Enter any dataset to instantly compute mean, sample standard deviation, population standard deviation, and variance — with step-by-step calculations.
Separate numbers with commas, spaces, or line breaks. Example: 10, 12, 23, 23, 16, 23, 21, 16
Observations
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Mean (x̄)
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Sample SD (s)
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Population SD (σ)
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Sample Variance (s²)
÷ (n−1)—
Population Variance (σ²)
÷ N—
Tip: Use population standard deviation when you have data for an entire group. Use sample standard deviation when your data is a subset of a larger population.
Standard deviation is a number that tells you how spread out your data is around the average. Think of it as the "typical distance" between each data point and the mean. A small standard deviation means your values are clustered tightly around the average. A large standard deviation means they're scattered widely.
Along with the mean, standard deviation is one of the two most important numbers in statistics. While the mean tells you where your data is centered, the standard deviation tells you how spread out it is [citation:1]. Both are needed to truly understand a dataset.
Entering your dataset is simple:
You can enter negative numbers (like -5) and decimals (like 3.14). The calculator handles single-value datasets correctly too — the standard deviation will be 0 since there's no variation.
This is the most common point of confusion in statistics, so let's make it clear:
Use this when your dataset contains every single member of the group you're interested in. For example, if you want to know the standard deviation of test scores for every student in a class, that's a population. The formula divides by N (the total number of items) [citation:7].
Use this when your dataset is a subset of a larger group, and you want to estimate the standard deviation of the whole group. For example, if you survey 50 people out of a city of 100,000, you have a sample. The formula divides by n − 1 instead of n [citation:7].
Why the difference? Dividing by n − 1 makes the sample standard deviation slightly larger, which corrects for the fact that a sample tends to underestimate the true variability of the population. This adjustment is called Bessel's correction [citation:18]. With large datasets, the difference becomes negligible, but with small samples it matters [citation:7].
Rule of thumb: If you're unsure, use the sample standard deviation. It's the conservative choice and won't understate uncertainty [citation:7].
Here's the formula for population standard deviation:
σ = √( Σ(xᵢ − μ)² / N )
And for sample standard deviation:
s = √( Σ(xᵢ − x̄)² / (n − 1) )
Variance is simply the standard deviation before you take the square root. It's the average of the squared deviations from the mean [citation:8].
The relationship is straightforward:
Variance = (Standard Deviation)²
Standard Deviation = √Variance
So why use standard deviation instead of variance? Because variance is in squared units, which are hard to interpret. If your data is in meters, variance is in square meters — but standard deviation is in meters, just like your original data [citation:11]. That makes standard deviation much easier to understand and communicate.
| Situation | Use |
|---|---|
| You have data for every member of a group | Population SD (σ) |
| You have a sample and want to estimate the population | Sample SD (s) |
| You're unsure which applies | Sample SD (s) — safer choice |
| Large dataset (100+ values) | Difference is negligible; either works |
Let's calculate the standard deviation for the dataset: 4, 8, 6, 5, 3
(4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2
| Value | Deviation (x − 5.2) | Squared |
|---|---|---|
| 4 | −1.2 | 1.44 |
| 8 | 2.8 | 7.84 |
| 6 | 0.8 | 0.64 |
| 5 | −0.2 | 0.04 |
| 3 | −2.2 | 4.84 |
1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.80
Population variance: 14.80 / 5 = 2.96
Population SD: √2.96 ≈ 1.72
Sample variance: 14.80 / 4 = 3.70
Sample SD: √3.70 ≈ 1.92
In a roughly normal (bell-shaped) distribution [citation:6][citation:8]:
This is why standard deviation is so useful — it gives you a concrete way to talk about how unusual or typical a particular value is.
If you need more statistical tools, try these free calculators:
Population standard deviation (σ) is used when your dataset contains every member of the group you're studying. Sample standard deviation (s) is used when you have a subset of a larger group and want to estimate the whole population's variability. The only mathematical difference is that sample SD divides by n−1 instead of N [citation:7]. This makes the sample SD slightly larger, correcting for the bias of estimating from a subset.
Use the sample standard deviation. It's the conservative choice because it won't understate the uncertainty in your data. In most real-world situations, you're working with a sample rather than an entire population [citation:7].
No. Standard deviation is always zero or positive. It's a square root, and the squared deviations inside are always non-negative [citation:5]. The minimum possible value is 0, which means all data points are identical.
The standard deviation is 0. With only one value, there's no variation — the single value is the mean. This is mathematically correct and not an error.
Variance is the square of standard deviation. If you know the variance, take its square root to get the standard deviation. Variance is harder to interpret directly because it's in squared units, but it's mathematically convenient for many statistical procedures [citation:8].
Yes. Standard deviation has the same units as your original data. If your data is in dollars, standard deviation is in dollars. If it's in kilograms, standard deviation is in kilograms [citation:1]. This is one advantage standard deviation has over variance.