Standard Deviation Calculator

Standard deviation is the most widely used measure of statistical dispersion. It tells you, on average, how far each number in a data set is from the mean. Because it is expressed in the same units as the original data, it is far more intuitive than variance, which is measured in squared units.

A low standard deviation means the values are clustered closely around the average. A high standard deviation means the values are spread out over a wider range. In fields like finance, manufacturing, and scientific research, this single number helps analysts decide whether data is reliable, consistent, or unusually volatile.

Paste or type your numbers into the input box. You can separate values with commas, spaces, or line breaks. The calculator instantly computes the count, sum, mean, range, population standard deviation, sample standard deviation, and both variances.

• Population standard deviation is best when your list includes every possible value.
• Sample standard deviation is best when your list is a subset of a larger population.
• Use the mean and range alongside standard deviation to get a fuller picture of your data.

Suppose five students scored 78, 82, 88, 90, and 92 on an exam. The mean score is 86. The squared differences from the mean are 64, 16, 4, 16, and 36, which sum to 136.

If these five students are the entire class, divide by 5 to get a population variance of 27.2 and a population standard deviation of about 5.22. If they are a sample from a larger school, divide by 4 to get a sample variance of 34 and a sample standard deviation of about 5.83. Choosing the right formula keeps your conclusions accurate.

Standard deviation appears almost everywhere data is analyzed. Investors use it to measure portfolio volatility: a higher standard deviation means returns swing more dramatically. Manufacturers track it to keep product dimensions within tolerance limits. Teachers use it to compare how evenly a class performed on a test. Scientists report it as error bars to show the uncertainty in repeated measurements.

When data follows a normal distribution, standard deviation powers the 68-95-99.7 rule. This makes it easy to estimate how likely a value is and to flag outliers that deserve a closer look.

• Always check whether your data is a sample or the full population before picking a result.
• Remove clear outliers only when you can justify them; outliers inflate standard deviation.
• Combine standard deviation with the mean or median for a balanced summary of your data.
• For comparing groups with different scales, consider the coefficient of variation, which divides standard deviation by the mean.
• Use a larger sample size when possible; small samples make Bessel’s correction more influential.