Mean, Median, Mode Calculator

Enter a list of numbers to find the mean, median, mode, range, variance, and standard deviation.

Separate with commas, spaces, or new lines
Mean (Average)
15.625
Median
12.5
Mode
10
Range
25
Count
8
Sum
125
Min
5
Max
30
Variance (Pop.)
60.234
Std Dev (Pop.)
7.761
Variance (Sample)
68.839
Std Dev (Sample)
8.297
Sorted Data
5, 10, 10, 10, 15, 20, 25, 30

Understanding Mean, Median, and Mode

These three measures of central tendency each describe the "center" of a data set differently. Mean is what most people think of as the average: add all numbers and divide by how many there are. Median is the middle number when values are sorted (or the average of the two middle numbers if the count is even). Mode is the value that appears most often.

Mean = Sum / Count
Median = Middle value of sorted data
Mode = Most frequent value

When to Use Each

Use the mean for symmetric data without extreme outliers (like test scores in a normal class). Use the median when data is skewed or has outliers (like household incomes, where a few billionaires would pull the mean way up). Use the mode for categorical data or when you need the most common value (like the most popular shoe size).

Variance and Standard Deviation

These measure how spread out the data is. Variance is the average of the squared differences from the mean. Standard deviation is the square root of variance, which puts the spread back into the same units as the data. This calculator shows both population (divide by N) and sample (divide by N-1) versions. Use population when you have all data points; use sample when working with a subset of a larger group.

What if there is no mode?
If every value appears only once, there is no mode. Some textbooks say the data has "no mode," and that is what this calculator reports. If multiple values tie for most frequent, all of them are listed.
Why are there two types of standard deviation?
Population standard deviation divides by N and is used when your data set IS the entire population. Sample standard deviation divides by N-1 (Bessel's correction) and is used when your data is a sample from a larger population. The N-1 correction makes the estimate unbiased. For homework, check whether the problem says "population" or "sample."
How do you find the median with an even number of values?
Sort the data. With an even count, take the two middle values and average them. For example, in the set {3, 7, 9, 12}, the two middle values are 7 and 9, so the median is (7 + 9) / 2 = 8.