What it is the middle score in the list after the scores are arranged in decreasing or increasing order?

Example 1 – Median time in 200 metres of a top running athlete

Imagine that a top running athlete in a typical 200-metre training session runs in the following times: 26.1 seconds, 25.6 seconds, 25.7 seconds, 25.2 seconds, 25.0 seconds, 27.8 seconds and 24.1 seconds. How would you calculate his median time?

Let’s start with arranging the values in increasing order:

Table 4.4.2.1
Rank associated with each value of 200-meter running times
Table summary
This table displays the results of Rank associated with each value of 200-meter running times. The information is grouped by Rank (appearing as row headers), Times (in seconds) (appearing as column headers).

There are n = 7 data points, which is an uneven number. The median will be the value of the data points of rank

(n + 1) ÷ 2 = (7 + 1)  ÷ 2 = 4.

The median time is 25.6 seconds.

Example 2 – Median time in 200 metres of a top running athlete (Part 2)

Now suppose that the athlete runs his eighth 200-metre run with a time of 24.7 seconds. What is his median time now?

Table 4.4.2.2
Rank associated with each value of 200-meter running times, updated
Table summary
This table displays the results of Rank associated with each value of 200-meter running times. The information is grouped by Rank (appearing as row headers), Times (in seconds) (appearing as column headers).

There are now n = 8 data points, an even number. The median is the mean between the data point of rank 

n ÷ 2 = 8 ÷ 2 = 4

and the data point of rank

(n ÷ 2) + 1 = (8 ÷ 2) +1 = 5

Therefore, the median time is (25.2 + 25.6) ÷ 2 = 25.4 seconds.

Example 3 – Median size of households of the students in the class

Imagine you ask the 30 students of your class how many people there are in their households. You summarize the data you collected in a frequency table, in which you include the relative frequencies and the cumulative relative frequencies.

Table 4.4.2.3
Frequency table of household sizes of the students
Table summary
This table displays the results of Frequency table of household sizes of the students. The information is grouped by Household size (appearing as row headers), Frequency (number of students), Relative frequency (%), Cumulative frequency (number of students) and Cumulative relative frequency (%) (appearing as column headers).

You can see that 10% of students (3 students) live in a household of size 2, 23% of students (7 students) live in a household of size 3 or less and 57% of students (17 students) live in a household of size 4 or less. The median will be equal to 4 because it’s the smallest value for which the cumulative relative frequency is higher than 50%. This is even more obvious if you visualize the cumulative relative frequency on a bar chart like on chart 4.4.2.1. The dotted line indicates the cumulative relative frequency of 50%.

Data table for chart 4.4.2.1
Table summary
This table displays the results of Data table for chart 4.4.2.1. The information is grouped by Household size (appearing as row headers), Cumulative relative frequency (%) (appearing as column headers).

The mean is the total number of people in the households of the students:

2 × 3 + 3 × 4 + 4 × 10 + 5 × 4 + 6 × 2 + 7 × 3 + 8 × 1 + 9 × 2 + 10 × 1 = 147

divided by the number of students, which is 30. The result is 147 ÷ 30 = 4.9 people per household.

In this example, the median (4) is lower than the mean (4.9).

Example 4 – Median size of households of the students in the class (Part 2)

A new student recently joined your class. You decide to ask him what the size of his household is in order to update your results. He replies to you that he lives in a large multi-generational house that includes 18 people!
Once updated, the mean is (147 + 18) ÷ 31 = 5.3 people per household. Just adding one new student increased the mean by 0.4 (5.3 – 4.9). The median is the same after the update. There are now 7 ÷ 31 = 22.6% of students in a household of size 3 or less, and 17 ÷ 31 = 54.8% of students that live in a household of size 4 or less. The value 4 is still the smallest value with a cumulative relative frequency of at least 50%.