How to Calculate Standard Deviation on a 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 (s2) 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.
The formula for variance for a population is:
Variance = \( \sigma^2 = \dfrac{\Sigma (x_{i} - \mu)^2}{n} \)
The formula for variance for a sample set of data is:
Variance = \( s^2 = \dfrac{\Sigma (x_{i} - \overline{x})^2}{n-1} \)
Take the square root of the population variance to get the standard deviation.
Population standard deviation = \( \sqrt {\sigma^2} \)
Take the square root of the sample variance to get the standard deviation.
Standard deviation of a sample = \( \sqrt {s^2} \)
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 x1 + x2 + x3 + ... + xn
\[ \text{Sum} = \sum_{i=1}^{n}x_i \]
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
\[ s^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1} \]
Acceptable Data Formats
Column (New Lines)
42
54
65
47
59
40
53
42, 54, 65, 47, 59, 40, 53
Comma Separated (CSV)
42,
54,
65,
47,
59,
40,
53,
or
42, 54, 65, 47, 59, 40, 53
42, 54, 65, 47, 59, 40, 53
Spaces
42 54
65 47
59 40
53
or
42 54 65 47 59 40 53
42, 54, 65, 47, 59, 40, 53
Mixed Delimiters
42
54 65,,, 47,,59,
40 53
42, 54, 65, 47, 59, 40, 53
How to Calculate Standard Deviation on a Calculator
Source: https://www.calculatorsoup.com/calculators/statistics/standard-deviation-calculator.php