Mean - Calculating the Average

The mean is what most people mean when they say "average". It is calculated by adding up all the values in a dataset and dividing by how many values there are. The mean takes every value into account, which makes it very useful but also sensitive to extremely large or small values.

The concept of averaging values to estimate a true measurement dates back centuries: 17th-century astronomers routinely took several separate readings of a star's position and averaged them, correctly reasoning that random measurement errors would tend to cancel out. The mathematical justification for this was formalised much later by Carl Friedrich Gauss in the early 1800s, whose work on the “normal distribution” (the bell curve) explained precisely why averaging many imperfect measurements gives a more reliable estimate than trusting any single one. The Greek letter used in the mean's formula, Σ (sigma), is simply the Greek capital S, chosen because it stands for “sum” – mathematicians have used Greek letters as compact shorthand for operations like this since the 18th century.

Mean as a Balance Point

Picture each value in a dataset as a small weight sitting on a number line. The mean is the exact point where that number line would balance perfectly, like a seesaw — the pull of the weights on one side exactly cancels the pull on the other.

The two blue weights left of 7 (at 3 and 4) are 4 and 3 units away — a total pull of 7. The two blue weights right of 7 (at 9 and 12) are 2 and 5 units away — also a total pull of 7. The green weight sits exactly at the balance point, contributing no pull at all. Because the pulls on each side are equal, the line balances perfectly at x = 7, which is exactly the mean of 3, 4, 7, 9 and 12. This is precisely why the mean formula works: it's the one point where the total distance to values above it always equals the total distance to values below it.

The Formula

Mean = Sum of all values ÷ Number of values

In symbols: Mean = Σx / n (where Σx means "the sum of all x values" and n is the count).

Worked Examples

Find the mean of: 4, 7, 9, 3, 12.

Sum = 4 + 7 + 9 + 3 + 12 = 35. Count = 5. Mean = 35 ÷ 5 = 7.

A student scores 72, 85, 60, 91, and 77 in five tests. Find the mean score.

Sum = 72 + 85 + 60 + 91 + 77 = 385. Mean = 385 ÷ 5 = 77.

The mean of four numbers is 9. Three of them are 6, 11, and 8. Find the fourth.

Total = 9 × 4 = 36. Known sum = 6 + 11 + 8 = 25. Fourth = 36 − 25 = 11.

Effect of an outlier: Find the mean of 3, 5, 4, 6, 4 and then of 3, 5, 4, 6, 100.

First: (3+5+4+6+4)/5 = 22/5 = 4.4. Second: (3+5+4+6+100)/5 = 118/5 = 23.6. An outlier (100) pulls the mean up dramatically.

Mean from a Frequency Table

Multiply each value by its frequency, sum those products, then divide by the total frequency.

Goals scored per match: 0 goals (3 matches), 1 goal (5), 2 goals (4), 3 goals (2).

Goals (x)	Frequency (f)	f × x
0	3	0
1	5	5
2	4	8
3	2	6
Total	14	19

Mean = 19 ÷ 14 ≈ 1.36 goals per match.

Key Takeaways

Mean = Sum of values ÷ Number of values.
The mean uses every value, so it is affected by outliers.
If the mean is known, the total sum = mean × count (useful for finding a missing value).
For a frequency table: mean = Σ(f × x) ÷ Σf.

Practice: Calculating the Mean

Find the Mean