Mean, Median and Mode Calculator
Enter any list of numbers separated by commas, spaces, or line breaks to instantly find the mean, median, mode, range, standard deviation, quartiles, and more. The calculator shows each step of the computation and draws a frequency chart so you can see the shape of your data at a glance.
What are mean, median and mode?
Mean, median and mode are the three classic measures of central tendency - single numbers that summarise where a data set is centred. The mean (arithmetic average) adds all values and divides by the count. The median is the middle value when the data is arranged in order; for an even count, it is the average of the two central values. The mode is the value that appears most often; a data set can have no mode (if every value is unique), one mode (unimodal), or several modes (bimodal, multimodal). Together with measures of spread such as the range and standard deviation, they give a compact picture of any data set.
How to calculate mean, median and mode - step by step
Mean: add all the numbers in the data set and divide the total by how many numbers there are. For the set {4, 7, 13, 2, 7, 9, 4, 7, 3, 11}, the sum is 67 and the count is 10, so the mean is 67 / 10 = 6.7. Median: sort the values in ascending order. If there is an odd count, the median is the middle value. If the count is even, average the two middle values. For the example set sorted as {2, 3, 4, 4, 7, 7, 7, 9, 11, 13}, the two middle values (positions 5 and 6) are 7 and 7, so the median is 7. Mode: count how often each value appears and keep the value(s) with the highest count. In the example, 7 appears three times, more than any other value, so the mode is 7. The range is the largest value minus the smallest: 13 - 2 = 11.
Standard deviation, variance and quartiles explained
Variance measures how far values spread from the mean on average. It is computed as the average of each value's squared distance from the mean. Use "population" variance (divide by N) when your data set is the entire population, and "sample" variance (divide by N-1, Bessel's correction) when it is a sample drawn from a larger group. Standard deviation is the square root of the variance and is expressed in the same units as the data, making it easier to interpret. Quartiles split the sorted data into four equal parts. Q1 is the 25th percentile (the median of the lower half), Q3 is the 75th percentile (the median of the upper half), and the interquartile range (IQR = Q3 - Q1) captures the spread of the central 50% of the data. Because the IQR ignores the top and bottom quarters, it is resistant to outliers where the standard deviation is not.
Outlier detection with the Tukey fence method
An outlier is a value that sits far from the bulk of the data and may distort summary statistics, especially the mean and standard deviation. This calculator uses the Tukey fence method: a value is flagged as an outlier if it falls below Q1 - 1.5 x IQR (the lower fence) or above Q3 + 1.5 x IQR (the upper fence). This rule catches roughly the bottom 0.7% and top 0.7% of a normal distribution. A flagged value may be a genuine extreme observation, a data-entry error, or simply a rare event - context always matters. If you suspect an outlier is an error, recalculate after removing it and compare results.
When does the mean mislead?
The mean is sensitive to extreme values. A single very large or very small number can shift it far from where most data points sit. Classic examples include household income (a few billionaires raise the mean far above what a typical household earns) and exam scores (one student who scores zero can drag the class mean below what most students achieved). In these cases the median is a more honest summary of the "typical" value. A quick check: if the mean is noticeably higher than the median, the data is right-skewed; if it is lower, the data is left-skewed.
Measures of central tendency: when to use which
| Measure | Best used when | Affected by outliers? |
|---|---|---|
| Mean | Data is roughly symmetric with no extreme outliers | Yes - strongly |
| Median | Data is skewed or has outliers (e.g. income, house prices) | No - very resistant |
| Mode | You need the most common value (e.g. shoe size, survey answers) | No |
| Range | You want a quick feel for overall spread | Yes - very strongly |
| IQR | You want spread that ignores the top and bottom 25% | No - robust |
| Std deviation | Quantifying spread around the mean in the same units | Yes |
A guide to choosing the right measure for your data.
Frequently asked questions
What is the difference between mean and median?
The mean divides the total of all values by the count and is pulled toward extreme values (outliers). The median is the physical middle of the sorted list and is unaffected by how extreme the highest or lowest values are. For symmetric data the two are close; for skewed data they can differ a lot. Use the median when your data contains outliers or has a long tail in one direction.
Can a data set have more than one mode?
Yes. If two values tie for the highest frequency the data set is bimodal, and if more than two values tie it is multimodal. A data set where every value appears exactly once has no mode. When no value repeats, the mode is not a useful summary statistic, and you should rely on the mean or median instead.
What is the IQR and why does it matter?
The interquartile range (IQR) is Q3 minus Q1 - the span of the middle 50% of your data. Because it ignores the top and bottom quarters, it is not distorted by outliers the way the range is. The IQR is also the basis of the Tukey fence method for detecting outliers: any value more than 1.5 times the IQR below Q1 or above Q3 is considered a candidate outlier.
When should I use population variance instead of sample variance?
Use population variance (divide by N) when your data set contains every member of the group you are studying - for example, the test scores of every student in one class. Use sample variance (divide by N-1) when your data is a subset drawn from a larger population - for example, scores from 30 students chosen from all students in the school. Dividing by N-1 (Bessel's correction) removes the downward bias that would otherwise occur when estimating a population variance from a sample.
What does standard deviation tell me?
Standard deviation measures how spread out values are around the mean, expressed in the same units as the data. A small standard deviation means the values cluster tightly around the mean; a large one means they are spread widely. For a bell-shaped (normal) distribution, roughly 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three.
How is the midrange different from the median?
The midrange is simply (minimum + maximum) / 2 - the centre of the total span. Because it uses only the two most extreme values, it is highly sensitive to outliers and rarely used as a primary summary. The median, by contrast, uses the middle of the sorted data and ignores how extreme the endpoints are. The midrange is occasionally useful for a quick range check but the median is almost always more meaningful.
Does order of the numbers matter?
Not for calculating mean, median, mode, variance or standard deviation. These statistics are all based on the set of values, not their order. The only step where ordering matters is sorting the data to find the median and quartiles, and this calculator does that automatically.