What is the difference between mean, median, and mode?

The mean is the sum of all values divided by the count (the average). The median is the middle value when the data is sorted. The mode is the value that appears most often. The mean uses every number and is sensitive to outliers, while the median and mode are more resistant to extreme values.

Can a data set have more than one mode?

Yes. A data set with two most-frequent values is bimodal, and one with three or more is multimodal. If every value appears the same number of times, the data set has no mode at all.

What is the difference between sample mean and population mean?

The calculation is identical — add the values and divide by how many there are. Only the notation changes: the Greek letter mu denotes the mean of an entire population, while x-bar denotes the mean of a sample drawn from a larger group. When a problem does not specify, treat the data as a sample.

Which measure of central tendency is most affected by outliers?

The mean is most affected because it uses every value, so a single extreme number pulls it up or down. The median is barely affected since it depends only on the middle position, which is why it is preferred for skewed data such as incomes and home prices.

How do I calculate mean, median, and mode in Excel?

Use =AVERAGE(range) for the mean, =MEDIAN(range) for the median, and =MODE.SNGL(range) for a single mode. For data with more than one mode, use =MODE.MULT(range) as an array formula to return all of them.

Mean, Median & Mode Calculator

Formula

Enter your values

Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

What Are Mean, Median, and Mode?

Mean, median, and mode are the three measures of central tendency — single numbers that describe the "center" or typical value of a data set. This mean, median, and mode calculator finds all three at once, plus the range, and shows the step-by-step work for your own numbers. Each measure answers a slightly different question about your data.

Mean (average): Add up all the values and divide by how many there are. It uses every value, so it is pulled toward unusually high or low numbers.

Median: The middle value when the data is sorted from smallest to largest. Exactly half the values fall below it and half above. It ignores how extreme the outer values are.

Mode: The value that appears most often. A data set can have one mode, several modes, or no mode at all. It is the only one of the three that works for non-numeric (categorical) data.

This calculator also reports the range (the gap between the largest and smallest value) and the midrange (the value halfway between them), which describe how spread out the data is rather than where its center lies.

How to Find the Mean (Average)

The mean is the most familiar average. You sum every value, then divide by the number of values.

x̄ = Σx / n = (sum of all values) / (number of values)

Symbol	What It Means
x̄	The mean (read "x-bar") of a sample
μ	The mean of a full population (read "mu")
Σx	The sum of all the values
n	The number of values (the count)

Worked Example — How to Find the Mean

Find the mean of the data set 4, 8, 15, 16, 23, 42:

Add every value. 4 + 8 + 15 + 16 + 23 + 42 = 108.
Count the values. There are 6 values, so n = 6.
Divide the sum by the count. 108 ÷ 6 = 18. The mean is 18.

How to Find the Median

The median is the middle value of an ordered data set. The first step is always the same: sort the values from smallest to largest. What you do next depends on whether you have an odd or even count.

Odd Number of Values

When the count is odd, there is one value exactly in the middle — that single value is the median. The position formula tells you which one: the median sits at position (n + 1) / 2.

Sort the data. For 14, 5, 20, 9, 11 → ordered: 5, 9, 11, 14, 20.
Find the middle position. n = 5, so (5 + 1) / 2 = position 3.
Read the value at that position. The 3rd value is 11, so the median is 11.

Even Number of Values

When the count is even, there is no single middle value — so you take the two middle values and average them.

Sort the data. For 9, 3, 12, 7 → ordered: 3, 7, 9, 12.
Identify the two middle values. With n = 4, the middle pair is 7 and 9 (positions 2 and 3).
Average them. (7 + 9) / 2 = 8. The median is 8, even though 8 never appears in the data.

How to Find the Mode

The mode is the most frequent value — the one that appears more often than any other. Build a frequency table (a count of how many times each value occurs) and the mode is the value with the highest count.

Single Mode Example

For the data set 3, 7, 7, 7, 9, 12:

Value	Frequency
3	1
7	3 ← most frequent
9	1
12	1

The value 7 appears three times — more than any other — so the mode is 7.

Bimodal, Multimodal, and "No Mode"

Mode is the one measure that does not always give a single answer:

Bimodal: two values tie for most frequent. In 2, 4, 4, 6, 8, 8, 9 both 4 and 8 appear twice, so the data has two modes: 4 and 8.
Multimodal: three or more values tie for the highest frequency.
No mode: if every value appears the same number of times — for example 1, 2, 3, 4, 5, where each appears once — the data set has no mode.

Categorical data: The mode is the only measure of central tendency that works for non-numeric data. For survey answers like Red, Blue, Blue, Green, the mode is "Blue" — you cannot take a mean or median of colors.

Range and Midrange

Range and midrange describe the spread of the data — how far apart the values are — rather than the center.

range = maximum − minimum

midrange = (maximum + minimum) / 2

For the data set 4, 8, 8, 12, 15, 16, 23, the maximum is 23 and the minimum is 4. So the range is 23 − 4 = 19, and the midrange is (23 + 4) / 2 = 13.5. The range is easy to compute but uses only two values, so a single outlier can inflate it dramatically. To measure spread using every value, use the standard deviation calculator.

Mean vs Median vs Mode — Which Should You Use?

No single measure is "best" — the right choice depends on the shape of your data and what you want it to tell you. Use this decision table to choose.

Your Situation	Best Measure	Why
Data is roughly symmetric, no extreme values	Mean	Uses every value and is the most informative summary when nothing distorts it.
Data is skewed or has outliers (e.g. incomes, home prices)	Median	The middle value is barely affected by a few extreme numbers, so it stays "typical."
Data is categorical / non-numeric (e.g. favorite color, shoe size)	Mode	It is the only measure that works without arithmetic on the values.
You need the single most common outcome	Mode	It reports the value that occurs most often, not an "average."
You need a balanced, all-purpose "typical" value	Median	It resists distortion and always lands on a real central position.

Skew and the mean–median gap: In a perfectly symmetric distribution, the mean, median, and mode are equal. When data is right-skewed (a long tail of high values), the mean is pulled above the median; when it is left-skewed, the mean falls below the median. A large gap between the mean and median is itself a signal that outliers or skew are present.

How Outliers Affect Each Measure

An outlier is a value far from the rest of the data. It illustrates the difference between the measures better than anything else. Start with the data set 20, 22, 23, 24, 25:

Mean = (20 + 22 + 23 + 24 + 25) / 5 = 114 / 5 = 22.8
Median = middle value = 23

Now replace the 25 with an outlier of 90, giving 20, 22, 23, 24, 90:

Mean = (20 + 22 + 23 + 24 + 90) / 5 = 179 / 5 = 35.8 — it jumped by 13 points.
Median = still 23 — completely unchanged.

The mean is dragged toward the outlier while the median holds steady, which is exactly why the median is the preferred "typical value" for skewed data like salaries or house prices. To measure how far an individual value sits from the mean in standard units, try the z-score calculator.

Sample vs population mean: The arithmetic is identical — sum divided by count. Only the notation differs: use μ when your data is the entire population and x̄ when it is a sample drawn from a larger group. If a homework problem does not specify, it is almost always a sample.

How to Use This Mean, Median, and Mode Calculator

Enter your values. Type or paste your numbers separated by commas, spaces, or new lines — pasting a column straight from a spreadsheet works.
Press Calculate. The mean, median, mode, and range appear instantly in the results card, along with the count, sum, minimum, and maximum.
Read the step-by-step solution. The Steps tab shows the sorted data set, the median position, and the sum-and-divide work for your exact numbers.
Check the dot plot. The Visual tab plots each value on a number line so you can see clustering, gaps, and any outliers at a glance.

Every result is computed in your browser, and the work shown always matches the numbers reported — nothing is rounded behind the scenes. Browse the full set of free math calculators for more step-by-step tools.

Real-World Applications

Different fields prefer different measures depending on how their data behaves.

Field	Typical Measure	Why
Education — test scores	Mean	Scores are usually fairly symmetric, so the average summarizes class performance well.
Real estate — home prices	Median	A few mansions skew the mean upward; the median reflects the "typical" home.
Business — salaries	Median	A handful of executive salaries inflate the mean, so the median is fairer.
Retail — clothing or shoe sizes	Mode	Stores stock the most commonly purchased size, which is the mode.
Weather — daily temperatures	Mean	Temperatures cluster without extreme outliers, so the average is representative.

Worked Example — All Four Measures on One Data Set

Let's compute the mean, median, mode, and range for a single data set: 16, 8, 23, 4, 15, 8, 12.

Sort the data. Ordered from smallest to largest: 4, 8, 8, 12, 15, 16, 23. There are n = 7 values.
Mean — sum, then divide. 4 + 8 + 8 + 12 + 15 + 16 + 23 = 86. Divide by 7: 86 ÷ 7 ≈ 12.29.
Median — find the middle position. With n = 7, the position is (7 + 1) / 2 = 4. The 4th value in the sorted list is 12, so the median is 12.
Mode — find the most frequent value. The value 8 appears twice; every other value appears once. The mode is 8.
Range — subtract min from max. The maximum is 23 and the minimum is 4, so the range is 23 − 4 = 19.

Reading the result: The mean (12.29) sits just above the median (12) because the largest value, 23, pulls the average slightly upward — a small, normal skew rather than a true outlier.

Common Mistakes to Avoid

Mistake 1 — Forgetting to sort before finding the medianThe median is the middle of the ordered data. Always sort smallest to largest first; reading the middle of an unsorted list gives the wrong answer.

Mistake 2 — Taking only one middle value when the count is evenWith an even number of values there is no single middle. Average the two middle values instead of picking just one of them.

Mistake 3 — Assuming every data set has exactly one modeA data set can be bimodal (two modes), multimodal (several), or have no mode at all if every value occurs equally often. Don't force a single answer.

Mistake 4 — Confusing the mode with its frequencyThe mode is the value that repeats, not how many times it repeats. In 5, 5, 5, 9 the mode is 5, not 3.

Mistake 5 — Reporting the mean for heavily skewed dataWhen outliers or strong skew are present (incomes, home prices), the mean misrepresents the typical value. Report the median instead.

Studying for AP Statistics? Mean, median, and mode are core to Unit 1 (Exploring One-Variable Data) — 20–30% of the exam.

Frequently Asked Questions

What is the difference between mean, median, and mode?

The mean is the sum of all values divided by the count (the average). The median is the middle value when the data is sorted. The mode is the value that appears most often. The mean uses every number and is sensitive to outliers, while the median and mode are more resistant to extreme values.
Can a data set have more than one mode?

Yes. A data set with two most-frequent values is bimodal, and one with three or more is multimodal. If every value appears the same number of times, the data set has no mode at all.
What is the difference between sample mean and population mean?

The calculation is identical — add the values and divide by how many there are. Only the notation changes: μ denotes the mean of an entire population, while x̄ denotes the mean of a sample drawn from a larger group. When a problem does not specify, treat the data as a sample.
Which measure of central tendency is most affected by outliers?

The mean is most affected because it uses every value, so a single extreme number pulls it up or down. The median is barely affected since it depends only on the middle position, which is why it is preferred for skewed data such as incomes and home prices.
How do I calculate mean, median, and mode in Excel?

Use =AVERAGE(range) for the mean, =MEDIAN(range) for the median, and =MODE.SNGL(range) for a single mode. For data with more than one mode, use =MODE.MULT(range) as an array formula to return all of them.