Cambridge O Level Statistics: Formation of Frequency Distributions (4040) – Revision Notes

These revision notes cover section 3.1 of the Cambridge O Level Statistics (4040) syllabus for examination in 2025, 2026 and 2027: class measures for grouped frequency distributions, meaning class limits, boundaries and mid-points, and class intervals. For the wider topic read the study guide. When you are ready to test yourself, use the practice questions, the Cambridge O Level Statistics hub, the printable checklist and the free diagnostics.

Drawing the polygons and histograms that come after this step is covered separately in the frequency polygons and histograms notes. Using mid-points to estimate a mean is in the averages study guide.

The four class measures

Measure	Meaning	How you find it
Class interval	The range of values that one row of the table covers	Read it from the table heading
Class limits	The lowest and highest values that can be written in that class, as printed	Read them from the table
Class boundaries	The exact edges of the class on the number line, with no space left between neighbouring classes	Move each limit out by half of the gap between classes
Class mid-point	The value halfway across the class	Add the two boundaries and halve

The words “class interval” are used in two ways. Sometimes they mean the class itself, such as 20-29. Sometimes they mean its width. Check which one the question wants. When the question says “width”, it wants a number.

Three ways a class is written

Style	Example	Used for	Boundaries
Limits with a hyphen	40-44	Data recorded in steps, such as whole kilograms	Not printed: work them out
Limits with decimals	2.0-2.4	Data recorded to a set number of decimal places	Not printed: work them out
Inequality	135 ≤ h < 145	Continuous measurements	Already printed: 135 and 145

A class written as an inequality needs no adjustment. The numbers you see are the boundaries. A class with a hyphen needs the half-gap step below.

Formulas to recall

Quantity	Rule
Gap between classes	Next lower limit minus this upper limit
Lower boundary	Lower limit minus half of the gap
Upper boundary	Upper limit plus half of the gap
Class width	Upper boundary minus lower boundary
Mid-point	(lower boundary + upper boundary) ÷ 2
Class width, shortcut	Next lower limit minus this lower limit (when classes touch or are separated by one equal gap)

Method in steps: from limits to measures

1. Write down the stated lower and upper limits.
2. Find the gap to the next class. For whole numbers it is 1, for one decimal place it is 0.1, for two decimal places it is 0.01.
3. Halve the gap. Subtract it from the lower limit and add it to the upper limit.
4. Subtract the boundaries to get the width.
5. Add the boundaries and halve to get the mid-point.
6. Check: the width of the class and the distance between its mid-point and each boundary should agree. The mid-point is exactly half a width from each edge.

Small worked reminders

Whole kilograms. Masses are recorded to the nearest kilogram and one class is 40-44. The next class starts at 45, so the gap is 1 and the half-gap is 0.5. Boundaries: 39.5 and 44.5. Width: 44.5 − 39.5 = 5. Mid-point: (39.5 + 44.5) ÷ 2 = 42.

One decimal place. Lengths in metres are recorded to 1 d.p. and one class is 2.0-2.4. The next class starts at 2.5, so the gap is 0.1 and the half-gap is 0.05. Boundaries: 1.95 and 2.45. Width: 0.5. Mid-point: 2.2.

Inequality form. For 135 ≤ h < 145 the boundaries are 135 and 145. Width: 10. Mid-point: 140.

Ages in completed years. A class of 10-14 contains people whose age last birthday is 10, 11, 12, 13 or 14. Someone who is 14 years 11 months is still in it. The exact ages covered run from 10 up to, but not including, 15. Width: 5. Mid-point: 12.5. The mid-point of the printed limits would be 12, so do not use limits for ages unless the question tells you to.

Unequal widths. Classes 0 ≤ x < 10, 10 ≤ x < 30 and 30 ≤ x < 60 have widths 10, 20 and 30 and mid-points 5, 20 and 45.

Must-know distinctions

Idea	Do not confuse it with	Test
Limits	Boundaries	Limits have gaps between classes. Boundaries meet.
Width	Upper limit minus lower limit	40-44 has width 5, not 4
Mid-point from limits	Mid-point from boundaries	For whole-number classes with a gap of 1 they give the same answer, but boundaries are safer
Discrete count	Continuous measurement	A count is exact. A measurement was rounded when it was recorded.
Mid-point	Class width divided by two	A mid-point is a position on the scale. A half-width is a distance.

Choosing class intervals

Classes have to follow some rules, whatever their width:

• Every value in the data must fit in one class.
• No value may fit in two classes. Classes like 10-20 and 20-30 break this, because 20 could go in either.
• There should be no missing class in the middle of the range, even if its frequency is zero.
• Equal widths make the table easy to read and compare. Unequal widths are used when some parts of the range are crowded and others are thin.

There is no official rule for the number of classes. A common working guide is somewhere around five to ten, but the right choice depends on how many values you have and how widely they are spread. Too many classes leave rows with one or no values and hide the shape. Too few classes squeeze different features into one row. When asked to comment on a choice, refer to the actual numbers: the range, the number of observations and the number of classes this gives.

Ungrouped tables list each separate value. They suit counts with a small number of different values. A grouped table suits a wide range of values or measurements.

Quick self-test

1. A class is written 35-39 and the next is 40-44. State the boundaries and width of 35-39.
2. Find the mid-point of the class 35-39 using its boundaries.
3. Lengths are recorded to 1 d.p. and a class is 7.5-8.4. State its boundaries.
4. A whole-number class has a lower boundary of 18.5 and a width of 5. State its limits.
5. A class has a mid-point of 62 and a width of 8. State its boundaries.
6. A table has classes 0-10, 10-20 and 20-30 for whole-number data. Give the fault.
7. State the width of the class 50 ≤ m < 65 and its mid-point.
8. A person aged 29 years 10 months is recorded by age last birthday. In which of the classes 20-29 and 30-39 does the person belong?
9. Why are boundaries needed even when the printed limits are clear?
10. A grouped table of 12 values has 11 classes. Comment.

Answers

1. The gap is 1, so the half-gap is 0.5. Boundaries are 34.5 and 39.5. Width is 5.
2. (34.5 + 39.5) ÷ 2 = 37.
3. The gap is 0.1, so the half-gap is 0.05. Boundaries are 7.45 and 8.45.
4. The upper boundary is 18.5 + 5 = 23.5. The limits are 19 and 23.
5. Half the width is 4. The boundaries are 62 − 4 = 58 and 62 + 4 = 66.
6. The classes overlap. A value of 10 or 20 could be placed in two rows.
7. Width is 65 − 50 = 15. Mid-point is (50 + 65) ÷ 2 = 57.5.
8. The class 20-29, because the age last birthday is 29.
9. Boundaries show where one class stops and the next starts, so there is no gap. Widths and mid-points are found from them.
10. Too many classes. With 12 values, most rows would hold one value or none, so the shape of the data would not show.

Where marks are usually lost

• Calculating the width as upper limit minus lower limit, giving 4 instead of 5 for a class like 40-44.
• Writing 44 and 45 as the boundaries, when the boundary between two classes is the half-way value 44.5.
• Using a half-gap of 0.5 for data recorded to one decimal place, instead of 0.05.
• Adjusting an inequality class that is already in boundary form.
• Treating ages as if they were rounded to the nearest year, when the question says age last birthday.
• Giving a mid-point that is not between the boundaries, usually from an arithmetic slip. Do a quick look-back to check.
• Saying “too many classes” or “too few” with no reference to the data.
• Writing a class width for unequal classes as one value for the whole table.

Frequently asked questions

How do you find class boundaries in a grouped frequency table? Find the gap between one class’s upper limit and the next class’s lower limit, halve it, then subtract the half-gap from the lower limit and add it to the upper limit. For a whole-number class of 40-44 this gives boundaries of 39.5 and 44.5.

How do you calculate the mid-point of a class? Add the lower and upper class boundaries and divide by two. For the class 40-44 recorded to the nearest kilogram, the mid-point is (39.5 + 44.5) ÷ 2 = 42.

What does “class interval” mean in statistics? The term is used in two ways: sometimes it means the class itself, such as 20-29, and sometimes it means the class width. When a question asks for the width, it wants a number.

How do you handle ages recorded as age last birthday? A class such as 10-14 holds exact ages from 10 up to, but not including, 15, so its width is 5 and its mid-point is 12.5. Using the printed limits would wrongly give a mid-point of 12.

How many classes should a grouped frequency table have? There is no official rule in Cambridge O Level Statistics 4040. A common working guide is around five to ten, but the right choice depends on how many values there are and how widely they are spread.

Official syllabus

Cambridge O Level Statistics (4040), syllabus for examination in 2025, 2026 and 2027, Cambridge University Press & Assessment (Cambridge International). Section 3.1: class measures for grouped frequency distributions, including class limits, boundaries and mid-points, and class intervals.