Mean, Median, Mode and Standard Deviation Explained
Understand descriptive statistics and learn to calculate the mean, median, mode, range, variance and standard deviation.
Descriptive statistics make data easier to understand
A set of numbers can be summarized with measures of center and spread. The mean, median and mode describe a typical or commonly occurring value in different ways. Range, variance and standard deviation help explain how dispersed values are.
Mean: the arithmetic average
Add all numbers and divide by the count. For 2, 4, 4, 4, 5, 5, 7, 9, the sum is 40 and there are 8 observations, so the mean is 5.
Median: the middle value
Sort the values first. With an even number of observations, take the average of the two middle values. In this example, those values are 4 and 5, giving a median of 4.5.
Mode: the most frequent value
Here 4 appears three times, more than any other number, so the mode is 4. Some datasets have multiple equally common modes; if all values are unique, this calculator reports no mode.
Range: maximum minus minimum
The maximum is 9 and minimum is 2, so the range is 7. Range is easy to calculate but can be sensitive to extreme observations.
Population variance and standard deviation
Population variance is the average squared deviation from the mean:
Population variance = Σ(x − mean)² ÷ n
For the eight-number example, the squared deviations add to 32, so population variance is 32/8 = 4. Population standard deviation is its square root, 2.
Sample variance and standard deviation
When a dataset is treated as a sample from a larger population, the common sample variance formula divides by n − 1 instead of n. In this example, sample variance is 32/7 ≈ 4.5714, and sample standard deviation is approximately 2.1381.
Which measure should you choose?
The mean reflects every data point but can be distorted by extremes. The median can be more representative of skewed data. Standard deviation is most useful when a meaningful comparison of variation is desired and the data context is known.
Data quality matters
Only compare datasets collected and measured consistently. Missing values, duplicate entries and measurement errors can distort the conclusions. Descriptive statistics alone do not establish statistical significance or causation.
Calculate your own values
Enter comma-separated values in the Statistics Calculator for mean, median, mode, range, sample/population variance and both standard deviations.