# Standard Deviation Calculator

## Calculator Use

Standard deviation is a statistical measure of diversity or variability in a data set. A low standard deviation indicates that data points are generally close to the mean or the average value. A high standard deviation indicates greater variability in data points, or higher dispersion from the mean.

This standard deviation calculator uses your data set and shows the work required for the calculations.

Enter a data set, separated by spaces, commas or line breaks. Click Calculate to find standard deviation, variance, count of data points
*n*, mean and sum of squares. You can also see the work peformed for the calculation.

You can copy and paste lines of data points from documents such as Excel spreadsheets or text documents with or without commas in the formats shown in the table below.

## Standard Deviation Formula

Standard deviation of a data set is the square root of the calculated variance of a set of data.

The formula for variance (s^{2}) is the sum of the squared differences between each data point and the mean, divided by the number of data points.

When working with data from a complete population the sum of the squared differences between each data point and the mean is divided by the size of the data set,
*n*. When working with a sample, divide by the size of the data set minus 1, *n - 1*.

Take the square root of the population variance to get the standard deviation.

Take the square root of the sample variance to get the standard deviation.

For additional explanation of standard deviation and how it relates to a bell curve distribution, see Wikipedia's page on Standard Deviation.

## Statistics Formulas and Calculations Used by This Calculator

### Sum

The sum is the total of all data values
*x _{1} + x_{2} + x_{3} + ... + x_{n}*

### Size, Count

Size or count is the number of data points in a data set.

\[ \text{Size} = n = \text{count}(x_i)_{i=1}^{n} \]### Mean

The mean of a data set is the sum of all of the data divided by the size. The mean is also known as the average.

For a Population

\[ \mu = \dfrac{\sum_{i=1}^{n}x_i}{n} \]For a Sample

\[ \overline{x} = \dfrac{\sum_{i=1}^{n}x_i}{n} \]### Sum of Squares

The sum of squares is the sum of the squared differences between data values and the mean.

For a Population

\[ SS = \sum_{i=1}^{n}(x_i - \mu)^{2} \]For a Sample

\[ SS = \sum_{i=1}^{n}(x_i - \overline{x})^{2} \]### Standard Deviation

Standard deviation is a measure of dispersion of data values from the mean. The formula for standard deviation is the square root of the sum of squared differences from the mean divided by the size of the data set.

For a Population

\[ \sigma = \sqrt{\dfrac{\sum_{i=1}^{n}(x_i - \mu)^{2}}{n}} \]For a Sample

\[ s = \sqrt{\dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1}} \]### Variance

Variance also measures dispersion of data from the mean. The formula for variance is the sum of squared differences from the mean divided by the size of the data set.

For a Population

\[ \sigma^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \mu)^{2}}{n} \]For a Sample

\[ s^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1} \]

Unit

Options

54

65

47

59

40

53

54,

65,

47,

59,

40,

53,

or

42, 54, 65, 47, 59, 40, 53

65 47

59 40

53

or

42 54 65 47 59 40 53

54 65,,, 47,,59,

40 53