Arithmetic Mean: Calculation Methods, Properties & Utility

1. What Is a Measure of Central Tendency?

In a large data set, individual observations tend to cluster around a central value. A Measure of Central Tendency (or Statistical Average) is a single representative value that summarizes the entire distribution.

Prof. Bowley’s Famous Quip:
“Statistics may rightly be called the science of averages.”

Requisites of an Ideal Average:

Rigidly Defined: Must have an exact mathematical formula, leaving no room for personal interpretation.
Based on All Observations: Every single value must contribute to the calculation.
Easy to Understand & Simple to Compute: Clear and intuitive for business executives.
Least Affected by Sampling Fluctuations: Different samples from the same population should yield similar averages.
Capable of Further Algebraic Treatment: Can be used in higher statistical formulas (e.g., Combined Mean, Variance).

2. Arithmetic Mean (Simple Average)

The Arithmetic Mean (commonly called the Mean and denoted by x̄, read as “X-bar”) is calculated by adding all observations together and dividing by the total number of observations.

Formulas for Arithmetic Mean Across All 3 Series

Individual Series

Direct: x̄ = (ΣX) ÷ N
Shortcut: x̄ = A + [(Σd) ÷ N]
where d = X - A

Discrete Series

Direct: x̄ = (ΣfX) ÷ N
Shortcut: x̄ = A + [(Σfd) ÷ N]
where N = Σf

Continuous Series

Direct: x̄ = (Σfm) ÷ N
Step-Deviation: x̄ = A + [(Σfd') ÷ N] × i
where d' = (m - A) ÷ i

3. Step-by-Step Calculation Guide

A. Individual Series Example

Problem: The monthly pocket expenses of 5 BBA hostel students are: ₹1,200, ₹1,500, ₹1,800, ₹2,000, and ₹2,500. Calculate Mean.

Calculation:
ΣX = 1200 + 1500 + 1800 + 2000 + 2500 = 9000
N = 5
x̄ = 9000 ÷ 5 = ₹1,800

B. Continuous Series via Step-Deviation Method (Exam Favourite!)

Problem: Calculate Mean marks of 50 students from the following data:

Marks	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50
Number of Students (f)	5	10	15	12	8

Solution Table:

Let Class Interval width i = 10.
Let Assumed Mean A = 25 (mid-point of middle class 20 - 30).

Class	Mid-point (m)	Frequency (f)	d = m - 25	d’ = d ÷ 10	fd’
0 - 10	5	5	-20	-2	-10
10 - 20	15	10	-10	-1	-10
20 - 30	25 (A)	15	0	0	0
30 - 40	35	12	+10	+1	+12
40 - 50	45	8	+20	+2	+16
Total	—	N = 50	—	—	Σfd’ = +8

Applying the Step-Deviation Formula:
x̄ = A + [(Σfd' ÷ N) × i]
x̄ = 25 + [(8 ÷ 50) × 10]
x̄ = 25 + [0.16 × 10] = 25 + 1.6 = 26.6 marks

4. Mathematical Properties of Arithmetic Mean

These two properties are frequently asked in DAVV viva and theory exams:

Property 1: The Algebraic Sum of Deviations from Mean is Always ZERO!
Σ(X - x̄) = 0
Example: For numbers 10, 20, 30 (Mean = 20):
Deviations: (10 - 20) + (20 - 20) + (30 - 20) = (-10) + 0 + (+10) = 0.
Property 2: The Sum of Squared Deviations is MINIMUM when taken from Mean!
Σ(X - x̄)² < Σ(X - Any other assumed number)²
Combined Arithmetic Mean Formula:
If two groups have sizes N1 and N2 with respective means x̄1 and x̄2:
x̄_combined = [(N1 × x̄1) + (N2 × x̄2)] ÷ (N1 + N2)

5. Merits, Demerits & Business Utility

Merits:

Rigidly defined by a mathematical formula.
Based on all items in the data series.
Easily amenable to further algebraic treatment (can compute combined mean).

Demerits:

Highly Sensitive to Extreme Outliers: If 4 workers earn ₹15,000 each and the CEO earns ₹10,00,000, the arithmetic mean suggests an average salary of ₹2,12,000 — giving a completely distorted picture!
Cannot be calculated for open-ended classes (e.g., “Below 10” or “Above 80”) without making arbitrary mid-point guesses.
Cannot be determined graphically or for qualitative attributes.

Arithmetic Mean: Calculation Methods, Properties & Utility | BBA Notes