This Standard Deviation Calculator is intended for calculation of the mean value, variance, standard deviation and standard error of the values in a given set of data. Enter the data set of input points in the appropriate field of the Standard Deviation Calculator and calculate the above parameters as well as the number of input points (dataset size). You can paste the data copied from a spreadsheet or csv-file or input manually using comma, space or enter as separators.
Sample standard deviation formula
In statistics, the standard deviation (represented by the Greek letter σ for the population standard deviation or by the Latin letter s for the sample standard deviation) is a measure of variation or dispersion of a set of data values. A low standard deviation indicates that the values of the data points are close to the mean value (also called the expected value) of the data set, while a high standard deviation indicates that the values of the data points are spread out over a wider range. The standard deviation is the square root of its variance. The standard deviation (unlike the variance) is expressed in the same units as the data.
The term sample standard deviation refers to either the available sample of data from a population or to the unbiased estimate of the entire population standard deviation.
In the case of a dataset of \(n\) values \(\{ { x }_{ 1 },{ x }_{ 2 },…,{ x }_{ n }\}\), the sample mean value is
$$\bar { x } =\frac { 1 }{ n } \sum _{ i=1 }^{ n }{ { x }_{ i } } .$$
The formula for the sample standard deviation (more precisely “corrected sample standard deviation“) is
$$s=\sqrt { \frac { 1 }{ n-1 } \sum _{ i=1 }^{ n }{ { ({ x }_{ i }-\bar { x } ) }^{ 2 } } } ,$$
and the sample variance accordingly is \({ s }^{ 2 }\).
The standard error of the mean is a measure of the dispersion of sample means around the population mean and can be expressed through the population standard deviation σ as follows:
$${ \sigma }_{ \bar { x } }=\frac { \sigma }{ \sqrt { n } } .$$
The standard error of the mean is usually estimated as the sample standard deviation divided by the square root of the sample size:
$${ s }_{ \bar { x } }=\frac { s }{ \sqrt { n } } .$$
The standard error of the mean can also be understood as the standard deviation of the error in the sample mean with respect to the true mean.
Related calculators
Check out our other statistics calculators such as Correlation Coefficient Calculator or Covariance Calculator.