Averages and spread

Two numbers describe most data sets: one for where the values sit, and one for how far they spread. Everything in this section is about keeping those two ideas separate, because they do not respond to the same things.

Shift it, or stretch it

Eight test marks. The slider on the left adds the same amount to every mark. The one on the right multiplies every mark. Watch which statistic reacts to which.

Add to every mark +0

Multiply every mark by ×1.0

Adding a constant slides every measure of position and leaves every measure of spread alone.

–mean

–standard deviation

–interquartile range

Nothing is stored and nothing is sent.

Adding moves everything together, so the gaps between values never change. The mean shifts by exactly what you added, and the spread does not move at all.

Multiplying stretches the gaps as well, so the mean and the spread both scale by the same factor.

Keep this. At Higher Level you will meet E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X), and the second one surprises people because the b has vanished. It vanished here too, on the slider, before any algebra was involved. Adding never changes spread.

Which average, and when

Mean. Uses every value, so it is dragged by extremes. Best when the data is roughly symmetric.
Median. The middle by position, so extremes do not move it. Best when the data is skewed or has an outlier.
Mode. The most common value. The only average that works for categories like favourite subject.

Worked example: standard deviation from a small set

Find the mean and the standard deviation of 4, 6, 8, 10, 12.

x̅= 4 + 6 + 8 + 10 + 125 = 405 = 8 the mean first, since the deviations are measured from it deviations= −4, −2, 0, 2, 4 each value minus the mean squares= 16, 4, 0, 4, 16 squared so the negatives stop cancelling σ²= 405 = 8 the variance: the mean squared deviation σ= √8 ≈ 2.83 back to the original units

In the exam you will use your calculator for this, and you should. Knowing the structure still matters, because it explains why variance is in squared units and why Var(aX) picks up an a² rather than an a.

Your turn

1. A set of marks has mean 54 and standard deviation 12. Every mark is increased by 8. What is the new standard deviation?

2. The same set, mean 54 and standard deviation 12, is instead multiplied by 1.5. What is the new mean?

3. A data set is strongly skewed, with a few very large values. Which average best represents a typical value?

The mean, because it uses every value. The median, because the extreme values do not move it. The mode, because it is the most common. The range, because it shows the spread.

Where the marks go

Transformation questions are worth easy marks and lost constantly. Write down which one you are doing before you calculate: adding shifts, multiplying scales both.

Standard deviation is never negative. If yours is, you have taken a square root of something that should not have been negative, and it is worth saying so rather than writing it down.

When asked to justify choosing an average, name the feature of the data. "The median, because the data is skewed" earns the mark. "The median, because it is better" does not.

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