Statistics: Standard Deviation & Variance

The mean tells you the average, but it doesn’t tell you how spread out the data is. Two classes could both average 60% but one has everyone at 58–62% while the other ranges from 20% to 100%. Variance and standard deviation measure that spread.

Quick Mean Revision

Mean = sum of all values ÷ number of values. For example, the mean of 4, 8, 6, 5, 7 is (4+8+6+5+7)÷5 = 30÷5 = 6. We use the symbol x̄(x-bar) for the mean.

Key Formulas

Variance (σ²): Σ(x_i− x̄)² / n
Standard Deviation (σ): √Variance = √[Σ(x_i − x̄)² / n]
Alternative formula: Variance = (Σx_i² / n) − x̄²

What Does Variance Actually Measure?

Variance measures how far each data point is from the mean, on average. We square the differences so negatives don’t cancel out positives. A small variance means the data is clustered close to the mean. A large variance means the data is spread out widely.

Step-by-Step Variance Calculation

Worked Example

Data: 2, 4, 4, 4, 5, 5, 7, 9

Find the mean: (2+4+4+4+5+5+7+9) ÷ 8 = 40 ÷ 8 = 5
Subtract the mean from each value:
−3, −1, −1, −1, 0, 0, 2, 4
Square each difference:
9, 1, 1, 1, 0, 0, 4, 16
Add them up: 9+1+1+1+0+0+4+16 = 32
Divide by n (8): 32 ÷ 8 = 4

Variance = 4 | Standard Deviation = √4 = 2

Standard Deviation = √Variance

Variance is in squared units which is hard to interpret. Taking the square root brings us back to the original units. So if we’re measuring heights in cm, the standard deviation is also in cm — much more useful.

The 68–95–99.7 Rule

For data that follows a normal distribution(bell curve):

68% of data falls within 1 standard deviation of the mean.
95% of data falls within 2 standard deviations.
99.7% of data falls within 3 standard deviations.

Worked Example

Exam scores have mean 70 and standard deviation 10.

68% of students scored between 60 and 80.
95% scored between 50 and 90.
99.7% scored between 40 and 100.

Normal Distribution — Explained Simply

A normal distribution is a bell-shaped curve where most values cluster around the mean and it tails off equally on both sides. It appears everywhere: heights, weights, exam scores, measurement errors.

We write it as X ~ N(μ, σ²) where μ is the mean and σ² is the variance. The curve is symmetric: the mean, median, and mode are all the same value.

Worked Example — Using the Alternative Formula

Data: 3, 5, 7

Mean = (3+5+7)/3 = 5
Σx² = 9 + 25 + 49 = 83
Σx²/n = 83/3 = 27.67
Mean² = 25
Variance = 27.67 − 25 = 2.67
SD = √2.67 = 1.63 (2 d.p.)

Try It Yourself

Want to check your answers or experiment with different data sets? Use our free standard deviation calculator to see variance and standard deviation calculated step by step.

Common Mistakes

Forgetting to square the differences (they’d just cancel out to zero).
Confusing σ (population) with s (sample) — check which formula the question wants.
Forgetting to square root the variance to get standard deviation.
Using n−1 when you should use n (or vice versa).

Practice Questions

Find the variance and standard deviation of: 10, 12, 14, 16, 18
A data set has mean 50 and standard deviation 8. Using the 68-95-99.7 rule, between which values do 95% of the data fall?
The heights of students are normally distributed with mean 165 cm and SD 7 cm. What percentage of students are between 158 cm and 172 cm?

Answers: 1) Variance = 8, SD = 2.83 2) 34 to 66 3) 68%

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