The standard deviation is the square root of the variance. They carry exactly the same information about how spread out a dataset is; the variance is expressed in squared units and the standard deviation in the units the data was measured in.
The one-line answer
If you have one, you have the other. Square a standard deviation and you get the variance; take the square root of a variance and you get the standard deviation. A dataset with a variance of 7 has a standard deviation of √7 ≈ 2.6458, and there is nothing further to learn from either number that the other does not already tell you.
The difference is entirely about presentation and arithmetic convenience. The standard deviation is the one you read, plot and report. The variance is the one you calculate with. Everything below expands on that division of labour.
Enter your own numbers to see both computed side by side, with the sum of squares that feeds them.
Separate values with commas, spaces, tabs or new lines — paste a column straight from a spreadsheet and it will parse. Decimals and negatives are fine.
The shaded bands are one, two and three standard deviations either side of the mean. 5 of 7 values — 71% — fall inside the innermost band.
Show the working, step by step
Side by side
| Variance | Standard deviation | |
|---|---|---|
| Symbol | s² (sample), σ² (population) | s (sample), σ (population) |
| Formula | Σ(x − x̄)² / (n − 1) | √[Σ(x − x̄)² / (n − 1)] |
| Units | Data units squared (kg², s², $²) | Same as the data (kg, s, $) |
| Typical use | Theory, derivations, ANOVA, combining errors | Reporting, error bars, z-scores, the 68–95–99.7 rule |
| Sensitivity to outliers | High — a point's contribution grows with the square of its distance | High, but the square root softens the headline number |
| Adds for independent variables | Yes: Var(X + Y) = Var(X) + Var(Y) | No: SD(X + Y) = √(SD(X)² + SD(Y)²) |
| Effect of doubling every value | Multiplied by 4 | Multiplied by 2 |
| Can be negative? | Never | Never |
The two "sensitivity" entries deserve a caveat. Because one is a monotonic function of the other, an outlier that inflates the variance inflates the standard deviation too, and by an equivalent amount in the sense that matters. The difference is only that the variance's raw number moves further, which can mislead if you are eyeballing figures rather than reasoning about them.
The formulas
For a sample of n values with mean x̄:
s² = Σ(x − x̄)² / (n − 1) s = √[Σ(x − x̄)² / (n − 1)]
For a complete population of N values with mean μ:
σ² = Σ(x − μ)² / N σ = √[Σ(x − μ)² / N]
Read left to right, the two lines of each pair are identical until the final step. Compute the deviations, square them, add them up, divide — that is the variance. Take the square root — that is the standard deviation. Whether you divide by n − 1 or N is a separate question about sample versus population, and it applies to both quantities in the same way.
A worked example
Take the seven values loaded in the calculator above: 6, 9, 11, 14, 8, 12, 10.
They sum to 70, so the mean is 70 ÷ 7 = 10.
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 6 | −4 | 16 |
| 9 | −1 | 1 |
| 11 | 1 | 1 |
| 14 | 4 | 16 |
| 8 | −2 | 4 |
| 12 | 2 | 4 |
| 10 | 0 | 0 |
| Σ | 0 | 42 |
The deviations sum to zero, as they always do, which is why the middle column is useless as a measure of spread and the squares are needed. The sum of squares is 42.
Sample variance s² = 42 / (7 − 1) = 42 / 6 = 7 Sample SD s = √7 = 2.6458 Population variance σ² = 42 / 7 = 6 Population SD σ = √6 = 2.4495
Notice what the square root does to the interpretation. "Variance 7" tells you little on its own. "Standard deviation 2.65 around a mean of 10" tells you at once that a typical value sits within about 2.65 of 10, that 14 is roughly one and a half standard deviations above the mean, and that a value of 20 would be extraordinary.
Converting between them
The conversion is a single operation in each direction:
standard deviation = √variance variance = standard deviation²
| Standard deviation | Variance |
|---|---|
| 0.5 | 0.25 |
| 1 | 1 |
| 1.5 | 2.25 |
| 2 | 4 |
| 5 | 25 |
| 10 | 100 |
Two things follow from the table. First, the variance is not "bigger" than the standard deviation in any general sense: below an SD of 1 it is smaller, and which side of 1 you land on depends on nothing more profound than whether you measured in metres or centimetres. Second, the conversion preserves the sample-or-population choice. The square root of a sample variance is a sample standard deviation; you never need to revisit the n − 1 when converting.
One trap: convert single values, not averages. The average of several standard deviations is not the square root of the average of their variances, so if you are combining groups, work in variance throughout and take the root at the very end. Pooled standard deviation is exactly that procedure.
Why variance exists at all
If the standard deviation is easier to read, why not drop the variance entirely? Because variances have an algebraic property that standard deviations lack: for independent quantities, they add.
Var(X + Y) = Var(X) + Var(Y) (X and Y independent)
Suppose a length measurement has two independent sources of error. The instrument itself has a standard deviation of 3 mm, and the way the part is positioned adds a further 4 mm of standard deviation. The overall standard deviation is not 3 + 4 = 7 mm. The variances are 9 mm² and 16 mm², they add to 25 mm², and the overall standard deviation is √25 = 5 mm.
The two errors are just as likely to partially cancel as to compound, which is why the combined spread is less than the sum. Variance captures that automatically; standard deviation does not. The same additivity is what lets analysis of variance split total variation into between-group and within-group parts, lets a portfolio's risk be built from the risks of its holdings, and gives the standard error of the mean its √n. All of that theory is written in variance, and the standard deviation is taken at the last line for the reader's benefit.
Variance is also the natural quantity in the mathematics. It is the second central moment of a distribution, it is what least-squares regression minimises, and it is the parameter that appears inside the normal distribution's formula. The square root is a convenience applied afterwards, not something the theory needs.
Why the standard deviation is what you report
Three reasons, all about interpretation.
It shares units with the data. A mean weight of 70 kg with a standard deviation of 4 kg can be read directly: most people in the group are within a few kilograms of 70. The equivalent variance, 16 kg², is not a quantity anyone has an intuition for.
It plugs into the 68–95–99.7 rule. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. That rule is stated in standard deviations because those are distances along the same axis as the data. There is no equivalent rule of thumb in variance.
It scales the way the data scales. Double every measurement and the standard deviation doubles, which is what a measure of "typical distance from the mean" should do. The variance quadruples, which is correct but not what a reader expects from a summary statistic.
When to use which
- Describing a dataset to a reader — standard deviation, alongside the mean.
- Error bars, control charts, z-scores, the empirical rule — standard deviation.
- Combining independent sources of variation — add the variances, then take the root.
- ANOVA, regression, hypothesis tests — the machinery runs on variance; the software reports both.
- Comparing spread across datasets in different units — neither; use the coefficient of variation.
- Data with outliers or heavy skew — consider the interquartile range instead of either.
If in doubt, compute the variance, report the standard deviation. That single habit covers almost every case.
Related quantities
The standard error is the standard deviation of a sample mean, rather than of the individual values: s ÷ √n. For the example above it is 2.6458 ÷ √7 = 1.0. It answers "how precisely have I estimated the mean?", which is a different question from "how spread out is the data?", and confusing the two is one of the commonest errors in reporting. Standard error calculator.
The coefficient of variation divides the standard deviation by the mean and is usually quoted as a percentage. For the example it is 2.6458 ÷ 10 = 26.5%. Because it is unit-free it lets you compare the relative spread of, say, heights in centimetres with weights in kilograms — something neither the SD nor the variance can do. Coefficient of variation calculator.
The mean absolute deviation skips the squaring altogether and averages the absolute distances from the mean. For the example the absolute deviations are 4, 1, 1, 4, 2, 2, 0, and their mean is 14 ÷ 7 = 2.0 — a little below the standard deviation, as it generally is. It is easier to explain but lacks the additivity that makes variance useful, which is why it never displaced the standard deviation. Mean absolute deviation calculator.
Related calculators
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Standard deviation calculator
Both statistics from the same input, with every step shown.
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Variance calculator
The variance page: formulas, a worked example and the Excel functions.
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The formula explained
Each symbol in the SD and variance formulas, taken apart.
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Sample vs population
The n − 1 question, which applies to both quantities equally.
Common questions
What is the difference between standard deviation and variance?
The standard deviation is the square root of the variance. Both measure how far the values in a dataset sit from their mean; the variance does it in squared units and the standard deviation converts back to the original units. Neither contains any information the other lacks — if you know one, you know the other.
How do I calculate standard deviation from variance?
Take the square root. A variance of 7 gives a standard deviation of √7 ≈ 2.6458; a variance of 2.25 gives a standard deviation of 1.5. Going the other way, square the standard deviation: an SD of 5 is a variance of 25.
The conversion does not care whether the figure was a sample or population value — it only matters that you keep the same kind on both sides.
Which is bigger, the variance or the standard deviation?
It depends on whether the standard deviation is above or below 1. If the SD is greater than 1, the variance is larger (SD 5 → variance 25). If the SD is less than 1, the variance is smaller (SD 0.5 → variance 0.25). They are equal only when the SD is exactly 1 or exactly 0. Since the comparison changes with the units you measure in, it is not a meaningful comparison — the two numbers are in different units.
Why take the square root at all?
To get back to the units the data was recorded in. Squaring the deviations is necessary to stop positive and negative deviations cancelling, but it leaves a result in kg², cm² or dollars², which cannot be compared with the mean or with any individual value. The square root undoes the change of units while keeping the ordering: a larger variance always means a larger standard deviation.
Can the variance be negative?
No. It is an average of squared deviations and every square is zero or positive, so the variance is zero or positive. A variance of exactly zero means every value is identical. If software reports a small negative variance, it has used the Σx² − (Σx)²/n shortcut and lost precision in the subtraction — a computational artefact, not a real result.
What units are variance and standard deviation in?
The standard deviation is in the same units as the data: seconds, grams, percentage points. The variance is in those units squared: seconds², grams², and — awkwardly — percentage points squared. That unit mismatch is the practical reason the standard deviation is the one that appears in reports, charts and error bars.