Why use this calculator
An average on its own hides the thing that usually matters. Two production lines can both average 500 g a pack while one fills every pack between 498 g and 502 g and the other swings from 460 g to 540 g, and only the second one will fail an inspection.
Standard deviation is the number that separates them, and it is the input to control limits, confidence intervals, process capability indices and almost every significance test. This calculator also settles the question people get wrong most often: whether to divide by n or by n − 1.
It shows both, labels which one applies to your situation, and reports the standard error so you can see how much your sample mean itself could move if you collected the data again.
How to use it
- Paste your numbers into the data field separated by commas, spaces or line breaks, straight from a spreadsheet column, a lab notebook or a meter log; text that is not a number is ignored rather than counted as zero.
- Then set whether the numbers are a sample or the whole population, because that single choice changes the divisor and therefore the answer.
- Choose sample whenever the numbers are a subset that you want to generalise from, which covers most surveys, test batches and experiments.
- Choose population only when the list is genuinely everything, such as the marks of all thirty students in one class when the class is all you care about.
What each result means
- Standard deviation
- Typical distance between a single value and the mean, in the same units as your data.
- Variance
- The standard deviation squared. It adds neatly across independent sources of variation, which is why theory uses it.
- Mean
- The arithmetic average of every value you entered, and the point the deviations are measured from.
- Count (n)
- How many numbers were successfully read from your data, after ignoring any text.
- Sum
- The total of all values, shown so you can check nothing was dropped or double-counted.
- Range
- Largest value minus smallest, with both endpoints. Useful as a sanity check but very sensitive to outliers.
- Standard error of the mean
- How much the sample mean itself would bounce around if you repeated the sampling.
- Coefficient of variation
- Standard deviation as a percentage of the mean, letting you compare spread across different scales.
- Sample standard deviation
- The n − 1 version, used when these numbers are a sample of something larger.
- Population standard deviation
- The n version, correct only when you have measured every member of the group.
- Typical range (mean ± 1 SD)
- Roughly two thirds of a normally distributed data set falls inside this band.
The formula
The mean
x̄ = Σx / n
The sum of squares
SS = Σ(x − x̄)²
Population variance
σ² = SS / N
Population standard deviation
σ = √(SS / N)
Sample variance
s² = SS / (n − 1)
Sample standard deviation
s = √(SS / (n − 1))
The standard error of the mean
SEM = s / √n
The coefficient of variation
CV = s / |x̄| expressed as a percentage
Units follow the data: the standard deviation carries the same unit as the values, while the variance carries that unit squared.
How this number is used in practice
In manufacturing the standard deviation is the process itself. A quality engineer measuring fill weights computes the mean and standard deviation, then builds control limits three standard deviations either side. Points outside those limits signal that something has changed: a worn nozzle, a new batch of material, a different operator.
Process capability indices such as Cp and Cpk are ratios of the specification width to six standard deviations, so a process can hold a perfect average and still fail because its spread is too wide. That is why inspectors ask for the standard deviation before they ask for the mean.
In laboratories and finance the same number is called precision or volatility. A chemist repeating an assay reports the mean plus or minus the standard deviation to show measurement precision, and a change in that spread usually points to an instrument problem rather than a real change in the sample.
An analyst annualises the standard deviation of daily returns to describe risk. In both cases the sample form with n − 1 is correct, because the data at hand is one draw from a much larger set of possible results, and using n instead would understate the true spread by a few per cent on small data sets.
Worked examples
Eight readings: 12, 15, 11, 18, 14, 16, 13, 17 treated as a sample
Mean 14.5000, sample standard deviation 2.4495, variance 6.0000.
The classic set 2, 4, 4, 4, 5, 5, 7, 9 as a full population
Mean 5.0000 and population standard deviation exactly 2.0000.
Five evenly spaced values 10, 20, 30, 40, 50 as a sample
Mean 30.0000, sample standard deviation 15.8114, population standard deviation 14.1421.
When people use it
- Setting statistical process control limits on a production line
- Reporting the precision of repeated laboratory measurements
- Describing the volatility of investment returns over a period
- Checking whether two groups differ by more than their natural spread
- Producing the standard error needed for a confidence interval
Tips
- Use n − 1 whenever the data is a sample. Dividing by n understates the spread, and the error grows as the data set gets smaller.
- The standard deviation carries the units of your data; the variance carries those units squared, which is why reports quote the standard deviation.
- About 68 per cent of a normal distribution lies within one standard deviation of the mean, 95 per cent within two and 99.7 per cent within three.
- Standard deviation is not robust: one mistyped digit can double it. Check the range and the largest squared deviation before you trust the answer.
- Compare spread across different scales with the coefficient of variation, not the raw standard deviation, since a large mean inflates the latter.
- Standard error shrinks with the square root of n, so quadrupling your sample size only halves the uncertainty in the mean.
What it assumes, and where it stops being accurate
- Every value is given equal weight. Weighted or frequency-grouped data needs a weighted formula instead.
- The data is a simple list of independent observations; clustered, paired or time-series data has structure this ignores.
- The interpretation of the 68 / 95 / 99.7 bands assumes an approximately normal distribution and is wrong for strongly skewed data.
- Outliers are included as entered. This is not a robust statistic, so a single bad reading distorts the result badly.
- Text that cannot be read as a number is skipped rather than treated as zero, which keeps the count honest but may hide a paste error.
Terms used on this page
- Variance
- The mean of the squared deviations from the mean; the square of the standard deviation.
- Bessel's correction
- Dividing a sample sum of squares by n − 1 instead of n to remove bias.
- Degrees of freedom
- The number of values free to vary once the mean is fixed, which is n − 1 for a sample.
- Coefficient of variation
- Standard deviation expressed as a percentage of the mean, for comparing across scales.
Questions people ask
- When do I use population instead of sample standard deviation?
- Use the population form only when your list contains every member of the group you care about, such as all the marks in one class where the class is the whole question. Use the sample form for anything you intend to generalise from, which covers surveys, test batches and experiments. When in doubt, the sample form is the safer choice.
- Why divide by n − 1 rather than n?
- Because the deviations are measured from the sample mean rather than from the unknown true mean, and the sample mean sits closer to its own data than the true mean does. That makes the sum of squares too small. Dividing by n − 1, known as Bessel's correction, removes the bias so the estimate is right on average.
- What does the standard deviation actually tell me?
- It is the typical distance between an individual value and the mean, in the same units as the data. A small standard deviation means values cluster tightly; a large one means they are scattered. It says nothing about whether the mean is correct or whether the distribution is symmetric, only about how widely the values spread.
- What is the difference between standard deviation and standard error?
- Standard deviation describes the spread of individual values. Standard error describes the spread of the sample mean itself across repeated samples, and equals the standard deviation divided by the square root of n. Confidence intervals for a mean are built from the standard error, not from the standard deviation.
- Can the standard deviation be negative?
- No. It is the square root of a sum of squared terms, so it is always zero or positive. A standard deviation of exactly zero means every value is identical. If software hands you a negative figure, it is reporting something else, most likely a difference between two standard deviations.
- How many data points do I need?
- Two is the mathematical minimum for a sample standard deviation, but estimates from fewer than about ten values are very unstable and can be out by a wide margin. Thirty or more gives a reasonably reliable figure. If your data set is small, report the count alongside the standard deviation so readers can judge it.
Use this with
Standard Deviation Calculator answers one part of the question. These pick up where it stops: Z-Score Calculator, Sample Size Calculator, Mean, Median, Mode Calculator and Probability Calculator.
Where these figures come from
This calculation is jurisdiction-neutral arithmetic. No country-specific rules are applied.
- NIST/SEMATECH e-Handbook of Statistical Methods. Reference treatment of descriptive statistics and hypothesis testing
- UK Office for National Statistics: methodology. Official guidance on survey design, sampling and uncertainty
Standard Deviation Calculator produces an estimate for general information. It is not financial, tax, medical, legal or engineering advice. read the full disclaimer. Spotted a wrong number? Report it and it gets fixed. The form picks up this calculator and the inputs you used.