Cumulative Frequency (Edexcel GCSE Maths): Revision Notes

Cumulative frequency

What is cumulative frequency?

Cumulative frequency is a way of keeping track of how many data values you've counted up to any particular point. Think of it as a running total - you start with the first group and keep adding the frequencies as you move through each group. This gives you the total number of values that fall at or below each point in your data set.

The key concept is that cumulative frequency shows you the total frequency accumulated so far, rather than just the frequency of individual groups.

Creating a cumulative frequency table

To create a cumulative frequency table, you start with your regular frequency table and add a new column. Here's how you build it step by step:

1. Start with your frequency table - this shows your data groups and how many values fall in each group
2. Add a cumulative frequency column - this will show your running totals
3. Fill in the running totals - for each row, add the current frequency to all the frequencies above it

Height (h cm)	Frequency	Cumulative Frequency
140 < h ≤ 150	4	4
150 < h ≤ 160	9	4 + 9 = 13
160 < h ≤ 170	20	13 + 20 = 33
170 < h ≤ 180	33	33 + 33 = 66
180 < h ≤ 190	36	66 + 36 = 102
190 < h ≤ 200	15	102 + 15 = 117
200 < h ≤ 210	3	117 + 3 = 120

The table demonstrates this process clearly. Notice how the cumulative frequency increases as you move down each row, showing the total count of all values up to that point.

Drawing a cumulative frequency graph

Creating an accurate cumulative frequency graph requires careful plotting and specific techniques:

Setting up the graph

• Plot the highest value from each class interval on the horizontal axis
• Plot the cumulative frequency on the vertical axis
• Start at zero using the lowest value from your first class interval

Plotting the points

Each point represents the cumulative frequency up to the upper boundary of each class interval. For example, if you have height ranges like 140-150cm, 150-160cm, etc., you would plot points at 150, 160, 170, and so on.

Drawing the curve

Once all points are plotted, connect them with a smooth curve rather than straight lines. This smooth curve is essential for accurate readings and represents the continuous nature of the data.

Finding vital statistics from the graph

The cumulative frequency graph is particularly useful for finding important statistical measures without complex calculations.

Finding the median

The median represents the middle value of your data set. To find it:

Finding the quartiles

Quartiles divide your data into four equal parts:

Lower quartile (Q1):

• Position = $\frac{1}{4} \times \text{total frequency}$
• Follow the same process as finding the median, but use the quarter position

Upper quartile (Q3):

• Position = $\frac{3}{4} \times \text{total frequency}$
• Again, use the same horizontal-to-curve-to-vertical reading process

Calculating the interquartile range

The interquartile range (IQR) shows the spread of the middle 50% of your data:

$\text{IQR} = \text{Upper quartile} - \text{Lower quartile}$

This gives you the range that contains the middle half of all your data values.

Making estimates from cumulative frequency graphs

Cumulative frequency graphs are excellent tools for making estimates about your data. You can work in two directions:

Estimating frequencies

If you want to know how many values are less than or equal to a particular value:

1. Find your target value on the horizontal axis
2. Go up vertically to meet the curve
3. Read across horizontally to find the cumulative frequency
4. This tells you how many values fall at or below your target

Estimating values

If you want to find the value that corresponds to a particular cumulative frequency:

1. Find your target frequency on the vertical axis
2. Go across horizontally to meet the curve
3. Read down vertically to find the corresponding value

This technique is particularly useful for finding percentiles or understanding the distribution of your data.

Practical tips for success