Standard Deviation Calculator

About the Standard Deviation Calculator

Standard deviation measures how spread out data values are around their mean. A small standard deviation means the values sit close to the average; a large one means they are widely scattered.

Use the population formula when your data includes every member of the group you care about, and the sample formula, which divides by $n-1$ , when the data is a sample used to estimate a larger population.

What does standard deviation measure?

Standard deviation measures how spread out a set of numbers is around its mean. A small standard deviation means the values sit close to the average; a large one means they are scattered widely. Two classes can both average 70 on a test, but if one class has a standard deviation of 3 and the other of 15, the first group performed consistently while the second ranged from struggling to excellent.

It is computed from the deviations, the distances of each value from the mean. Those deviations always add to zero, so they are squared before being averaged, which gives the variance. Taking the square root of the variance brings the result back to the original units, so a standard deviation of heights in centimetres is itself in centimetres.

How to use the Standard Deviation Calculator

Enter or paste the numbers separated by commas, spaces or line breaks.
Choose Sample or Population.
Press Calculate to see the mean, the squared deviations, the variance and the standard deviation.

Formulas

Population standard deviation

\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}

Sample standard deviation

s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}

Variance

\sigma^2,\ s^2 = (\text{standard deviation})^2

Worked example: 4, 8, 6, 5, 3, 7 (sample)

The worked example treats 4, 8, 6, 5, 3, 7 as a sample. The mean is $\frac{33}{6} = 5.5$ . The deviations are $-1.5, 2.5, 0.5, -0.5, -2.5, 1.5$ , and their squares add up to 17.5. Dividing by $n - 1 = 5$ gives the sample variance 3.5, and its square root is the sample standard deviation $s \approx 1.870828693$ . For comparison, treating the same six numbers as a whole population divides by 6 instead and gives $\sigma \approx 1.707825128$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

s = 1.870828693

How we read your problem

Treat the data as a sample from a larger population (Bessel's correction, divide by n − 1).

Step-by-step solution

Find the mean
$\bar{x} = \frac{33}{6} = 5.5$
Square each deviation from the mean
$(4 - 5.5)^2 = 2.25 \\ (8 - 5.5)^2 = 6.25 \\ (6 - 5.5)^2 = 0.25 \\ (5 - 5.5)^2 = 0.25 \\ (3 - 5.5)^2 = 6.25 \\ (7 - 5.5)^2 = 2.25$
Add the squared deviations
$\sum (x_i - \bar{x})^2 = 17.5$
Divide by n − 1 to get the sample variance
$s^2 = \frac{17.5}{5} = 3.5$
Take the square root
$s = \sqrt{3.5} = 1.870828693$

Formulas used

Sample standard deviation
$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$
Population standard deviation
$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}$

Standard deviation measures how spread out the values are around the mean. A small value means the data are clustered near the mean.

Verification

Variance = 3.5, and 1.870828693² = 3.5
Population SD for comparison: 1.707825128

Detailed explanation

Sample versus population in practice

The two formulas differ only in the divisor, but the choice matters for small data sets. If the numbers are every member of the group you care about, such as the scores of all 25 students in one class, use the population formula with $N$ . If they are a sample used to estimate the spread of a larger group, such as 25 students chosen from a whole school, use the sample formula with $n - 1$ . The second example in the calculator, 2, 4, 4, 4, 5, 5, 7, 9 as a population, has mean 5 and squared deviations summing to 32, so $\sigma = \sqrt{\frac{32}{8}} = 2$ .

Interpreting the number

For data that are roughly bell-shaped, the empirical rule says about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three. A value more than two standard deviations from the mean is therefore unusual. Dividing a value's distance from the mean by the standard deviation gives its z-score, which lets you compare results measured on different scales, such as a test score and a race time.

Standard deviation is sensitive to outliers because the deviations are squared, so one extreme value can inflate it considerably. For strongly skewed data the interquartile range is often a better summary of spread.

Common mistakes

Using $n$ instead of $n - 1$ for a sample, which underestimates the spread.
Forgetting to square the deviations, so positive and negative ones cancel to zero.
Reporting the variance when the question asks for the standard deviation.
Rounding the mean before computing the deviations, which shifts every squared term.
Forgetting the final square root and reporting the variance as the standard deviation.

Tips

For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean.
Standard deviation has the same units as the data; variance has squared units.
Adding a constant to every value leaves the standard deviation unchanged.
Add a constant to every value and the standard deviation does not change; multiply every value by $k$ and it is multiplied by $|k|$ .
Use the z-score calculator to see how far a single value sits from the mean in standard deviations.

Frequently asked questions

What is standard deviation?

It is the square root of the average squared distance from the mean. It tells you how far a typical value lies from the average.

Should I use sample or population?

Use population when you have data for the entire group. Use sample when your data is a subset used to describe a larger group; this is the more common case.

Why does the sample formula divide by n − 1?

The sample mean is closer to the sample values than the true mean is, which shrinks the deviations. Dividing by $n-1$ (Bessel's correction) compensates and gives an unbiased variance estimate.

Can the standard deviation be zero?

Yes, exactly when every value is the same. It can never be negative.

How many values do I need?

One value is enough for a population standard deviation (it is 0). The sample formula needs at least two values.

What is the difference between variance and standard deviation?

Variance is the mean of the squared deviations; standard deviation is its square root, which brings the measure back to the units of the data.

What is a good or high standard deviation?

There is no universal threshold; it depends on the scale of the data. Compare it with the mean, or use the coefficient of variation $\frac{s}{\bar x}$ , to judge whether the spread is large relative to typical values.

How does an outlier affect the standard deviation?

Strongly, because each deviation is squared. Replacing 7 by 70 in the example raises the sample standard deviation from about 1.87 to more than 26.

Find the spread of your data

Paste your numbers above, choose sample or population, and get the mean, variance and standard deviation with every step.

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