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Grouped frequency and class boundaries

One thing that you may have noticed in this lesson is the class boundaries for some of the groups. In a previous post we learnt that each boundary should have individual values so that data cannot be put into two categories, but here we have used categories such as 0 – 60 and 60 – 90. This is because the data is known as continuous and a value will never be allowed to go into both groups. This is because, even though a value may be 60 minutes, this has really been rounded. A true time will always be either a little bit more or a little bit less than 60 minutes.

Alternative notation for grouped frequency

To avoid any confusion we can instead use different notation in these problems. By using x to represent the specific value we can make use of inequalities for the groups so that we are not leaving any groups ambiguous.

The use of < and > can help us to show that we are looking at a group where the values are less than or more than some value and the symbols ≤ and ≥ will tell us that the frequency includes values that are equal to the value too.


Change the following frequency table so that the groups use inequalities:

Weight of package (kg) Frequency
0 – 10 47
10 – 20 29
20 – 30 18
30 – 40 11
40 – 50 3

We must alter the groups slightly so that if we had a package that is exactly 10kg then we would know which it would be counted in. At the moment a package like this could be in either the first or second group. Inserting inequalities and to represent the individual weight we can have:

Weight of package (kg) Frequency
0<x \leq10 47
10<x \leq20 29
20<x \leq30 18
30<x \leq40 11
40<x \leq50 3

This table has then eliminated any ambiguity and enabled us to categorise a package that is exactly on the weight of 10kg, 20kg, 30kg etc.

Time 0 – 60 60 – 90 90 – 120 120 – 180
Frequency 5 11 29 12
Cumulative Freq 5 16 45 57
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