Maths⏱ 6 min read

What Is Standard Deviation and What Does It Actually Tell You?

Standard deviation is the most commonly used measure of spread in statistics — but most people only vaguely understand what it means. Here's a plain-English explanation with real examples.

Standard deviation appears on exam results, medical studies, financial reports, and quality control charts. If you've ever seen "mean ± SD" and wondered what the ± part actually means, this is the explanation.

What Standard Deviation Measures

Standard deviation measures how spread out values are around the mean (average). A low standard deviation means values are clustered tightly around the average. A high one means they're scattered widely.

Consider two classes of 30 students who both scored an average of 65% on a test:

Class A: Most students scored 60–70%, with a few slightly higher and lower. SD = 5.
Class B: Scores ranged from 20% to 100%, all over the place. SD = 22.

Same mean. Completely different distributions. Standard deviation captures this difference; the mean alone doesn't.

The Formula

1. Find the mean of all values 2. Subtract the mean from each value, then square the result 3. Find the mean of all those squared differences (this is the variance) 4. Take the square root of the variance = standard deviation

Worked example: Test scores: [60, 65, 65, 70, 70]

Mean = (60+65+65+70+70) ÷ 5 = 66 Differences from mean: [−6, −1, −1, 4, 4] Squared: [36, 1, 1, 16, 16] Mean of squares (variance) = 70 ÷ 5 = 14 Standard deviation = √14 ≈ 3.74

Population vs Sample Standard Deviation

You'll notice calculators offer two options: population SD (divides by n) and sample SD (divides by n-1).

Use population SD when you have data for every member of the group you're describing (e.g., all 30 students in the class)
Use sample SD when your data is a sample meant to represent a larger population (e.g., 30 people from a poll of the whole country)

Sample SD uses n-1 (called Bessel's correction) because a small sample tends to underestimate the spread of the full population. For most everyday purposes with large datasets, the difference is negligible.

The 68–95–99.7 Rule

If data follows a normal distribution (the classic bell curve), standard deviation has a very useful property:

RangeContainsExample

Mean ± 1 SD~68% of all valuesMiddle 68% of heights

Mean ± 2 SD~95% of all valuesAlmost all typical values

Mean ± 3 SD~99.7% of all valuesVirtually everything

Practical example: Adult male height in the UK has a mean of ~175cm and SD of ~7cm.

68% of men are between 168–182cm
95% are between 161–189cm
Only 0.3% are shorter than 154cm or taller than 196cm

Where Standard Deviation Appears in Real Life

Finance: Volatility in investing is measured as standard deviation of returns. A fund with SD of 15% has more variable returns than one with SD of 5%. Higher SD = higher risk and higher potential reward.

Manufacturing: If bolts need to be 10mm with an SD of 0.1mm, quality control can predict exactly what percentage will fall outside acceptable tolerances.

Medicine: "Normal range" for blood tests is often defined as mean ± 2 SD — the range that covers 95% of healthy individuals. Values outside this trigger further investigation.

Education: Standardised test scoring often converts raw scores to standard deviations from the mean to compare performance across different test forms.

Standard Deviation vs Variance

Variance is simply standard deviation squared (or equivalently, SD is the square root of variance). Both measure spread. SD is preferred in most situations because it's in the same units as the original data — if you're measuring heights in centimetres, SD is also in centimetres, making it intuitively interpretable. Variance would be in cm², which is harder to visualise.

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