Mean, Median, Mode Calculator — Range & Steps Shown

About the Mean Median Mode Calculator

The mean, median and mode are three ways to describe the center of a data set. The mean is the arithmetic average, the median is the middle value once the data is sorted, and the mode is the value that occurs most often. The range, largest minus smallest, gives a first idea of spread.

Each measure reacts differently to unusual values. A single very large number pulls the mean but barely moves the median, which is why both are often reported together.

What are the mean, median and mode?

The mean, median and mode are three measures of central tendency: three different answers to the question what is a typical value in this data set. The mean is the arithmetic average, the sum divided by the count. The median is the middle value once the data are sorted. The mode is the value that occurs most often. Each captures a different idea of typical, and they agree only for perfectly symmetric data.

The range, the largest value minus the smallest, is reported alongside them as a first look at how spread out the data are. Together these four numbers give a quick summary that is often enough to compare two groups or to spot a value that does not fit.

How to use the Mean Median Mode Calculator

Enter the numbers separated by commas, spaces or line breaks.
Press Calculate.
Read the sorted list, the sum and count for the mean, the middle position for the median, and the frequency count for the mode.

Formulas

Mean

\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}

Median position

\text{position} = \frac{n + 1}{2} \text{ in the sorted list}

Range

\text{range} = x_{\max} - x_{\min}

Worked example: 3, 7, 7, 2, 9, 4

The worked example uses 3, 7, 7, 2, 9, 4. Sorted, the values are 2, 3, 4, 7, 7, 9. They add to 32, so the mean is $\frac{32}{6} \approx 5.333333333$ . With six values there are two middle values, 4 and 7, so the median is $\frac{4 + 7}{2} = 5.5$ . The value 7 appears twice and every other value once, so the mode is 7, and the range is $9 - 2 = 7$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

Mean = 5.333333333; Median = 5.5; Mode = 7; Range = 7

How we read your problem

Measures of central tendency for a list of numbers.

Step-by-step solution

Sort the data from smallest to largest
$2, 3, 4, 7, 7, 9\quad (n = 6)$
Mean: add the values and divide by how many there are
$\bar{x} = \frac{32}{6} = 5.333333333$
Median: the middle value of the sorted list
With 6 values (even), the median is the average of values 3 and 4.
$\text{median} = \frac{4 + 7}{2} = 5.5$
Mode: the value(s) that appear most often
Most frequent: 7.
Range: largest minus smallest
$9 - 2 = 7$

Formulas used

Mean
$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$
Range
$\text{range} = x_{\max} - x_{\min}$

The mean balances all values, the median splits the sorted data in half and is less affected by outliers, and the mode is the most common value.

Verification

Sum 32 = mean × n = 5.333333333 × 6 = 32

Detailed explanation

How outliers pull the mean

The mean uses every value, so a single extreme value moves it. If the 9 in the example were 90, the sum would become 113 and the mean would jump to about 18.83, while the median would stay at 5.5 because the middle of the sorted list is unchanged. This is why reports on incomes and house prices usually quote the median: a few very large values would make the mean unrepresentative of a typical household.

When the mean is noticeably larger than the median, the data are usually skewed to the right, with a long tail of high values; when it is smaller, they are skewed to the left.

When the mode is the right choice

The mode is the only one of the three that works for categorical data. You cannot average shoe colours or favourite subjects, but you can say which one is most common. For numbers it is useful when values repeat, such as the most common shoe size a shop should stock. A data set can have one mode, several modes, or none at all when every value appears the same number of times, and the calculator reports each of those cases explicitly.

Common mistakes

Finding the median without sorting the data first.
Picking one of the two middle values when $n$ is even; the median is their average.
Saying there is no mode when two values tie; the data is then bimodal.
Taking the middle value of the unsorted list as the median.
Counting how many different values there are instead of how many values in total when dividing for the mean.

Tips

Use the median for skewed data such as incomes or house prices.
If every value appears once, there is no mode.
The mean times the count gives the total, which is handy for missing-value problems.
Check that the mean lies between the smallest and largest values; if it does not, a value was mistyped.
Multiply the mean by the count to recover the total, which is handy when a missing value must be found.

Frequently asked questions

How do I calculate the mean?

Add all the values and divide by how many there are. For 3, 7, 7, 2, 9, 4 the sum is 32 and the mean is $\frac{32}{6} \approx 5.33$ .

How do I find the median of an even number of values?

Sort the data and average the two middle values. For 2, 3, 4, 7, 7, 9 the median is $\frac{4 + 7}{2} = 5.5$ .

Can a data set have more than one mode?

Yes. If two or more values share the highest frequency, all of them are modes. If every value occurs equally often, there is no mode.

Which measure of center should I use?

The mean uses every value and suits symmetric data. The median resists outliers. The mode is the only one that also works for categories.

Does the calculator accept negative numbers and decimals?

Yes. Any real numbers can be entered, separated by commas, spaces or new lines.

What is the range used for?

It is the simplest measure of spread. For a more robust measure, use the standard deviation calculator.

How do I find a missing value from the mean?

Multiply the target mean by the number of values to get the required total, then subtract the sum of the known values. To average 80 over five tests after scoring 72, 85, 90 and 78, you need $400 - 325 = 75$ .

Why are the mean and median different?

The mean is pulled toward extreme values while the median depends only on the middle of the sorted list. A gap between them is a sign of skewed data or outliers.

Summarise your data set

Enter your numbers above to get the mean, median, mode and range, with the sorted list and every calculation shown.

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