Standard deviation FAQ

It is the typical distance between the values in a dataset and their mean, measured in the same units as the data. A small standard deviation means the values bunch up near the mean; a large one means they are spread out.

What is standard deviation? covers the idea with pictures and examples.

The raw deviations from the mean always add up to zero, so their average says nothing. Squaring makes every deviation positive, and the square root at the end brings the result back to the original units.

Averaging the absolute deviations works too: that is the mean absolute deviation. It is always smaller than the SD, because squaring gives far-out values extra weight.

No. It is the square root of an average of squares, so it is zero or positive. It is exactly zero only when every value is identical. A negative figure means a formula error, usually a missing square or a subtraction in the wrong order.

The same units as the data. Heights in centimetres give an SD in centimetres. The variance is in squared units (cm²), which is why the standard deviation is the one people usually report. See SD vs variance.

Greek letters describe a whole population and Latin letters describe a sample: σ is the population SD and s the sample SD; μ is the population mean and x̄ the sample mean. The symbols page lists the rest and how to type them.

No, it measures precision: how closely repeated values agree with each other. A scale that always reads 2 kg too heavy can have a tiny standard deviation. Accuracy is about the distance between the mean and the true value, which the SD cannot see.

Ask whether you want to describe only these values or draw conclusions about a larger group. If these values are every member you care about (every employee, every product in the batch), it is a population. If they stand in for something bigger, it is a sample. In practice, most data is a sample.

The sample vs population guide walks through borderline cases.

A sample's values sit closer to their own mean than to the unknown population mean, so the squared deviations come out too small. Dividing by n − 1 instead of n corrects that bias in the variance. It is called Bessel's correction.

The sample SD is the population SD multiplied by √(n ÷ (n − 1)). With 5 values that is about 12% larger; with 30 values about 1.7%; with 1,000 values it is negligible. The choice only really matters for small datasets.

Excel's STDEV.S, R's sd(), pandas' .std() and the TI-84's Sx all give the sample SD. NumPy's np.std() gives the population SD unless you pass ddof=1. See the guides for Excel, Python and the TI-84.

Yes. The standard deviation calculator shows the other mode under the headline result. There are also dedicated sample and population calculators.

Find the mean. Subtract it from each value and square the result. Add the squares. Divide by n − 1 for a sample (or n for a population) to get the variance. Take the square root. How to calculate standard deviation works a full example.

Yes: s² = (Σx² − (Σx)² ÷ n) ÷ (n − 1). It saves a column of subtractions by hand, but on a computer it can lose most of its precision when the values are large and close together. This site uses Welford's algorithm instead. The formula page explains both.

Weight each value's squared deviation by its frequency, divide by Σf − 1 (sample) or Σf (population), and take the root. The frequency table calculator takes value–count pairs directly. For class intervals, use midpoints in the grouped data calculator.

Roughly. The classroom rule is range ÷ 4, which works for about 25–30 normal values. A better estimate adjusts the divisor for the sample size; the range estimate section has a calculator for it.

Do not average the SDs. Average the variances, weighted by each group's degrees of freedom, then take the square root. That is the pooled standard deviation.

Multiply the standard error by √n: s = SE × √n. The standard error calculator does this for you.

For a confidence interval of a mean, first recover the SE: it is the interval's half-width divided by the critical t value on n − 1 degrees of freedom (2.045 for a 95% interval with n = 30). Then multiply by √n as above.

Only relative to the mean and to what was measured. An SD of 5 is tiny for incomes and huge for shoe sizes. To compare spread across different scales, divide by the mean to get the coefficient of variation.

For roughly normal data, about 68% of values fall within one SD of the mean, 95% within two and 99.7% within three. It does not hold for skewed data. The empirical rule calculator gives the bands and the exact percentages.

Strongly, because deviations are squared. One extreme value can double the SD of a small dataset. If outliers are a concern, check them with the outlier and IQR calculator and consider the IQR as a steadier measure of spread.

As “M = 72.4, SD = 8.1” in APA style, or “72.4 ± 8.1” with the ± labelled as the SD (not the SE). Round both to the same decimal place, usually one more than the raw data. The psychology guide has examples.

Not systematically. A bigger sample estimates the population SD more precisely, but the SD itself does not shrink. What shrinks is the standard error of the mean, s ÷ √n.

The variance is the average squared deviation. The standard deviation is its square root. They carry the same information, but the SD is in the data's units, so it is easier to read. Use the variance calculator for s² and σ².

The SD describes how spread out individual values are. The standard error, s ÷ √n, describes how precisely the sample mean estimates the population mean. The SE is always smaller once n is above 1. See the standard error calculator.

A value's distance from the mean in standard deviations: z = (x − mean) ÷ SD. A z of 2 means two SDs above the mean. The z-score calculator also converts it to a percentile.

Yes. Both are SD ÷ mean; %RSD is the same ratio multiplied by 100 and is the usual name in analytical chemistry. See the %RSD calculator.

The margin of error is a critical value times the standard error: E = t* × s ÷ √n for a mean. It is half the width of a confidence interval. The margin of error calculator handles means and proportions.

The difference between two group means divided by their pooled standard deviation. It expresses an effect in SD units, so it can be compared across studies. Use the Cohen’s d calculator.

No. The calculators run in your browser, and nothing you type is uploaded, logged or stored. The page keeps working if you disconnect from the network after it loads. Details are in the methodology.

Numbers separated by commas, spaces, tabs, semicolons or new lines, so a column pasted from a spreadsheet works as is. Negatives, decimals and scientific notation such as 1e3 are read. Words, units and currency symbols are skipped, and the calculator tells you which entries it skipped.

For files with several columns, the CSV calculator can read a single column.

It is shorthand for a value that repeats: 10:3 is read as 10, 10, 10. It saves typing out a frequency table. 10x3 and 10*3 work the same way.

Yes, when the layout makes it unambiguous: values separated by semicolons, spaces or new lines, with no full stops anywhere. 12,5; 13,1 is read as 12.5 and 13.1. With commas between values (12, 5) the comma is a separator.

Yes. Add ?q= and your values to a calculator's address, for example /?q=2,4,4,4,5,5,7,9, and add &mode=population if needed. The page opens with that data already calculated. Every calculator also has a Copy results button.

Results are shown to six significant figures. Internally everything runs at full double precision, about 15–17 significant digits, and nothing is rounded between steps. If your textbook answer differs in the last digit, it probably rounded an intermediate value.

Tens of thousands of values work comfortably in a modern browser. The step-by-step table shows only the first 200 rows, but every value is used in the result. See the large dataset calculator.

There are several accepted quartile definitions. This site uses linear interpolation (R's default type 7, Excel's QUARTILE.INC). Other methods give slightly different values on small datasets. The methodology lists them.