The mean is the arithmetic average (add the values, divide by the count), the median is the middle value when the data is sorted, and the mode is the most frequent value. When data is symmetric they are close together. When it is skewed, they split apart: for home prices of $210k, $235k, $240k, $255k, $260k, $270k and $1.2M, the mean is $381k but the median is $255k, and the median is the better description of a typical home.
Choosing the right average is less about arithmetic and more about the question you are answering. This guide shows how to find each one, why they disagree, and which to report.
How to find each average
Mean
Add every value and divide by how many there are. For shoe sizes 8, 9, 9, 10, 10, 10, 11: the sum is 67, so the mean is 67 ÷ 7 ≈ 9.57.
Median
- Sort the values from smallest to largest.
- If the count is odd, the median is the middle value, at position (n + 1) ÷ 2.
- If the count is even, average the two middle values.
For the seven shoe sizes, the 4th value is 10. For 3, 7, 8, 12, the median is (7 + 8) ÷ 2 = 7.5.
Mode
Count how often each value appears and pick the most frequent. In the shoe sizes, 10 appears three times, so the mode is 10. A data set can have one mode, several (bimodal or multimodal), or none if every value appears equally often.
The mean, median, mode calculator finds all three at once and also reports the range.
Why the averages disagree: outliers and skew
Look at what happens to the home price example with and without the $1.2M house.
| Data | Mean | Median |
|---|---|---|
| Six homes, $210k to $270k | $245k | $247.5k |
| Same six plus one $1.2M home | $381k | $255k |
Adding one expensive home moved the mean by $136,000 but the median by only $7,500. The mean uses every value’s size, so a single extreme value drags it. The median only cares about order, so the outlier counts as “one more value above the middle,” no matter how far above.
An even starker case: nine employees earn $50,000 and the owner takes $1,000,000. The mean salary is $145,000, which nobody at the company actually earns, while the median is $50,000.
Reading skew from the averages
| Shape of the data | Typical relationship | Examples |
|---|---|---|
| Symmetric | mean ≈ median ≈ mode | Heights, many test scores, measurement error |
| Right-skewed (long tail of high values) | mean > median | Incomes, home prices, wait times, insurance claims |
| Left-skewed (long tail of low values) | mean < median | Age at retirement, scores on an easy exam |
If the mean is well above the median, a few large values are stretching the right tail. That gap is a quick diagnostic before you even draw a histogram. To flag the specific values responsible, use the outlier calculator or the five number summary calculator.
Which average should you use?
| Situation | Best choice | Why |
|---|---|---|
| Roughly symmetric numeric data | Mean | Uses all the information; basis for standard deviation |
| Skewed data or outliers | Median | Resistant to extreme values |
| Ordinal data (ratings, rankings, survey scales) | Median | Order is meaningful, distances are not |
| Categories (colors, brands, majors) | Mode | The only average that exists for categories |
| Most popular size, option or product | Mode | Answers “what is most common?” |
| Total matters (budgets, averages that must add back up) | Mean | mean × count = total |
Official statistics follow this logic. The U.S. Census Bureau reports median household income, and housing reports usually quote median sale prices, because both distributions are strongly right-skewed. Scientific measurements with symmetric errors are usually summarized with the mean.
Survey scales are a gray area. A 1-to-5 satisfaction scale is ordinal, so the median is the textbook choice, but means are widely reported for large samples because they detect small shifts. If you report a mean, report the distribution too.
A worked comparison
Nine customer ratings on a 1-to-6 scale: 2, 3, 3, 4, 4, 4, 5, 5, 6.
Mean = 36 ÷ 9 = 4
Median = 5th value = 4
Mode = 4 (appears three times) = 4
All three agree because the data is symmetric around 4. That agreement is itself informative: when the three averages match, the data has no strong skew and the mean is a safe summary.
Now a bimodal set: 1, 1, 1, 2, 2, 5, 5, 5, 9. The modes are 1 and 5, the median is 2 and the mean is 3.44. No single number describes this data well; the two modes suggest two different groups mixed together, which is worth investigating before summarizing.
Other averages worth knowing
- Weighted mean: each value counts according to a weight, as in course grades or GPA. See how to calculate a weighted average or use the weighted average calculator.
- Trimmed mean: drop a fixed percentage of the highest and lowest values, then average the rest. It sits between the mean and median in its resistance to outliers. Olympic judging and some economic indicators use it.
- Midrange: (minimum + maximum) ÷ 2. Easy, but very sensitive to outliers.
- Geometric mean: the right average for growth rates and ratios.
Report the spread too
An average alone hides how varied the data is. Pair the mean with the standard deviation, and the median with the interquartile range. Two classes can both average 75 on a test while one has scores from 70 to 80 and the other from 40 to 100. See how to calculate standard deviation by hand and how to find quartiles and the IQR. The descriptive statistics calculator reports every average and spread measure in one table, and the mean calculator handles quick averages.
Frequently asked questions
What is the difference between mean, median and mode?
The mean is the sum of the values divided by how many there are. The median is the middle value when the data is sorted. The mode is the value that appears most often. All three describe the center of a data set, but they react very differently to outliers and skew.
When should I use the median instead of the mean?
Use the median when the data is skewed or has outliers, such as incomes, home prices or response times. A few extreme values pull the mean toward them, while the median stays with the typical value.
How do you find the median of an even number of values?
Sort the values and average the two in the middle. For 3, 7, 8 and 12, the middle pair is 7 and 8, so the median is (7 + 8) ÷ 2 = 7.5.
Can a data set have more than one mode or no mode?
Yes. If two values tie for the most frequent, the data is bimodal; with more ties it is multimodal. If every value appears the same number of times, there is no mode.
Which average is best for categories like colors or brands?
Only the mode. You cannot add or sort categories such as eye color or favorite brand, so the mean and median do not exist for them. The mode, the most common category, is the only meaningful average.