If you are looking for a fairer way to summarize a set of data, this median calculator is for you. The mean, or average, of a dataset, can be significantly affected by a few extreme values, whereas the median is less sensitive. Read on to learn how to find the median, how to find the median of a set of numbers using the median formulas, and what the median symbols used in statistics books mean.
What is the median?
The median value of a set of numbers is the value at which half of the numbers in the set are below it, and the other half are above it. It is a measure of the center of a sample or population and is sometimes called the "middle" number.
It is similar to the average (or mean) value. However, if you have a dataset with a few extremely large or small values compared to the rest, the median is a better measure of the "typical" value.
Let's look at an example that illustrates the difference between the median and the mean. For the data set 4, 5, 6, 7, the mean and median are the same with a value of 5.5. If we add the number 88, the mean jumps up to 22, whereas the median only increases slightly to 6. So, for skewed data sets, such as household income, the median is a better measure of the typical value. If you're not sure what this means, our skewness calculator has all the information.
Median vs mode
What about median vs mode? The mode is the value from a dataset that appears the greatest number of times. For a normal distribution, the mode will have the same value as the median and the mean. For skewed distributions, these three values can differ greatly.
The mode calculator can help you if you're interested in this quantity as well.
There is no standard median symbol, but some commonly used ones are x᷉, μ1/2, and M.
How to find the median number using median formulas?
We now know how the median number is defined, so we should look at how to calculate the median. The first step is to sort the values into numerical order (or reverse numerical order - you'll get the same result!).
The second step is to find the middle number or numbers in the sorted dataset. How you do this depends on whether there's an odd or even number of values in your dataset.
If there are an odd number of values, the median is simply the middle number. For the dataset 3, 5, 7, 9, 11, the number 7 is the middle number, with two values on either side. So the median is 7.
For a dataset with an even number of values, you take the mean of the two center values. So, if the dataset has the values 1, 4, 7, 9, the two center values are 4 and 7. The mean of these middle values is
(4 + 7) / 2 = 5.5, so the median is 5.5.
We can also write down two formulas for finding the median, one for the odd case and another for the even scenario.
median (odd dataset) = x(n+1)/2
median (even dataset) = (xn/2 + x(n+2)/2) / 2
xis a value in the sorted dataset, with the subscript indicating its position in the sorted list; and
nis the number of values in the dataset.
So, the odd dataset median formula says, add one to the number of values and divide by 2 to find the index of the median number. The even median formula says to take the
n/2'th and the
(n+2)/2'th values and calculate their mean value to find the median.
How to use this calculator to find the median?
Here's how to use our median calculator to find the median of a dataset. It can also show you the step-by-step procedure to manually calculate the answer.
- Enter your data row by row, with one number in each row of the median calculator. As you enter the numbers, a new row will appear for you to enter the next value. The calculator supports datasets with up to 50 values.
- The median number will be displayed for you as you go along.
- If you want to see the steps used to reach the answer, select "Yes" from the drop-down menu where the calculator asks, "Show step-by-step solution?".
- If you want to analyze another dataset, click the reload button at the bottom of the calculator.
How to find the median of a set of numbers?
Let's show an example of a step-by-step solution for a dataset with the following 15 values:
58, 47, 55, 6, 5, 14, 60, 3, 39, 6, 28, 15, 87, 31, 19
Sorting the numbers, we get:
3, 5, 6, 6, 14, 15, 19, 28, 31, 39, 47, 55, 58, 60, 87
There are 15 values, so using the formula
(n + 1) / 2 and using
n = 15, we find that we need the 8th number in the sorted dataset. So the median is 28.
Another dataset has 16 values:
71, 71, 5, 18, 98, 23, 53, 92, 74, 82, 65, 74, 97, 75, 87, 13
Sort them to get:
5, 13, 18, 23, 53, 65, 71, 71, 74, 74, 75, 82, 87, 92, 97, 98
Using the formula for the median, when there is an even number of values, we need to take the mean value of the
(n+2)/2'th values. So that's the 8th and 9th values, which are 71 and 74, respectively. Then we need to take the mean of these values:
(71 + 74) / 2 = 145 / 2 = 72.5. So the median is 72.5.
💡 Throughout this article, we've been dealing with the median of a sample. If you've encountered median in statistics and need to test for the equality of medians in two populations, you may want to take a look at our Wilcoxon-Mann-Whitney U test calculator
How do I calculate the median?
To determine the median of a dataset, take the following steps:
- Sort the observations in ascending order.
- Determine (e.g., count) how many observations you have in your dataset.
- If you have an odd number of observations, there is a number right in the middle of your (sorted!) dataset. It's the median.
- If you have an even number of observations, two numbers are in the middle. Add them together and divide the result by
2: the result is the median!
What is the median of the set 0, 1, 1, 18?
The answer is 1. Our data is already sorted, and it has four elements. We clearly see the two values in the middle are
1. Calculating the average of the two identical numbers poses no problem: it's again
1, and this is our median.
When should I use median vs mean?
Both the median and the mean are measures of central tendency, i.e., we use them to describe where the center of a dataset lies.
- Use the median for skewed distributions or if you see clear outliers (i.e., observations that lie well outside of the remaining data set). These factors would distort the result if you used the mean.
- Use the mean for symmetric distributions with no clear outliers.
When is the median equal to the mean?
The mean and the median coincide for symmetric distributions (for instance, the normal distribution). If this distribution has only one mode, then this mode coincides with both the median and the mean.