sum of values of the variables and their corresponding frequencies
∑f= sum of frequencies.
Example 1.
𝑿̄=(∑𝒇𝑿)/𝒇
=391/150
=2.61
Short cut method: 𝑿̄̅ = 𝑨 +
∑𝒇𝒅
∑𝒇
X̄= arithmetic mean
A= Assumed mean
No. of
chlorophyll
deficient plants
No. of plants fx
0 34 0
1 14 14
2 20 40
3 24 72
4 25 100
5 33 165
∑f=150 ∑fX=391
5. d= deviation of items from the assumed mean (x-A)
∑fd= sum of the deviations from the assumed mean and the respective frequencies
∑f= sum of the frequencies
3. Calculation of Arithmetic Mean in Continuous Series
In a continuous series, the arithmetic mean may be calculated after taking into
consideration the mid point of various classes. The method will be the same for both
inclusive class-intervals as well as for exclusive class-intervals.
Direct Method:
𝑋 =
∑𝑓𝑚
∑𝑓
X̄= arithmetic mean
∑fm= sum of values of midpoints multiplied by the respective frequencies of each class
∑f= sum of frequencies
m= midpoint of various classes.
Mid-Point (m)= (Lower limit + Upper limit)/ 2
6. Plant Height
Classes
No. of
varieties (f)
Mid-points
(m)
fm
0-10 5 5 25
10-20 10 15 150
20-30 25 25 625
30-40 30 35 1050
40-50 20 45 900
50-60 10 55 550
∑f= 100 ∑fm=3300
Example 1. Compute the arithmetic mean of the following data
𝑋 =
∑𝑓𝑚
∑𝑓
X̄= 3300/100
=33
7. Short cut method: 𝑋 = 𝐴 +
∑𝑓𝑑
∑𝑓
X̄= arithmetic mean
A= Assumed mean
d= Deviation of midpoints from the assumed mean (m-A)
∑f= sum of frequencies
f= frequency of each class
4. Calculation of arithmetic mean in series having open-end classes
No. of pods No. of plants
Below 10 4
10-20 6
20-30 20
30-40 12
40-50 10
50-60 5
Above 60 4
No. of pods No. of plants
0-10 4
10-20 6
20-30 20
30-40 12
40-50 10
50-60 5
60-70 4
‗
> In this data the class interval is uniform therefore the lower limit if the first class
would be zero and the last limit would be 70
8. References
Khan, I. A., & Khanum, A. (1994). Fundamentals of biostatistics. Ukaaz.
Sharma, A. K. (2005). Text book of biostatistics I. Discovery Publishing House.
Daniel, W. W., & Cross, C. L. (2018). Biostatistics: a foundation for analysis in
the health sciences. Wiley.
