Median and quartiles

Median and quartiles

What is a median?

The median is the middle number found when the list is ordered. Using a cumulative frequency we can quickly and easily find the median of a data set since the groups are already ordered and the cumulative frequency increases. However, this method may not always be hugely accurate as the curve has been drawn on by hand and so every person may draw this slightly different.

The median is found by halving the total cumulative frequency (which is the same as the total frequency) and then drawing a line through this point on the graph that cuts the drawn curve. Then a vertical line is drawn from the point, and the value on the axis is read off which is the median for the data.

Example

Find the median from the following cumulative frequency graph:

Median from cumulative frequency

The cumulative frequency goes up to 45 and so we must draw a line across that cuts the vertical axis at half here (since the median is the middle point) and then find the mark at the point where the line cuts this curve:

Median from cumulative frequency curve

The point that is found on the x axis is 36. So the median for this data must be 36 marks.

Quartiles

After we have found the median, the horizontal line has really just divided the graph into two equal parts. This can be done again for each of these parts to split the graph up into quarters which are called quartiles. These are found by dividing the frequency up by 4 and then each quartile will have this many pieces of data within it. The first quartile is called the ‘lower quartile’ and the last is known as the ‘upper quartile’. Quartiles split the data into 4 different sections each with 25% of the frequency.

Example

Find the lower and upper quartiles of the cumulative frequency graph shown here as well as the median.

Quartiles from a curve

Doing this we use the fact that the entire frequency is 60. So the quartiles will be at every 15, meaning we must draw lines for the 15th, 30th and 45th.

This gives us a median of 32 when reading off the value corresponding to the 30th piece of data. The upper and lower quartiles marks are 26 and 36. So we then know that one quarter of people gained a score of 26 marks or less and that three quarters of people scored 36 marks or less.

Measures of dispersion

Dispersion is also known as the ‘spread’ of data. The simplest way of measuring this is by finding the range, which is defined as:

range = highest value – lowest value

This then gives us a range which all the results must be in since no result will ever exceed the highest or fall below the lowest.

Example

Find the ranges of the following data sets:

1) 25, 48, 36, 29, 22, 41, 46, 33, 27, 30, 45, 41, 30

2) 5, 4, 7, 6, 6, 3, 4, 8, 8, 6, 7, 5, 7, 4, 6

3) 109, 111, 103, 103, 105, 106, 107, 104, 110, 106

1) Here we have a list of numbers and the highest is the number 48 and the lowest is 22. By taking 22 from 48 we get the range which is 48-22=26

2) Now the largest number is 8 and the smallest is 3. Giving a range of 8-3=5

3) The largest number is 111 and the smallest 103 so the range must be 111-103=8

Example

Glen carries out a survey and asks people of different ages some questions, as shown below. What is the range of ages of people that he asked?

Survey results

Since there were no people over 61 or under 11 that were asked we can call 60 the highest value and 11 the lowest (since we do not know the exact ages of people in a group). So the range must be 60-11=49 years.

Like in the example above, when we find data that is grouped then the highest value is the highest possibility in the group and the lowest value is the lowest possibility. Therefore, the highest value in the group 30 – 40 is 40 and the lowest is 30. It is these highest and lowest values that must be used to calculate the range.

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