Calculate the sample and population variance of a data set instantly, with a step-by-step procedure and formulas to calculate the mean deviation. Use a period as the decimal separator.
Sample Variance
s² =
Population Variance
σ² =
1. Arithmetic mean (x):
Sum: () / =
2. Subtract the mean from each value and square it:
(xi − x)2 for each number
3. Sum of squared differences:
Σ(xi − x)2 =
4. Division for final variance:
Sample Variance (s²): / ( − 1) =
Population Variance (σ²): / =
| # | xᵢ | xᵢ - x̄ | (xᵢ - x̄)² |
|---|---|---|---|
| Sum (Σ) | 0 |
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Variance is a statistical measure that quantifies the dispersion or variability of a data set with respect to its statistical mean. Essentially, variance indicates how far individual values are from the average of the data set.
In simpler terms, if all data in a set are very similar to each other, the variance will be low, indicating little dispersion. Conversely, if the data vary significantly, the variance will be high, signaling greater dispersion.
There are two main types of variance used in statistics to measure data dispersion: sample variance and population variance. Each applies in different contexts depending on whether you are working with a sample or a complete population.
Sample variance is calculated when only a sample of the population is available. It is used to estimate the population variance and is calculated by dividing the sum of the squared differences between each data point and the sample mean by the number of data points in the sample minus one (n-1). This adjustment, known as Bessel’s correction, corrects the bias in estimating the population variance.
Population variance is calculated when data for the entire population is available. It is obtained by dividing the sum of the squared differences between each data point and the population mean by the total number of data points in the population (N). This formula does not require correction, as it is based on all available data.
To calculate variance, first you need to find the statistical mean of your data. Then, subtract the calculated mean from each individual value, square the result, and sum these squares. If you are working with a sample, divide the sum of the squares by the total number of data points minus one (n-1) to obtain the sample variance. If you are working with the entire population, divide by the total number of data points (N) to obtain the population variance.
Where:
Where:
The statistical mean x̄ is calculated by summing all values in the sample and dividing by the total number of data points.
Where: