Standard Deviation Formula Calculator

This calculator works out the standard deviation of a set of numbers you enter. Type or paste your values (comma-separated) and choose sample or population mode; you get the standard deviation, variance, and count instantly. The formula and a step-by-step example are explained below so you can verify the result by hand.

Formula

For a sample of n values x₁, x₂, …, x_n with mean x̄, the sample standard deviation s is:

s = √[ Σ(x_i − x̄)² / (n − 1) ]

For a population, the standard deviation σ uses n in the denominator instead of n − 1:

σ = √[ Σ(x_i − x̄)² / n ]

Where:

• x_i – each individual value in the dataset
• x̄ – arithmetic mean of all values
• n – number of values
• Σ – sum over all values

The variance is the square of the standard deviation (the expression inside the square root). The calculator uses the sample formula by default, which is the standard choice when your data is a random sample from a larger population.

Worked Example

Using the default values: 4, 8, 15, 16, 23, 42 (sample mode).

1. Count: n = 6
2. Mean: (4 + 8 + 15 + 16 + 23 + 42) / 6 = 108 / 6 = 18
3. Deviations: (4−18) = −14, (8−18) = −10, (15−18) = −3, (16−18) = −2, (23−18) = 5, (42−18) = 24
4. Squares: 196, 100, 9, 4, 25, 576
5. Sum of squares: 196 + 100 + 9 + 4 + 25 + 576 = 910
6. Sample variance: 910 / (6−1) = 182
7. Sample standard deviation: √182 ≈ 13.4907

The calculator shows 13.4907 (rounded to 4 decimals) for the default data, matching the hand calculation.

When to Use Sample vs. Population

Use the population formula when your data includes every member of the group you are studying. For example, the test scores of all 30 students in one class, or the daily high temperature for every day of a specific year. Use the sample formula when your data is a subset – e.g., 100 voters polled out of millions, or 5 lightbulbs tested from a factory batch. The sample formula uses n − 1 (Bessel’s correction) to give an unbiased estimate of the population standard deviation. If you are unsure, choose sample; it is the safer default for most real-world data.

How to Read the Result

The standard deviation tells you how spread out the numbers are around the mean. A low value (close to 0) means the data points cluster tightly; a high value means they are scattered widely. For roughly bell-shaped distributions, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. But this rule of thumb works poorly for skewed or multi-modal data, so always look at a histogram or boxplot alongside the number.

The units of the standard deviation are the same as the data itself. If your values are in meters, the standard deviation is in meters; the variance is in square meters. That is why standard deviation is usually more interpretable than variance.

Common Mistakes and Pitfalls

• Using sample formula on a full population – overestimates spread slightly; for small n the difference is noticeable.
• Forgetting to square deviations – negative deviations cancel out, giving zero if you sum them directly.
• Mixing units – if some values are in feet and others in meters, the result is meaningless. Convert everything first.
• Rounding too early – round only at the end; intermediate rounding can change the final answer noticeably.
• Using median instead of mean – the standard deviation is defined around the mean; using a different center changes the result.

Limitations and Alternatives

Standard deviation is sensitive to outliers – a single extreme value can inflate it dramatically. For skewed data, the interquartile range (IQR) or median absolute deviation (MAD) are more robust. Also, standard deviation assumes the mean is a meaningful measure of center; for data with multiple modes, it can be misleading. When comparing datasets with different means, the coefficient of variation (standard deviation divided by mean) is often more useful.

For grouped data (e.g., values given in frequency tables), the exact standard deviation requires the raw values; approximations can be made using class midpoints, but that introduces error. The calculator here works only with raw, ungrouped numbers.

Frequently Asked Questions

What is the difference between standard deviation and variance?

Variance is the average of the squared deviations from the mean. Standard deviation is the square root of the variance. The standard deviation is in the same units as the original data, making it easier to interpret; variance is in squared units.

Why do we use n-1 for sample standard deviation?

Using n − 1 (Bessel’s correction) makes the sample variance an unbiased estimator of the population variance. If you used n, the sample variance would tend to be slightly too low, especially for small samples.

Can standard deviation be negative?

No. Standard deviation is a square root of a sum of squares, so it is always zero or positive. A zero standard deviation means all values are identical.

How do I calculate standard deviation in Excel?

Use STDEV.S for sample data and STDEV.P for population data. For example, =STDEV.S(A1:A10) gives the sample standard deviation of the numbers in cells A1 through A10.

What does a standard deviation of 1 mean?

It means the typical distance of a data point from the mean is 1 unit. Whether that is large or small depends on the context – a standard deviation of 1 cm for heights is tiny, but for the length of a paperclip it is huge.

Is standard deviation the same as standard error?

No. Standard deviation measures the spread of individual data points. Standard error (of the mean) measures how much the sample mean varies from sample to sample; it equals the sample standard deviation divided by the square root of the sample size.

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in list | Data values (comma separated) | 4, 8, 15, 16, 23, 42 | 0
sel mode | Mode | Sample=1, Population=0
out sd | Standard deviation | - | sqrt(var) | 4
out variance | Variance | - | var | 4
out count | Count | - | count | 0