What Standard Deviation Measures
Standard deviation describes how spread out a set of numbers is. A small value means the data cluster tightly around the mean; a large value means they are scattered widely. It is expressed in the same units as the original data, which is what makes it more interpretable than variance.
Two data sets can share an identical mean and be entirely different. {10, 10, 10} and {0, 10, 20} both average 10, but the first has a standard deviation of 0 and the second of about 8.2. The mean alone tells you almost nothing without a measure of spread beside it.
The Formulas
Population: σ = √( Σ(xi − μ)2 ÷ N )
Sample: s = √( Σ(xi − x̄)2 ÷ (n − 1) )
The procedure is the same either way:
- Find the mean.
- Subtract the mean from each value to get the deviations.
- Square each deviation — this removes the signs, which would otherwise cancel to zero.
- Sum the squared deviations.
- Divide by N for a population, or by n − 1 for a sample. This is the variance.
- Take the square root, returning to the original units.
Why n − 1 for a Sample?
This is Bessel's correction, and it exists because a sample's own mean is always closer to its own data than the true population mean is. Deviations measured from the sample mean are therefore systematically too small, and dividing by n would under-estimate the population's spread.
Dividing by n − 1 inflates the result just enough to correct the bias. The intuition is about degrees of freedom: once the mean is known, only n − 1 of the values are free to vary, since the last one is determined by the requirement that the deviations sum to zero.
| Use | Divide by | When |
|---|---|---|
| Population (σ) | N | You have every member of the group you care about |
| Sample (s) | n − 1 | You have a subset and want to infer about the whole |
The correction matters most for small samples. With n = 5 it changes the result by about 12%; with n = 1,000 by about 0.05%. When in doubt, use the sample version — most real data is a sample.
Variance vs. Standard Deviation
Variance is the standard deviation squared. It has better mathematical properties — variances of independent variables add, standard deviations do not — so it dominates the theory. But its units are squared, which makes it unintuitive: the variance of a set of heights in centimetres is in square centimetres. Standard deviation is the version you report.
The 68-95-99.7 Rule
For data that follows a normal distribution:
| Interval | Proportion of data | Roughly |
|---|---|---|
| mean ± 1σ | 68.27% | 2 in 3 |
| mean ± 2σ | 95.45% | 19 in 20 |
| mean ± 3σ | 99.73% | 369 in 370 |
The calculator reports how much of your data actually falls in each band. A large discrepancy from these figures is a sign the data are not normally distributed — skewed, bimodal, or dominated by outliers.
For any distribution at all, Chebyshev's inequality gives a weaker guarantee: at least 75% of data falls within 2σ and at least 89% within 3σ, regardless of shape.
Standard Deviation vs. Standard Error
These are constantly confused. Standard deviation measures the spread of the data. Standard error measures the precision of the mean, and shrinks as the sample grows:
SE = s ÷ √n
Collecting more data does not reduce the standard deviation — people's heights vary by the same amount however many you measure — but it does reduce the standard error, because the average of a larger sample is a more reliable estimate. The √n means quadrupling the sample halves the standard error.
The Coefficient of Variation
A standard deviation of 5 is large for data averaging 10 and trivial for data averaging 10,000. The coefficient of variation expresses spread relative to the mean:
CV = s ÷ |mean| × 100%
This makes data sets with different units or scales comparable. It is meaningless when the mean is near zero, and undefined when it is zero.
Where It Is Used
- Finance — standard deviation of returns is the standard definition of volatility and the basis of risk measures such as the Sharpe ratio.
- Manufacturing — Six Sigma aims to fit specification limits six standard deviations from the mean, giving roughly 3.4 defects per million.
- Testing and grading — standardised test scores are reported as deviations from a mean, and curves are frequently set in standard-deviation units.
- Science — error bars, confidence intervals and significance thresholds are all expressed through it. Particle physics requires 5σ before claiming a discovery.
- Weather and climate — anomalies are reported relative to the historical standard deviation.
Limitations
- Outliers dominate it. Because deviations are squared, one extreme value can double the standard deviation. The interquartile range is more robust when outliers are present.
- It assumes symmetry is meaningful. For heavily skewed data such as income, the mean and standard deviation both mislead; median and IQR describe the distribution better.
- It says nothing about shape. Very different distributions can share a mean and a standard deviation. Always look at the data as well as the summary.
Frequently Asked Questions
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is always zero or positive. Zero means every value is identical.
Which should I use, sample or population?
Sample, unless you genuinely have every member of the population — every student in one specific class, every product in one specific batch. Data collected to learn about a wider group is a sample.
What is a "good" standard deviation?
There is no universal answer; it depends entirely on context and units. Compare it to the mean using the coefficient of variation, or against a known benchmark for the same measurement.
How do I find standard deviation in a spreadsheet?
In Excel and Google Sheets, STDEV.S for a sample and STDEV.P for a population. The older STDEV is the sample version.