Mean, Median, and Mode
Choose a summary that matches the distribution.
MathGPT Editorial TeamUpdated · Verify with another methodStatistics is the math of data. You collect numbers, describe what they show, and then decide how much of the pattern could just be chance.
Mean, Median, and Mode: the central idea
Mean, median, and mode answer different questions about center: balance point, middle ordered position, and most frequent value.
Words you need
- Mean
- The mean is the total of all the values divided by how many values there are, so it acts like the balance point of the data.
- Median
- The median is the value in the middle position once the data is sorted, or the average of the two middle values when the count is even.
- Mode
- The mode is the value that appears more often than any other value in the data set.
- Outlier
- An outlier is a value that sits much higher or much lower than the rest of the data, and it pulls the mean toward itself while leaving the median almost still.
- Bimodal data
- Bimodal data is a data set with two different values tied for the highest count, which usually means two groups got mixed together.
- Skewed data
- Skewed data is a data set with a long tail on one side, and in skewed data the mean is dragged out toward that tail while the median stays near the bulk of the values.
What to know before this lesson
Be able to order numerical data, count observations, add accurately, and identify the study variable.
If one of those prerequisites is uncertain, use the Statistics subject guide to locate the earlier concept before memorizing a procedure.
Mean, median, and mode: a worked example
Every step, with the arithmetic
- Step 1 - Write the seven quiz scores7, 9, 4, 9, 6, 7, 9
- Step 2 - Sort them from small to large4, 6, 7, 7, 9, 9, 9
- Step 3 - Add them for the mean4 + 6 + 7 + 7 + 9 + 9 + 9 = 51
- Step 4 - Divide the total by the countmean = 51 / 7 = 7.285... which rounds to 7.3
- Step 5 - Find the middle positionn = 7, so the middle spot is (7 + 1) / 2 = the 4th score
- Step 6 - Read the 4th sorted scoremedian = 7 (three scores sit below it and three sit above it)
- Step 7 - Count how often each score appears4 appears once, 6 once, 7 twice, 9 three times, so the mode = 9
- Step 8 - Check what happens with an even countdrop the 4: 6, 7, 7, 9, 9, 9 has n = 6, so median = (7 + 9) / 2 = 8
For $2,3,3,4,18$, the median is $3$ while the mean is $30/5=6$. The large value $18$ pulls the balance point upward.
Which measure of center to report, with a small data set showing each one at work
| Measure | Best when | Example data set | Result |
|---|---|---|---|
| Mean | The values are packed close together with nothing far away | 5, 6, 7, 8, 9 | 35 / 5 = 7 |
| Median | One value sits far from all the rest | 5, 6, 7, 8, 90 | middle value = 7 (the mean is 23.2) |
| Median | The count is even, so you average the two middle values | 4, 6, 8, 10 | (6 + 8) / 2 = 7 |
| Mode | You want the most common answer | 2, 3, 3, 3, 8 | 3, because it appears three times |
| Mode | The data is labels instead of numbers | red, blue, red, green, red | red (a mean of colors makes no sense) |
| Mean | You will need the total again later | 12, 15, 18 | 45 / 3 = 15, and 15 x 3 gives the total back |
The step-by-step method for mean, median, and mode
- Sort the data and inspect its shape, extreme values, and repeated observations.
- Compute the mean from the total and count; locate the median by position; count frequencies for the mode.
- Choose the measure that represents the context and report units rather than presenting all three without interpretation.
How to check your answer
Check that the mean lies between the minimum and maximum and that exactly half the ordered observations lie on each side of the median position.
Verify that probabilities stay between 0 and 1, measures of spread are nonnegative, and numerical conclusions match the shape, units, and direction visible in the data.
A mistake that changes the mathematics
A data set may have no mode or several modes. The mode is not automatically the largest number.
Explain why the tempting step is invalid, then write the condition or definition that prevents it. This turns the error into a rule you can recognize in a new problem.
Where you will actually use this
Housing reports
News stories quote the median home price rather than the mean because a handful of multi-million-dollar sales would push the mean far above what a normal buyer pays.
Grading a class
A teacher averages test scores to get a course mean, but checks the median too, because one student who scored 0 on a missed test can drag the class mean down by several points.
Stocking a shoe store
A store orders the most pairs in the modal size, the size that sells most often, since the mean size might be 8.4 and no one wears a 8.4.
How mean, median, and mode connects to the rest of statistics
- Standard deviation — The mean tells you where the data sits, and the standard deviation tells you how far the values usually stray from that mean.
- Linear regression — Every least-squares regression line is forced to pass through the point made from the mean of x and the mean of y.
- Probability foundations — Expected value is just a mean where each outcome is weighted by how likely it is instead of counted once.
Try a transfer problem
Compare incomes $32,34,35,36,250$ using mean and median, then defend which center better describes a typical value.
Show the worked answer
Sort the incomes first: 32, 34, 35, 36, 250 (thousands of dollars). The total is 32 + 34 + 35 + 36 + 250 = 387, so the mean is 387 / 5 = 77.4 thousand. There are five values, so the median is the 3rd sorted value, which is 35 thousand. The median is the better description of a typical income here. Four of the five people earn between 32 and 36 thousand, and not a single person earns anything close to the mean of 77.4. The lone 250 is an outlier: it adds 214 to the total and therefore about 43 to the mean all by itself, while it changes the median not at all. Swap that 250 for 2,500 and the mean jumps to 527.4 while the median stays exactly 35.
Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to statistics practice questions.
Questions about mean, median, and mode
Can a data set have more than one mode?
Yes. If two values tie for the highest count the data is bimodal, like 2, 2, 5, 7, 7, where both 2 and 7 appear twice. If every value appears the same number of times, such as 3, 5, 8, 9, statisticians usually say the set has no mode at all.
Why do news reports use median income instead of mean income?
Income has a long tail of very large values. A few people earning millions raise the mean far above what most people take home, but the median only cares about who is standing in the middle of the line, so it keeps describing an ordinary earner.
Can the mean be a number that never appears in the data?
Yes, and that is normal. The seven scores 4, 6, 7, 7, 9, 9, 9 have a mean of about 7.3, and no student scored 7.3. The mean is a balance point, not one of the observations.
Does adding the same amount to every value change all three measures?
It shifts all three by that amount and changes nothing else. Add 10 to 4, 6, 7, 7, 9, 9, 9 and the mean goes from about 7.3 to about 17.3, the median goes from 7 to 17, and the mode goes from 9 to 19, while the spread of the data stays exactly the same.