Mode and median (AQA GCSE Statistics): Revision Notes

Mode and median

When working with grouped data in frequency tables, you cannot determine exact values for the mode and median. Instead, you can only identify the class intervals where these averages are located and estimate their values using specific methods.

Understanding grouped data limitations

In grouped data, individual data values are organised into class intervals rather than being listed separately. This means you only know which interval each data point falls into, not the precise value. As a result, you cannot calculate exact averages but must work with class intervals and estimates instead.

Modal class

The modal class is the class interval that contains the mode. Since you cannot determine the exact mode value from grouped data, identifying the modal class is the best you can achieve.

To find the modal class, examine your frequency table and identify which class interval has the highest frequency. This interval contains the most data values and is therefore the modal class.

For example, in the time late data shown above, the modal class is 0 < T ≤ 5 because it has the highest frequency of 12.

Class containing the median

The median class is the class interval that contains the median value. To find this interval, you need to use cumulative frequency.

Finding the median position

First, calculate the position of the median using the formula:

Median position = $\frac{n + 1}{2}$

Where n is the total number of data values (the sum of all frequencies).

Using cumulative frequency

Create a cumulative frequency column by adding up the frequencies as you go down the table. The median class is the first interval where the cumulative frequency reaches or exceeds the median position.

In the example above:

• Total frequency (n) = 43
• Median position = $\frac{43 + 1}{2} = 22$
• Looking at cumulative frequencies: 12, 20, 31, 39, 43
• The 22nd data value lies in the interval 10 < T ≤ 15 (cumulative frequency 31)

Estimating the median value

Once you've identified the median class, you can estimate the actual median value using this formula:

Estimated median = $L + \frac{\frac{1}{2}n - F}{f} \times w$

Where:

• L = lower boundary of the median class interval
• n = total number of values
• F = cumulative frequency of intervals before the median class
• f = frequency of the median class interval
• w = width of the median class interval

Worked example

Key examination tips

Common mistakes to avoid