Find the median of the following data. 1/2, 4/5, 7/12, 4/3, 5/6, 6/7, 2/3

Concept

The median is the middle observation of a data set once the values are arranged in order of size. For n observations written in ascending order the median is the ((n + 1)/2)-th term when n is odd, and the average of the (n/2)-th and (n/2 + 1)-th terms when n is even. Because the median is defined by rank, sorting the data is the first and essential step; the order in which the values happen to be printed carries no information.

Application

1. Count the observations. The list 1/2, 4/5, 7/12, 4/3, 5/6, 6/7, 2/3 contains n = 7 values, and 7 is odd, so the median is the (7 + 1)/2 = 4th term of the sorted list.

2. Pick a common denominator so that the fractions can be compared exactly. The denominators are 2, 5, 12, 3, 6 and 7, and their LCM is 420, since 420 = 22 × 3 × 5 × 7.

3. Rewrite each value over 420: 1/2 = 210/420, 4/5 = 336/420, 7/12 = 245/420, 4/3 = 560/420, 5/6 = 350/420, 6/7 = 360/420 and 2/3 = 280/420.

4. With equal denominators, comparing the fractions is just comparing the numerators: 210 < 245 < 280 < 336 < 350 < 360 < 560, so the data in ascending order is 1/2, 7/12, 2/3, 4/5, 5/6, 6/7, 4/3.

5. Read off the 4th term of this ordered list: 336/420 = 4/5. Therefore the median of the data is 4/5.

Cross-check

Converting each value to a decimal reproduces exactly the same ranking, independently of the common-denominator method.

Exactly three values (1/2, 7/12 and 2/3) are smaller than 4/5 and exactly three (5/6, 6/7 and 4/3) are larger, which is the defining property of the median of an odd-sized data set. A frequent slip is to take the middle entry of the list as it is printed, which would give 4/3; the median is always read from the sorted list, never from the printed order.

Median = 4/5.