Standard Deviation Calculator
Calculate the mean, variance, and standard deviation for a population or sample data set.
Standard Deviation
5.2372
Mean
18
For this set of 8 numbers, the mean is 18.00 and the sample standard deviation is 5.2372 — meaning individual values typically differ from the mean by about 5.24.
Count (n)
8
Variance
27.4286
Sum
144
Min / Max
10 / 23
Coefficient of Variation
29.10%
Empirical Rule Ranges for Your Data
| Range | Low | High | Expected Share (if normal) |
|---|---|---|---|
| Mean ± 1σ | 12.76 | 23.24 | ~68% |
| Mean ± 2σ | 7.53 | 28.47 | ~95% |
| Mean ± 3σ | 2.29 | 33.71 | ~99.7% |
These ranges only reflect real proportions of your data if it's roughly normally distributed — see the empirical rule section below.
What is Standard Deviation?
Standard deviation measures variation, or dispersion, between values in a data set — how spread out the numbers are from their average. The lower the standard deviation, the closer individual data points sit to the mean; the higher it is, the wider the spread of values. It's typically written as σ (sigma) for a population and s for a sample.
Two data sets can have identical averages and still tell very different stories once standard deviation is factored in — this calculator shows both the raw statistics and what they mean in plain terms for your specific numbers.
Population vs. Sample Standard Deviation
Population standard deviation (σ) applies when you can measure every member of the group you care about — every value that exists, not just a subset.
Sample standard deviation (s) applies when it's not possible or practical to measure every member of a population, so you work from a representative sample instead. The only formula difference is the denominator: N − 1 instead of N — known as Bessel's correction, which removes some of the bias that comes from estimating a population's spread from a smaller sample. This "corrected sample standard deviation" is the most commonly used estimator for population standard deviation, though it still carries meaningful bias for very small samples (fewer than about 10 values).
For example, for the data set 1, 3, 4, 7, 8: the mean is 4.6, and the population standard deviation works out to σ = √[(1−4.6)² + (3−4.6)² + (4−4.6)² + (7−4.6)² + (8−4.6)²] / 5 ≈ 2.577.
Real-World Uses: Quality Control, Weather, and Finance
In manufacturing quality control, standard deviation helps establish a minimum and maximum value within which some product measurement should fall a high percentage of the time — a tighter standard deviation means a more consistent, predictable process.
In climate data, two cities can share an identical average temperature of 75°F yet feel completely different to live in: a coastal city might swing only between 60–85°F year-round (low standard deviation), while an inland city swings between 30–110°F (high standard deviation) — the average alone hides that difference entirely.
In finance, standard deviation is a standard measure of risk in an asset's returns. Two stocks might both average a 7% annual return, but one with a 10% standard deviation is far more predictable than one with a 50% standard deviation — the second is much more likely to post a dramatically better, or dramatically worse, year than its average suggests.
The Empirical Rule (68-95-99.7)
For data that follows a roughly normal (bell-curve) distribution, standard deviation predicts exactly how values cluster around the mean: about 68% of values fall within one standard deviation of the mean, about 95% fall within two, and about 99.7% fall within three. This is why standard deviation is so useful for spotting outliers — a value more than 2–3 standard deviations from the mean is unusual enough to warrant a closer look in most normally distributed data.
This rule doesn't hold for every data set — skewed or multi-peaked distributions won't follow these percentages closely. The empirical rule ranges shown above are computed directly from your entered numbers' mean and standard deviation, but treat the "expected share" column as a reference for roughly bell-shaped data, not a guarantee for any arbitrary data set.
Example — Your Current Data
For this set of 8 numbers, the mean is 18.00 and the sample standard deviation is 5.2372 — meaning individual values typically differ from the mean by about 5.24.
Additional Example — Two Classrooms, Same Average
Classroom A's test scores: 78, 80, 82, 79, 81 — mean 80, sample standard deviation about 1.58. Classroom B's test scores: 50, 95, 100, 60, 95 — mean also 80, but sample standard deviation about 22.5. Both classes averaged exactly the same score, but Classroom A's students performed almost uniformly, while Classroom B had a huge spread between struggling and excelling students — information the average alone completely hides.
A teacher looking only at the class average would conclude both classes are performing identically. Standard deviation reveals that Classroom B likely needs a very different teaching approach — probably differentiated instruction — while Classroom A's students are all roughly on the same page.
About These Parameters
- Numbers
- Your full data set, in any order — separated by commas, spaces, or new lines. At least two values are required, since a single value has no variation to measure.
- Data Type (Sample vs. Population)
- Choose Sample if your numbers are a subset representing a larger group you haven't fully measured — this is the far more common case, and what most statistical software defaults to. Choose Population only when your numbers are every single member of the group you're studying, with nothing left out.
Frequently Asked Questions
Should I use sample or population standard deviation?
Use sample standard deviation unless you are certain your data set represents every single member of the group you're studying. In practice, sample standard deviation is the correct choice the vast majority of the time — for example, test scores from one class are a sample of that student's overall ability, not the entire population of possible scores.
What's the difference between variance and standard deviation?
Variance is the average of the squared differences from the mean; standard deviation is simply the square root of variance. Standard deviation is used far more often in practice because it's expressed in the same units as your original data (variance is in squared units), which makes it much easier to interpret directly against the mean.
What counts as a "high" or "low" standard deviation?
There's no universal threshold — it depends entirely on the scale and context of your data. The coefficient of variation (standard deviation divided by the mean, shown as a percentage above) is often more useful for comparing variability across different data sets or units, since it's scale-independent. A coefficient of variation under about 15% is generally considered low variability; above 30% is generally considered high, though this varies by field.
Why does sample standard deviation divide by N-1 instead of N?
Using the sample mean (rather than the true, usually unknown, population mean) to compute deviations slightly underestimates the true spread, because the sample mean is calculated to minimize those very deviations. Dividing by N − 1 instead of N — Bessel's correction — inflates the result just enough to correct for that bias, producing a better estimate of the true population standard deviation from a limited sample.