Linear Interpolation (Edexcel GCSE Statistics)

• The formula can be tricky to remember correctly

  • It’s better to understand how the method works

  • Then you don’t need to remember the formula

• Remember that the median found this way is an estimate

  • You can’t find the exact median without knowing all the data values

A student collected data about the length of time (x hours) students in his school spent listening to music in a given week. He collected data from 50 students in total.  The following table summarises the data:

Work out an estimate for the median amount of time spent listening to music by the students.

STEP 1: Identify the class interval containing the median

Divide the total number of values, n, by 2

Here n = 50

So we’re looking for the interval with the 25^th value

Note that this is different from finding the median from a set of data values

In that case we would be looking for the value halfway between the 25^th and 26^th values

With linear interpolation we don’t have to worry about that!

Add a cumulative frequency column to the table and work out the cumulative frequencies

The second class interval goes up to the 22^nd data value and the third class interval goes up to the 34^th data value

So the median is in the third class interval

The median is in the class interval

STEP 2: Find how far into the class interval the ^th data value is

The interval with the median contains the 23^rd through 34^th data values

The 25^th data value is ‘3 values in’ to the interval (23, 24, 25)

And there are 12 data values in the interval

So the median is 1/4 of the way into the interval

STEP 3: Multiply the class width by the fraction found in Step 2

Subtract the lower boundary from the upper boundary to find the class width of the interval

30-20=10

Multiply by the fraction in Step 2

STEP 4: Add the result from Step 3 to the lower boundary of the class interval

20 + 2.5 = 22.5

Don’t forget the units in your answer!

Estimated median = 22.5 hours