Standard deviation is a single number that tells you how spread out your data is around the average — a small standard deviation means most values sit close to the mean, while a large one means they're scattered widely. If you're staring at a spreadsheet wondering why your professor keeps asking for "the SD," you're in the right place: this is the number that turns a lonely average into a picture of your whole dataset.
Key Takeaways
- Standard deviation (SD) measures spread: it's the typical distance between each data point and the mean of the dataset.
- A small SD means consistent, tightly clustered data; a large SD means variable, widely scattered data — even when two datasets share the same average.
- For roughly bell-shaped data, the 68-95-99.7 rule applies: about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.
- Standard deviation is the square root of the variance, and it's reported in the same units as your original data (e.g. kg, IQ points, seconds).
- In APA 7 style, you report it as M = 24.5, SD = 3.2 alongside the mean.
What does standard deviation actually mean?
Standard deviation answers a question the average alone can't: how typical is the average? Imagine two study groups both score a mean of 70% on an exam. In Group A everyone scored between 68% and 72% — the mean describes them well. In Group B, scores ranged from 40% to 100% — the mean of 70% describes almost nobody. The standard deviation is what separates these two stories: Group A has a small SD, Group B has a large one.
Put simply, standard deviation is the average distance of your data points from the mean. The bigger that average distance, the more your data varies. That's the entire intuition — everything else is just how we calculate it precisely.
How is standard deviation calculated?
The formula looks intimidating, but each step maps onto a plain idea. To find the standard deviation:
- Find the mean (the average) of your data.
- Subtract the mean from each value to get how far each point sits from the centre (the deviation).
- Square each deviation — this removes negatives so distances above and below the mean don't cancel out.
- Average the squared deviations — this gives you the variance.
- Take the square root of the variance to get back to your original units. That's the standard deviation.
One practical note: there are two versions. The sample standard deviation divides by n − 1 (used when your data is a sample from a larger population — almost always the case in research), while the population standard deviation divides by n. Most thesis and dissertation work uses the sample version, and it's the default in SPSS, R, and StatRyx.
A worked example with real numbers
Suppose you measured the reaction times (in milliseconds) of 5 participants: 420, 460, 480, 500, 540.
Step 1 — Mean: (420 + 460 + 480 + 500 + 540) ÷ 5 = 480 ms.
Step 2 — Deviations from the mean: −60, −20, 0, +20, +60.
Step 3 — Square them: 3600, 400, 0, 400, 3600.
Step 4 — Variance (sample, divide by n − 1 = 4): (3600 + 400 + 0 + 400 + 3600) ÷ 4 = 8000 ÷ 4 = 2000.
Step 5 — Standard deviation: √2000 ≈ 44.7 ms.
So you'd report the reaction times as M = 480 ms, SD = 44.7 ms. In plain terms: participants' reaction times typically fell about 45 ms away from the 480 ms average. If a sixth participant clocked in at 600 ms, you'd immediately know that's unusual — it sits more than 2 SD above the mean.
How do I interpret standard deviation once I have it?
The SD only makes sense relative to your data and its units. An SD of 44.7 ms is meaningful for reaction times; an SD of 44.7 would be enormous for a variable measured on a 1–5 Likert scale and tiny for annual income in dollars. Always interpret it against the scale you're working with.
Three practical anchors for interpretation:
- Compare SD to the mean. A rough gauge called the coefficient of variation is SD ÷ mean. In our example, 44.7 ÷ 480 ≈ 0.09, meaning the spread is about 9% of the average — fairly consistent data.
- Use it to spot outliers. Values more than about 2–3 SDs from the mean are often flagged as potential outliers worth investigating.
- Use the 68-95-99.7 rule for bell-shaped (normal) data, explained next.
What is the 68-95-99.7 rule?
For normally distributed data, about 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations, and about 99.7% fall within 3 standard deviations. With IQ scores (mean = 100, SD = 15), that means roughly 68% of people score between 85 and 115, and about 95% score between 70 and 130. This rule is why SD is so powerful: it lets you translate "spread" into concrete probability statements about where data lands.
Standard deviation vs standard error: what's the difference?
These two get confused constantly, and mixing them up will change your conclusions. Here's the distinction at a glance.
| Standard deviation (SD) | Standard error (SE) | |
|---|---|---|
| What it measures | Spread of individual data points around the mean | Precision of the sample mean as an estimate |
| Gets bigger when | Your data is more variable | (Gets smaller as your sample grows) |
| Formula link | √variance | SD ÷ √n |
| Use it to describe | How varied your participants are | How confident you are in the average |
| Reported with | M and SD in descriptives | Confidence intervals, error bars |
The key insight: standard deviation describes your data; standard error describes your estimate of the mean. If you want to say "my sample varies a lot," use SD. If you want to say "I'm confident where the true mean lies," use SE and confidence intervals.
How do I report standard deviation in APA 7 style?
APA 7 pairs the standard deviation with the mean, both italicised, with two decimal places and no percent sign inside the statistic. The correct format is:
Reaction times were slower in the caffeine condition (M = 480, SD = 44.70) than in the placebo condition (M = 512, SD = 51.30).
Report SD in the same units as your variable, and keep decimal places consistent across a table. If you're building a descriptives table, standard deviation belongs right next to the mean for every variable — and if you're then running a t-test or ANOVA on those groups, see our related guide on choosing the right statistical test.
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Frequently Asked Questions
What is a good standard deviation value?
There is no universally "good" standard deviation — it depends entirely on your variable and its scale. A small SD relative to the mean indicates consistent, reliable data, while a large SD indicates high variability. Compare the SD to the mean (via the coefficient of variation, SD ÷ mean) to judge whether spread is small or large for your context.