How To Calculate The Median

If observations of a variable are ordered by value, the median value corresponds to the middle observation in that ordered list. The median value corresponds to a cumulative percentage of 50% (i.e., 50% of the values are below the median and 50% of the values are above the median). The position of the median is

The Median is the score found at the exact middle of the set of values. One way to compute the median is to list all scores in numerical order, and then locate the score in the center of the sample. For example, if there are 500 scores in the list, score #250 would be the median. If we order the 8 scores shown above, we would get:

15,15,15,20,20,21,25,36

There are 8 scores and score #4 and #5 represent the halfway point. Since both of these scores are 20, the median is 20. If the two middle scores had different values, you would have to interpolate to determine the median.

Example 1 – Raw data (discrete variables)

Imagine that a top running athlete in a typical 200-metre training session runs in the following times:

26.1, 25.6, 25.7, 25.2 and 25.0 seconds.

How would you calculate his median time?

First, the values are put in ascending order: 25.0, 25.2, 25.6, 25.7, 26.1. Then, using the following formula, figure out which value is the middle value. Remember that n represents the number of values in the data set.

Median = (n + 1) ч 2th value
= (5 + 1) ч 2
= 6 ч 2
= 3

The third value in the data set will be the median. Since 25.6 is the third value, 25.6 seconds would be the median time.

= 25.6 seconds

 

Example 2 – Raw data (discrete variables)

Now, if the runner sprints the sixth 200-metre race in 24.7 seconds, what is the median value now?

Again, you first put the data in ascending order: 24.7, 25.0, 25.2, 25.6, 25.7, 26.1. Then, you use the same formula to calculate the median time.

Median = (n + 1) ч 2th value
= (6 + 1) ч 2
= 7 ч 2
= 3.5

Since there is an even number of observations in this data set, there is no longer a distinct middle value. The median is the 3.5th value in the data set meaning that it lies between the third and fourth values. Thus, the median is calculated by averaging the two middle values of 25.2 and 25.6. Use the formula below to get the average value.

Average = (value below median + value above median) ч 2
= (third value + fourth value) ч 2
= (25.2 + 25.6) ч 2
= 50.8 ч 2
= 25.4

The value 25.4 falls directly between the third and fourth values in this data set, so 25.4 seconds would be the median time.

 

Example 3 – Ungrouped frequency table (discrete variables)

Imagine that your school baseball team scores the following number of home runs in 10 games:

4, 5, 8, 5, 7, 8, 9, 8, 8, 7

If you were to place the total home runs in a frequency table, what would the median be?

First, put the scores in ascending order:

4, 5, 5, 7, 7, 8, 8, 8, 8, 9

Then, make a table with two columns. Label the first column "Number of home runs" and then list the possible number of home runs the team could get. You can start from 0 and list up until the number 10, but since the team never scored less than 4 home runs, you may wish to start listing at the number 4.

Label the second column "Frequency." In this column, record the number of times 4 home runs were scored, 5 home runs were scored and so on. In this case, there was only one time that 4 home runs were scored, but two times that 5 home runs were scored. If you add all of the numbers in the Frequency column, they should equal 10 (for the 10 games played).

To find the median, again use the same formula:

Median = (n + 1) ч 2^th value
= (10 + 1) ч 2
= 11 ч 2
= 5.5
= the median is the 5.5^th value in the data set

To get the median, add up the numbers in the Frequency column until you get to 5 (and since the total number of games is 10, the remaining numbers in that column should also equal 5). You will reach 5 after adding all of the frequencies up to and including those for the 7 home runs. The next set of five will begin with the frequencies for 8 home runs. The median (the 5.5^th value) lies between the fifth value and the sixth value. Thus, the median lies between 7 home runs and 8 home runs.

If you calculate the average of these values (using the same formula used in Example 2), the result is 7.5.

Average = (middle value before + middle value after) ч 2
= (fifth value + sixth value) ч 2
= (7 + 8) ч 2
= 15 ч 2
= 7.5

Technically, the median should be a possible variable. In the above example, the variables are discrete and always whole numbers. Therefore, 7.5 is not a possible variable—no one can hit 7 and a half home runs. Thus, this number only makes sense statistically. Some mathematicians may argue that 8 is a more appropriate median.