Standard Deviation Using the Mean: Formula and Example

This calculator finds the standard deviation of a list of values using their mean, showing how far the values typically lie from that average. Enter your numbers as a comma-separated list to get the sample standard deviation, along with the count, mean, and range.

Formula

For a sample of n values, the sample standard deviation is the square root of the sum of the squared differences from the mean, divided by n − 1:

s = √[Σ(xᵢ − x̄)² / (n − 1)]

• xᵢ is an individual observation, in the same units as the data.
• x̄ is the arithmetic mean, calculated as Σxᵢ / n, in those same units.
• n is the number of observations, a count.
• s is the sample standard deviation, in the same units as the observations.
• Σ means to add the indicated values across all observations.

The calculator uses the sample formula. It requires at least two values. For a complete population rather than a sample, divide the squared deviations by N instead of n − 1; that is the population standard deviation, often written σ.

Worked Example

Using the calculator’s default values, 2, 4, 4, 4, 5, 5, 7, and 9:

1. There are n = 8 observations. Their sum is 40, so the mean is x̄ = 40 / 8 = 5.
2. Subtract 5 from each value and square the differences: 9, 1, 1, 1, 0, 0, 4, and 16.
3. Add the squared differences: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
4. For a sample, divide by n − 1 = 7: the sample variance is 32 / 7 ≈ 4.5714.
5. Take the square root: s = √(32 / 7) ≈ 2.27.

The calculator therefore returns a sample standard deviation of 2.27 units, a mean of 5 units, 8 values, and a range of 9 − 2 = 7 units. The unit is called “units” because the example has no specified measurement scale; if the values represent centimeters, for instance, the standard deviation is 2.27 centimeters.

Why Standard Deviation Uses the Mean

The mean gives a single central reference point for the data. Standard deviation measures dispersion around that point: it finds each value’s difference from the mean, squares those differences so negative and positive deviations cannot cancel, averages the squared differences, and takes a square root.

Squaring gives more influence to large deviations than small ones. The final square root changes the result back into the original data units. Variance, by contrast, remains in squared units, such as square centimeters or square points. Standard deviation is often easier to interpret because it can be compared directly with the scale of the observations.

When to Use Sample or Population Standard Deviation

Use sample standard deviation when your observations are a subset intended to represent a larger group. The denominator n − 1 adjusts the calculation because the sample mean was estimated from those same observations. This is commonly called Bessel’s correction.

Use population standard deviation when you have every member of the group you want to describe and are summarizing that group itself. Its formula is σ = √[Σ(xᵢ − μ)² / N], where μ is the population mean and N is the population count. The two answers differ because the denominators differ; the sample result is usually larger for the same data. Choose based on what the data represent, not on which answer seems preferable.

How to Interpret the Result

A standard deviation of 2.27 units means the data have a typical spread around the mean on a scale of about 2.27 units. It is not the maximum distance from the mean, nor does it say that every value is within 2.27 units. In the example, the mean is 5 and the values run from 2 to 9, so observations can be farther away than one standard deviation.

For data that are roughly symmetric and bell-shaped, a common rule of thumb is that about 68% of observations fall within one standard deviation of the mean, and about 95% within two. This is an approximate pattern, not a guarantee, and it does not generally apply to strongly skewed data or distributions with unusual shapes. Standard deviation by itself also cannot reveal whether the data are skewed, clustered, or contain outliers.

Common Calculation Mistakes and Limits

• Mixing up the denominator: dividing by n gives the population formula; dividing by n − 1 gives the sample formula.
• Skipping the square root: the mean of squared deviations is variance, not standard deviation.
• Rounding too early: keep the full mean and squared-deviation values through the calculation, then round the final result. Early rounding can change the answer.
• Combining incompatible units: do not calculate one standard deviation from values such as meters and feet unless you first convert them to a common unit.
• Using it with too little information: one observation is not enough to calculate a sample standard deviation. Even with many values, the result can be strongly affected by extreme observations.

Standard deviation is most informative when numerical differences are meaningful, as with lengths, test scores, or measurements. It is less suitable for categories without a numerical scale. For data with major outliers or a pronounced skew, also consider the median and interquartile range, which describe the center and spread differently and are less sensitive to extreme values.

Frequently Asked Questions

How do you calculate standard deviation from the mean?
Subtract the mean from each value, square each difference, add the squares, divide by n − 1 for a sample or N for a population, and take the square root.

Why do you subtract the mean when finding standard deviation?
The differences show how far each observation is from the data’s average. Squaring them makes all contributions nonnegative and makes larger departures count more.

Is standard deviation the same as variance?
No. Variance is the mean squared deviation, using the appropriate sample or population denominator. Standard deviation is the square root of that variance and has the original data units.

Should I divide by n or n − 1?
Divide by n − 1 when calculating the sample standard deviation to estimate spread in a larger population. Divide by N when describing the entire population you have measured.

Can standard deviation be negative?
No. Squared differences cannot be negative, and their square root is nonnegative. A standard deviation of zero means all recorded values are identical.

Related

• Probability for Independent Events: Formula and Examples
• Two Dice Probability: Calculate Sums and Outcomes
• Precision From a Confusion Matrix: Formula and Example

list values | Data values | 2, 4, 4, 4, 5, 5, 7, 9
out sd | Sample standard deviation | units | sd | 2
out avg | Mean | units | mean | 2
out n | Number of values | - | count | 0
out spread | Range | units | range | 2