Mean, median and mode explained, with worked examples

Your child brings home a worksheet headed mean median mode and range, next to a list of seven numbers. The three answers come out different, which makes it look like two of them are wrong. All three are correct, because each one summarizes the same list in a different way.

What mean, median and mode each measure

The mean, median and mode are three summaries of one list of numbers, and each answers a different question about it. The mean answers what every value would be if the total were shared out evenly.

The median answers what sits in the middle once the values are put in order. The mode answers which value shows up most often.

Take the six numbers 2, 5, 7, 9, 9 and 10. The mean is 42 ÷ 6 = 7, the median is the average of the two middle values 7 and 9, which is 8, and the mode is 9, the only value that repeats.

The three answers are 7, 8 and 9, and none of them is a mistake.

Why all three are called an "average"

The word "average" in everyday English covers all three measures, which is why a parent and a textbook can mean different things by it. In school math and in statistics, "average" with no qualifier means the arithmetic mean, the sum divided by the count.

The Common Core mathematics standards avoid the ambiguity by calling them measures of center: sixth graders are asked to give "quantitative measures of center (median and/or mean) and variability" for a data set (6.SP.B.5c).

Why the three answers differ

The three answers differ because each measure looks at a different feature of the data.

The mean uses every value, so every number pulls on it. The median uses only the position of the values, so it ignores how far away the extremes sit. The mode uses only repetition, so it ignores everything that happens once.

What one outlier does to the mean

An outlier is a value far away from the rest, and it moves the mean while leaving the median almost still. Five quiz scores of 78, 82, 85, 85 and 90 give a mean of 420 ÷ 5 = 84, a median of 85 and a mode of 85.

Replace that 78 with a 0 for a missed test and the sorted list becomes 0, 82, 85, 85, 90. The sum drops to 342, so the mean falls to 342 ÷ 5 = 68.4, a drop of 15.6 points.

The median stays at 85, because the middle position still holds an 85. That stability is what statisticians mean when they call the median resistant.

The Census Bureau and HUD reported a median sales price of $393,800 for new houses sold in July 2026 and an average sales price of $508,800, a gap of $115,000 opened by the most expensive houses in the sample.

How to work out the mean, median and mode, step by step

The mean, median and mode each need one pass through the data, and the order of those passes matters. Work through the minutes one child read on seven days: 12, 5, 9, 5, 15, 7 and 3.

1. Count the values first. Write the list down and count how many numbers it holds. This set has seven values, so n = 7.
2. Sort from smallest to largest. Rewrite the list in order on a fresh line: 3, 5, 5, 7, 9, 12, 15. Every later step reads from this sorted line.
3. Add the values for the mean. The sum is 3 + 5 + 5 + 7 + 9 + 12 + 15 = 56.
4. Divide the sum by the count. The mean is 56 ÷ 7 = 8 minutes, which sits between the smallest value 3 and the largest value 15.
5. Take the middle value for the median. With an odd count the middle position is (7 + 1) ÷ 2 = 4, so the median is the fourth sorted value, 7 minutes.
6. Average the two middle values when the count is even. Six values such as 2, 5, 7, 9, 9 and 10 put the middle at positions 3 and 4, so the median is (7 + 9) ÷ 2 = 8.
7. Count repeats for the mode, then subtract for the range. The value 5 appears twice and every other value once, so the mode is 5, and the range is 15 − 3 = 12 minutes.

The part people get wrong: sort before you touch the median

Sorting comes before the median every single time, and skipping it is the most common error on this topic. The unsorted list 12, 5, 9, 5, 15, 7, 3 holds a 5 in the fourth position, so anyone who counts to the middle without sorting reads the median as 5.

The sorted list puts a 7 in the fourth position, and 7 is the median. Putting numbers in order is the skill the comparing numbers game drills, and dividing a sum by a count is the step the division facts game makes automatic. The find the average game puts both together in timed rounds.

Play free in your browserFind the AverageGrade 4–6

Mean vs median vs mode vs range

The mean, median, mode and range answer four separate questions about one data set. Picking one of them is a decision about what you want that single number to say.

Which one to report

The shape of the data decides the measure, and official statistics show the rule in action. Income and house prices have a long tail of high values, so the Census Bureau reports a median household income of $83,730 for 2024 rather than a mean.

Pay inside a single profession is closer to symmetric, so the National Center for Education Statistics reports an average base salary of $61,600 for full-time public school teachers in 2020–21. Both agencies are reporting an average; they chose different ones because the data has a different shape.

Percent change rests on the same kind of decision about which number goes underneath, which the guide to calculating percentages works through.

Worked examples from real life

Three everyday data sets show which measure tells the truth in each case.

Test scores, house prices and shoe sizes

Test scores usually call for the mean, because a report card should count every score that was earned. Quiz scores of 78, 82, 85, 85 and 90 add to 420, and 420 ÷ 5 = 84. A teacher who drops the lowest score is admitting that the mean is sensitive to one bad day.

House prices call for the median, because a handful of very expensive homes sit far above the rest. The July 2026 gap between the $393,800 median and the $508,800 average is $115,000, and the median is the figure a buyer can plan around.

Shoe sizes call for the mode, because a shop needs the size it sells most, not a decimal. Seven pairs sold in sizes 6, 7, 7, 8, 9, 9 and 10 have a mean of 8 and a median of 8, while the two modes are 7 and 9, which is what the buyer has to reorder.

The arithmetic under all three examples is addition, division and putting numbers in order. The printable mixed worksheet for 5th graders drills those operations with an answer key.

Free printable worksheetMixed Operations — 5th GradersWorksheet · PDF

How to check your child has it

A child who understands mean, median and mode does four things without being prompted.

• Sorts before finding a median. The child rewrites the list in order on a new line instead of pointing at the middle of the original.
• Handles an even-sized list. Given six values, the child adds the two middle values and halves the total instead of choosing one of them.
• Names a set with no mode. Given 3, 5, 8 and 11, the child says there is no mode rather than writing 0 or guessing.
• Picks the right measure for a story. Asked which average fits house prices or shoe sizes, the child can name one and say why in a sentence.

A five-minute check at the kitchen table

One list of six numbers tests every rule in this guide. Write 4, 6, 6, 8, 10 and 14 on paper and ask for all four measures.

The answers are a mean of 48 ÷ 6 = 8, a median of (6 + 8) ÷ 2 = 7, a mode of 6 and a range of 14 − 4 = 10. A child who answers 6 for the median has skipped the even-count rule, and a child who answers 10 for the mode has given the range.

Averages arrive in fifth and sixth grade in most US schools, and the guide to what math each grade should know sets out what comes before and after them.

Common mistakes, edge cases and your checklist

Five mistakes account for most of the lost marks on a mean, median and mode question.

• Forgetting to sort before the median. An unsorted middle value is the wrong answer far more often than it is the right one, so rewrite the list in order first.
• Dividing by the wrong number for the mean. The divisor is how many values the list holds, never how many different values it holds and never the largest value.
• Taking one middle value from an even-sized list. A list with an even count has two middle values, and the median is their mean, so 2, 5, 7, 9, 9 and 10 give (7 + 9) ÷ 2 = 8.
• Calling the range an average. The range is the largest value minus the smallest, which describes spread, so it never answers a question about the middle.
• Writing 0 when there is no mode. A set where every value appears once has no mode at all, and 0 is a value that claims something else.

When a data set has two modes or none

A data set can have one mode, several modes or no mode, and all three outcomes are normal. The sizes 6, 7, 7, 8, 9, 9 and 10 have two modes, 7 and 9, which makes that set bimodal.

The numbers 3, 5, 8 and 11 have no mode, because nothing repeats. A second peak is information worth keeping: two modes in a class's test scores often mean two groups of learners sitting in one room.

The checklist that follows turns the whole method into an order you can run on any list.

• Count the values. Write down n before anything else, because both the mean and the median depend on it.
• Sort the list. Put the numbers in order on a new line and leave the original list alone.
• Mean: add, then divide. Total every value and divide by n, then check that the answer sits between the smallest and largest values.
• Median: middle, or the mean of two middles. Take the middle value when n is odd and the mean of the two middle values when n is even.
• Mode and range last. Count repeats for the mode, accept none or several, and subtract the smallest value from the largest for the range.

Twenty-two of MathIt's roughly 90 game types play free in a browser with no sign-up, and the free iOS and Android app adds adaptive difficulty, so averages get harder only once the easy sets come out right. Adults rebuilding their own arithmetic can start from the math games for adults page.