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SECTION 05 / 7

Spread: range, variance and standard deviation

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Two groups can have the same mean but feel very different. Spread tells us whether values stay close together or vary widely.

Both groups have mean 3, but only the second group varies.
Both groups have mean 3, but only the second group varies.

Range: largest minus smallest

Range=max⁡(x)−min⁡(x)\text{Range}=\max(x)-\min(x)

For 3,3,3,3,3, range = 3−3 = 0. For 1,2,3,4,5, range = 5−1 = 4. Replacing the last value by 15 increases the range to 15−1 = 14. Range is easy to calculate but only uses the extremes.

Why not simply add deviations?

A deviation is a value minus its mean. Positive and negative deviations cancel: for 2,4,6 with mean 4, deviations are −2,0,2, whose sum is zero. Zero total deviation does not mean zero spread. Squaring makes all contributions nonnegative.

For 2,4,6, squared deviations total 8; divide by 3 for population variance or 2 for sample variance.
For 2,4,6, squared deviations total 8; divide by 3 for population variance or 2 for sample variance.

Population variance versus sample variance

σ2=∑i=1N(xi−μ)2N,s2=∑i=1n(xi−xˉ)2n−1\sigma^2=\frac{\sum_{i=1}^{N}(x_i-\mu)^2}{N},\qquad s^2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}

Use N when describing the whole population as given. The course uses n−1 for sample variance when the data are a sample. Sample variance needs at least two observations. After estimating the sample mean, the deviations have to sum to zero, so one deviation is fixed by the others; this helps explain the n−1 correction.

Worked example: the ten marks

MarkDeviation from 59Squared deviation
68981
7920400
38-21441
68981
35-24576
7011121
6124
47-12144
58-11
66749
Total01898

Population variance = 1898/10 = 189.8 marks². Sample variance = 1898/9 ≈ 210.889 marks². The dataset is the same; the denominator depends on the requested interpretation.

Source correction: the slide writes the deviation of 66 as −7. It is +7 because 66−59 = 7. Its square is still 49, so the displayed squared-deviation total 1898 remains correct.

Standard deviation returns to the original unit

σ=σ2,s=s2\sigma=\sqrt{\sigma^2},\qquad s=\sqrt{s^2}

Population SD = √189.8 ≈ 13.777 marks. Sample SD = √(1898/9) ≈ 14.522 marks. If the measurements were kilograms, variance would be kg² and standard deviation would be kg. Standard deviation is a measure of spread around the mean, not a statement that every observation lies exactly that far away.

What happens when every value changes?

y=a+bx:Var⁡(y)=b2Var⁡(x),SD⁡(y)=∣b∣SD⁡(x)y=a+bx:\quad \operatorname{Var}(y)=b^2\operatorname{Var}(x),\quad \operatorname{SD}(y)=|b|\operatorname{SD}(x)
Change to every observationCenterRange / SDVariance
Add 5Increases by 5UnchangedUnchanged
Multiply by 0.4Multiplied by 0.4Multiplied by 0.4Multiplied by 0.16
Multiply by −2Multiplied by −2Multiplied by 2Multiplied by 4

For the marks multiplied by 0.4, sample variance = 0.16 × (1898/9) ≈ 33.7422 and sample SD ≈ 5.8088. Adding a constant shifts every point and its mean together, so distances do not change.

Try it yourself

For the population 1,3,5, find range, variance and SD.

Show explained answer

Mean = 3. Range = 4. Squared deviations are 4,0,4; population variance = 8/3 ≈ 2.667 and SD ≈ 1.633.

Try it yourself

Now treat 1,3,5 as a sample. What changes?

Show explained answer

Sample variance = 8/(3−1) = 4. Sample SD = 2. Range and mean are unchanged.

Try it yourself

If SD is 4 cm and each value is doubled then increased by 10 cm, what is the new SD?

Show explained answer

8 cm. Doubling doubles SD; adding 10 changes location only.

Source: Lecture 3.4. Extra examples and diagrams are teaching material prepared for these notes.

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