Standard Deviation

Two data sets can have exactly the same mean but look completely different — one tightly clustered, one wildly spread out. To describe a data set fully you need a measure of spread alongside an average.

The range = highest − lowest value: simple, but entirely determined by the two most extreme values, so a single freak result distorts it. The interquartile range (IQR) = $Q_3 - Q_1$ ignores the top and bottom quarter, so it's more robust to outliers — but it still only uses two data points.

Standard deviation measures the typical distance every value lies from the mean, so it uses the whole data set. In the exam you'll almost always find it using the statistical mode on your calculator (input the data as a frequency table, then read off $\sigma_x$) rather than the raw formula, but you should know the formula exists: $\sigma = \sqrt{\frac{\sum (x-\bar{x})^2}{n}}$. A small standard deviation means the data is tightly bunched around the mean; a large one means it's spread out.

On its own, ‘σ = 6’ means nothing — you must compare it to something (another data set, or the mean). Always answer in context: e.g. ‘Branch A has a higher standard deviation than Branch B, meaning its daily sales are more variable/inconsistent.’ Never just state the numbers without saying what they mean for the situation.

(a) Both classes have the same mean (28), so on average they performed identically.

(b) Class X's standard deviation (3.1) is much smaller than Class Y's (8.4), so Class X's marks were far more tightly clustered around 28, while Class Y's marks were much more spread out — some students did much better than 28, and presumably some did much worse.

(c) The teacher's statement is misleading: the means are equal, so overall performance was the same; Class Y simply has more variable results (a wider spread), not a genuinely higher-achieving class — its highest marks are offset by lower marks elsewhere.

1. Standard deviation uses every value in the data set (not just the two extremes like the range), so it gives a more complete/representative measure of spread and is less affected by a single outlier.
2. It tells you that, typically, data values lie about 2.8 away from the mean (45) — the data is fairly tightly clustered around 45; a larger σ would mean more spread out.
3. Data set Q has the larger standard deviation — its values are much further from the mean of 11 (e.g. 2 and 20 are far from 11) than P's values, which are all close to 11.
4. Route 1 — it has the smaller standard deviation (1.5 vs 6 minutes), meaning its journey times are much more consistent/predictable around the mean, even though both routes take the same average time.
5. Squaring ensures all deviations are positive (so they don't cancel each other out when summed — the positive and negative deviations from the mean always sum to zero otherwise), and it gives more weight to values that are further from the mean.