Introduction to Average

What is an average?

The average (arithmetic mean) of a set of numbers is their sum divided by how many numbers there are.

Average = Sum of observations ÷ Number of observations

So, Sum = Average × Number.

Examples

• Average of 10, 20, 30, 40 = 100 ÷ 4 = 25.
• Average of first n natural numbers = (n + 1)/2. First 10 → 5.5.
• Average of first n even numbers = n + 1; first n odd numbers = n.
• Average of consecutive numbers = (first + last) ÷ 2.

Deviation method (fast)

Assume a value, find deviations, and correct. Marks 72, 75, 78, 71: assume 75 → deviations −3, 0, +3, −4 = −4 → average = 75 − 4/4 = 74.

Weighted average

If group A (n₁ items) has average a₁ and group B (n₂ items) has average a₂, combined average = (n₁a₁ + n₂a₂)/(n₁ + n₂).

Example: 30 boys average 60, 20 girls average 70 → (1800 + 1400)/50 = 64.

Analogy

An average is like pouring all glasses of juice into one jug and refilling them equally. Some glasses were fuller, some emptier — after levelling, every glass holds the same amount. That common level is the average.