Calculators

Statistics

Calculator Use

Variance is a measure of dispersion of data points from the mean. Low variance indicates that data points are generally similar and do not vary widely from the mean. High variance indicates that data values have greater variability and are more widely dispersed from the mean.

The variance calculator finds variance, standard deviation, sample size n, mean and sum of squares. You can also see the work peformed for the calculation.

Enter a data set with values separated by spaces, commas or line breaks. You can copy and paste your data from a document or a spreadsheet.

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

How to Calculate Variance

1. Find the mean of the data set. Add all data values and divide by the sample size n.
   \( \overline{x} = \dfrac{\sum_{i=1}^{n}x_i}{n} \)
2. Find the squared difference from the mean for each data value. Subtract the mean from each data value and square the result.
   \( (x_{i} - \overline{x})^{2} \)
3. Find the sum of all the squared differences. The sum of squares is all the squared differences added together.
   \( SS = \sum_{i=1}^{n}(x_i - \overline{x})^{2} \)
4. Calculate the variance. Variance is the sum of squares divided by the number of data points.
   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 = \( s^2 = \dfrac{\Sigma (x_{i} - \overline{x})^2}{n-1} \)

Variance Formula

The formula for variance of a is the sum of the squared differences between each data point and the mean, divided by the number of data values. This calculator uses the formulas below in its variance calculations.

For a Complete Population divide by the size n

\[ \text{Variance} = \sigma^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \mu)^{2}}{n} \]

For a Sample Population divide by the sample size minus 1, n - 1

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

The population standard deviation is the square root of the population variance.

The sample standard deviation is the square root of the calculated variance of a sample data set.