Q.What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face?

Arithmetic Mean: The "Fair Share" Intuition

Imagine you have a group of friends with different amounts of pocket money. One has ₹50, another ₹80, a third ₹120, and the last ₹150. If you wanted everyone to have the same amount — a perfectly equal distribution — how much would each person get?

You'd add up all the money: ₹50 + ₹80 + ₹120 + ₹150 = ₹400. Then you'd split it equally among the 4 friends: ₹400 ÷ 4 = ₹100 each.

That ₹100 is the arithmetic mean (or simply the mean or average). It's the single number that represents the "centre" of a set of numbers — the value you'd get if you redistributed the total equally.

The Precise Definition

For a set of $n$ numbers $x_1, x_2, x_3, \dots, x_n$, the arithmetic mean $\bar{x}$ (read as "x-bar") is:

In words: sum of all observations divided by the number of observations.

The symbol $\sum$ (sigma) means "sum of". So $\sum_{i=1}^{n} x_i$ means "add up all the $x$ values from the first to the $n$th".

Why It Works — The Balancing Point

The mean has a beautiful physical interpretation. If you imagine each data point as a weight placed on a number line at its value, the mean is the balance point — the position where the seesaw would perfectly balance.

For example, with numbers 2, 4, and 6:

• Mean = (2+4+6)/3 = 12/3 = 4
• The deviations from the mean are: 2−4 = −2, 4−4 = 0, 6−4 = +2
• The sum of deviations is always zero: (−2) + 0 + (+2) = 0

This is a key property: the sum of deviations from the mean is always zero. The positive and negative differences cancel out perfectly.

Step-by-Step Calculation

Let's calculate the mean marks of 5 students: 72, 85, 91, 68, 79.

Step 1: Add all the values.

Step 2: Count the number of values.

Step 3: Divide the sum by the count.

The mean mark is 79.

Common Mistakes to Avoid

When to Use the Arithmetic Mean

Use the mean when:

• The data is roughly symmetric (no extreme outliers)
• You want a single representative value that accounts for every data point
• You're working with continuous numerical data (heights, weights, scores, temperatures)

Avoid the mean when:

• There are extreme outliers (use median instead)
• The data is categorical (use mode instead)
• The data is skewed heavily in one direction

The arithmetic mean is the most commonly used measure of central tendency — and now you know exactly what it means and why it works.