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Variance and standard deviation

Build variance from squared deviations, use the shortcut formula and distinguish descriptive denominator n from sample denominator n minus one. Original model, animation and practice.

Before you startMeans, squared numbers and square roots.

01 / Measure distance from the mean

Signed deviations cancel; squared deviations do not.

Descriptive variance = Σ(x − x̄)²/n. Standard deviation = √variance.

Here n is the number of observations being described.

Distances on opposite sides of the mean have opposite signs. Their sum is zero, even for a very spread-out dataset. Squaring prevents that cancellation, and the square root brings the final spread measure back to the original units.

Same mean, different squared distancesExplore
xx − 10(x − 10)²
8−24
9−11
1000
1111
1224

Mean = 10; deviations sum to 0; squared deviations sum to 10.

Variance = 10/5 = 2; SD ≈ 1.414214.

We describe these five observations using denominator n = 5. A zero SD means every value equals the mean.

02 / Start with deviations

Find the mean before measuring distances.

Data: 2, 4, 6.Worked example

Mean = (2 + 4 + 6)/3 = 4

This is the common reference point.

Deviations: −2, 0, 2

They sum to zero, not a useful spread measure.

Squared deviations: 4, 0, 4

Their sum is 8.

01 · Cancellation

For values 1, 5, 9, calculate the mean and sum of deviations.

Hint

Measure every deviation from the same mean.

Worked solution

Mean = 5. Deviations −4, 0, 4 sum to zero, although the observations differ.

02 · Square signs correctly

What are (−3)² and the sum of squared deviations for deviations −3, 0, 3?

Hint

Parentheses include the negative sign in the square.

Worked solution

(−3)² = 9. The squared deviations sum to 9 + 0 + 9 = 18.

03 / Average the squared deviations

Take the square root only at the end.

Watch: cancellation disappears when deviations are squared

Pause, replay or seek freely. The notes explain the same idea and stay in view.

Continue with 2, 4, 6.Worked example

Variance = 8/3

Divide the squared-deviation sum by n = 3.

SD = √(8/3) ≈ 1.633

Keep the exact fraction until the final calculation.

If measurements are cm, variance is cm² and SD is cm

The units explain why the square root matters.

03 · A full calculation

Find descriptive variance and SD for 4, 4, 8, 8.

Hint

Mean is 6; each squared deviation is 4.

Worked solution

Squared-deviation sum = 16. Variance = 16/4 = 4; SD = 2.

04 · Constant data

Find the variance and SD for 7, 7, 7.

Hint

All deviations are zero.

Worked solution

Both are zero. This means no variation among these observations, not missing information.

04 / Use sums and squared sums

Square each value before adding.

Variance = Σx²/n − (Σx/n)²

Σx² is not the same as (Σx)².

Expanding Σ(x − x̄)² gives Σx² − 2x̄Σx + nx̄². Since Σx = nx̄, this simplifies to Σx² − nx̄². Dividing by n gives the shortcut formula.

05 · Keep the two squares distinct

For 2, 4, 6, calculate Σx² and (Σx)², then use the shortcut.

Hint

Square each value for the first expression; sum first for the second.

Worked solution

Σx² = 4 + 16 + 36 = 56; (Σx)² = 12² = 144. Variance = 56/3 − 4² = 8/3.

06 · Supplied summaries

Five observations have Σx = 30 and Σx² = 200. Find mean, variance and SD.

Hint

Mean = 30/5.

Worked solution

Mean = 6; variance = 200/5 − 6² = 4; SD = 2.

05 / Recognise Sxx

A sum of squares is not yet a variance.

Sxx = Σ(x − x̄)² = Σx² − (Σx)²/n

Descriptive variance = Sxx/n.

Sxx is the total squared deviation, whereas variance is its average under the chosen denominator convention. Notation varies between sources; read each definition.

07 · Find Sxx

For the five-observation summaries in question 6, find Sxx.

Hint

Subtract (Σx)²/n from Σx².

Worked solution

Sxx = 200 − 900/5 = 20. Dividing by five gives descriptive variance 4.

08 · Recover a squared sum

There are ten observations with mean 3 and descriptive variance 2. Find Σx and Σx².

Hint

Σx = n×mean; Σx²/n = variance + mean².

Worked solution

Σx = 30. Σx² = 10×(2 + 9) = 110.

06 / Choose the intended denominator

Describing the data and estimating population variance are different tasks.

This lesson uses n to describe the observed dataset, even if those observations were collected as a sample. Another common quantity is sample variance s² = Sxx/(n − 1), for n > 1. Under standard independent sampling assumptions, that is an unbiased estimator of population variance. Its square root is commonly called sample standard deviation; it is not generally an unbiased estimator of population SD.

Calculator labels often distinguish σx (denominator n) from sx (denominator n − 1). Check the requested definition and calculator documentation rather than choosing a button by habit.

09 · Compare denominators

For 2, 4, 6, Sxx = 8. Give descriptive variance and sample variance with denominator n − 1.

Hint

n = 3.

Worked solution

Descriptive variance = 8/3. Sample variance = 8/2 = 4. Their SDs are √(8/3) and 2 respectively.

10 · One observation

What can you say about these two variance formulas when n = 1?

Hint

The observed value equals its own mean.

Worked solution

Descriptive variance is 0. Sxx/(n − 1) is undefined because its denominator is zero.

07 / Interpret scale and sensitivity

SD is a spread summary, not a fixed distance for every observation.

11 · Extreme observation

For 0, 0, 0, 0, 10, calculate the descriptive mean and variance.

Hint

Mean 2; squared deviations are 4, 4, 4, 4, 64.

Worked solution

Mean = 2; squared-deviation sum = 80; variance = 16; SD = 4. The extreme value contributes 64 of the total 80 squared deviations.

12 · Compare like units

Two comparable groups have the same mean time, but SDs of 2 and 5 minutes. What does that support?

Hint

The SD measures spread around each group’s mean.

Worked solution

The first group has the smaller squared-deviation spread about its mean. This alone does not show its exact range, shape, or the proportion within any threshold.

08 / Use arithmetic checks

A true variance cannot be negative.

Check n, sum, squared sum and units. Avoid rounding the mean before squaring it. With very large, nearly equal values, subtracting close rounded quantities can cause numerical error; direct deviations or suitable coding can improve stability.

13 · Impossible summaries

Four observations are claimed to have Σx = 20 and Σx² = 80. Show that these cannot both be correct.

Hint

Apply the variance shortcut.

Worked solution

Mean = 5; variance = 80/4 − 25 = −5. A sum of squared deviations cannot be negative, so the summaries are inconsistent.

14 · Wrong units

Mass data in grams have variance 9. What are the variance units and SD?

Hint

Variance has squared units.

Worked solution

Variance = 9 g²; SD = 3 g. Do not label variance as 9 g.

09 / Square, average, root

Keep the formula and meaning together.

Identify the denominator convention, keep exact intermediate arithmetic, state variance in squared units and SD in the original units. The mean describes centre; SD describes a particular kind of spread.

Section 1 of 9 · Measure distance from the mean