MISSION · Data and probability
Spread and Standard Deviation
The average alone hides everything that matters
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Standard deviation answers “how far from the mean is a typical value?” — the question the mean itself cannot answer.
Why the centre isn’t enough
Two classes both average 70% on a test. In the first, every score is between 68 and 72. In the second, half scored 40 and half scored 100. Same mean, completely different classes, and any decision you make about teaching them should differ.
Spread is the missing half of the description. The simplest measure is the range — highest minus lowest — but it uses only two values and one freak result destroys it. Standard deviation uses every value, so it doesn’t have that weakness.
How it is built
The logic is worth following once, because each step fixes a problem with the step before.
- Find each value’s distance from the mean. These are the deviations.
- Square them. Otherwise the positives and negatives cancel and the total is always zero. Squaring also makes far-out values count for much more than near ones.
- Average the squares. That average is the variance.
- Take the square root. Squaring inflated the units — squared marks, squared pounds — and the root brings the answer back to the units of the original data.
The formula looks forbidding and is four plain steps: subtract, square, average, root.
Reading it
Standard deviation is in the same units as the data, which makes it directly interpretable. A mean of 70 with \(\sigma = 3\) says most scores sit within a few marks of
- The same mean with \(\sigma = 25\) says the class is scattered.
For data with the familiar bell shape, the guideline is sharp:
- About 68% of values fall within 1 standard deviation of the mean.
- About 95% fall within 2.
- About 99.7% fall within 3.
That’s why \(\sigma\) is the natural yardstick for “unusual”. A value three standard deviations out isn’t merely high, it’s in the rarest 0.3% — and that is how quality control, medical reference ranges, and standardised test scores all decide what counts as an outlier.
A note on dividing by n − 1
You’ll meet two versions of the formula, one dividing by \(n\) and one by \(n - 1\). Use \(n\) when your data is the entire population you care about. Use \(n - 1\) when it is a sample being used to estimate a larger population’s spread — a sample tends to underestimate it, and the smaller denominator compensates.
Common mix-ups
- Skipping the square root and reporting the variance as if it were the standard deviation.
- Averaging the raw deviations. They sum to zero, always. That’s why they’re squared.
- Comparing spreads across different units. A \(\sigma\) of 5 kg and one of 5 cm are not comparable.
- Assuming the 68–95–99.7 rule always applies. It’s for bell-shaped data; skewed data breaks it.
Check yourself
A set has mean 10 and values 8, 9, 10, 11, 12. Work out the variance and the standard deviation.
Deviations are \(-2, -1, 0, 1, 2\); squared they are 4, 1, 0, 1, 4, totalling 10. The variance is \(\frac{10}{5} = 2\) and \(\sigma = \sqrt{2} \approx 1.41\).
LEVEL 3 · REFLECT
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