Mean (Average)

Mean (Average) 

Introduction 

• The mean is one of the most widely used measures of central tendency, offering a single value that represents the average of a dataset. 

• It’s applied across fields like economics, psychology, education, and pharmaceutical sciences to summarize a dataset with a single representative value. 

• There are different types of mean, each suited for specific types of data and analysis contexts. 

Types of Mean 

1. Arithmetic Mean 

• The most common form of mean. 

• Calculation: Sum all the values and divide by the total number of values. 

• Formula 

x−=∑XN𝐱-=∑𝐗𝐍

 

• Sensitive to outliers. 

1. Geometric Mean

• Used for averaging percentages, ratios, or growth rates. 

• Formula:  

GM=x1×x2×⋯×xn−−−−−−−−−−−−−−−−√nGM=𝒙𝟏×𝒙𝟐×⋯×𝒙𝒏𝒏

 

• Less influenced by large outliers or skewed data. 

1. Harmonic Mean 

• Suitable for rates (e.g., speed, density). 

• Formula:  

HM=n∑ni=11xiHM=𝒏∑𝒊=𝟏𝒏𝟏𝒙𝒊

 

• Especially helpful when values are defined per unit (e.g., km/hour). 

 

Arithmetic Mean 

Overview 

• The Arithmetic Mean is the most widely used measure of central tendency. 

• It provides a single value that represents the average of a dataset. 

• It is calculated by summing all values in a dataset and dividing by the number of values. 

Formula 

• • For a dataset with values 

 x1,x2,…,xn x1,x2,…,xn

•  

xˉ=∑XN𝒙ˉ=∑𝑿𝑵

 

Arithmetic Mean for Individual Series (Simple Data) 

1. Direct Method 

• Steps: 

1. Add all observations. 

1. Divide the sum by the total number of observations. 

• Formula: 

AM=∑XN𝑨𝑴=∑𝑿𝑵

 

• Example: Reduction in Blood Pressure 

Individual	Reduction (mmHg)
1	8
2	10
3	7
4	5
5	9

• Calculation: 

∑X=8+10+7+5+9=39∑X=8+10+7+5+9=39

•  

• N = 5 

• AM = 39 / 5 = 7.8 

• Result: Mean reduction = 7.8 mmHg 

1. Shortcut Method 

• Uses an assumed mean to simplify calculations. 

• Steps: 

1. Choose a value as the assumed mean (A). 

1. Compute deviations d = X – A. 

1. Use the formula to find the mean. 

• Formula: 

AM=A+∑dNAM=A+∑dN

 

• Example: Using assumed mean A = 8 

Individual	X (Reduction)	A	d = X – A
1	8	8	0
2	10	8	2
3	7	8	-1
4	5	8	-3
5	9	8	1

• Calculation: 

∑d=0+2−1−3+1=−1∑d=0+2−1−3+1=−1

•  

AM=8+(−1/5)=7.8AM=8+−1/5=7.8

•  

• Result: Same mean, 7.8 mmHg 

Arithmetic Mean for Discrete Series (Ungrouped Frequency Data) 

1. Direct Method 

• Frequency is associated with each data point. 

• Formula: 

AM=∑fXN𝑨𝑴=∑𝒇𝑿𝑵

 

• Example: Days to Recover 

X (Days)	f (Patients)
3	2
4	5
5	3
6	4

• Calculation: 

∑fX=(3×2)+(4×5)+(5×3)+(6×4)=6+20+15+24=65∑fX=3×2+4×5+5×3+6×4=6+20+15+24=65

•  

N = 2 + 5 + 3 + 4 = 14N = 2 + 5 + 3 + 4 = 14

•  

AM = 65 / 14 ≈ 4.64AM = 65 / 14 ≈ 4.64

•  

• Result: Mean recovery time = 4.64 days 

1. Shortcut Method 

• Formula: 

AM=A+∑fdN

 

• Assumed Mean (A) = 5 

X	f	A	d = X – A	fd
3	2	5	-2	-4
4	5	5	-1	-5
5	3	5	0	0
6	4	5	1	4

• Calculation: 

∑fd=−5

•  

• N = 14 

AM=5+(−5/14)≈4.64

•  

• Result: Mean recovery time = 4.64 days 

Arithmetic Mean for Continuous Series (Grouped Data) 

1. Direct Method 

• Use midpoint (m) of each class interval. 

• Formula: 

AM=∑f⋅mN

 

• Example: Cholesterol Reduction 

Class Interval	f	Midpoint (m)
10 – 19	5	14.5
20 – 29	12	24.5
30 – 39	8	34.5
40 – 49	5	44.5

• Calculation: 

∑fm=(14.5×5)+(24.5×12)+(34.5×8)+(44.5×5)=865∑fm=14.5×5+24.5×12+34.5×8+44.5×5=865

•  

N = 30N = 30

•  

AM = 865 / 30 = 28.83AM = 865 / 30 = 28.83

•  

• Result: Mean cholesterol reduction = 28.83 mg/dL 

1. Shortcut Method 

• Formula: 

AM=A+∑fdNAM=A+∑fdN

 

• Assumed Mean (A) = 24.5 

Class Interval	f	m	d = m – A	fd
10 – 19	5	14.5	-10	-50
20 – 29	12	24.5	0	0
30 – 39	8	34.5	10	80
40 – 49	5	44.5	20	100

• Calculation: 

∑fd=−50+0+80+100=130∑fd=−50+0+80+100=130

•  

N = 30N = 30

•  

AM=24.5+(130/30)=28.83AM=24.5+130/30=28.83

•  

• Result: Mean cholesterol reduction = 28.83 mg/dL 

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