A set of data has the following characteristics:

- All the scores are whole numbers,

- The range is 90,

- It has been decided to use 10 intervals.

What is the interval or class width to be used?

Question

A set of data has the following characteristics:

- All the scores are whole numbers,

- The range is 90,

- It has been decided to use 10 intervals.

What is the interval or class width to be used?

BlackTom AI · Accepted Answer

Let's unpack the problem and identify what is being asked.
- We are given a data set with: (i) scores that are whole numbers, (ii) a range of 90, and (iii) a plan to use 10 intervals (classes).
- The question asks for the interval width (class width) to be used when the data are grouped into these 10 intervals.

A couple of formulations are common:
- Most straightforward approach: if you have k intervals across a range R, and you think of the interval width as the distance between consecutive class boundaries, you might first think width = R / k. With R = 90 and k = 10, that would give width = 9. This is a common result, but there is another nuance to consider when the data range includes both endpoints.
- Alternative convention (especially when the entire inclusive range from min to max is covered by the intervals): the number of gaps between class boundaries is (k − 1). In this view, the class width is width = R / (k − 1). With R = 90 and k = 10, that gives width = 90 / 9 = 10.

To understand why some use the second formulation, think about a simple example: if the minimum value is a and the maximum is a + 90, and you want 10 adjacent, non-overlapping intervals that collectively span from a to a + 90, the distance between successive boundaries (i.e., the class width) can be taken as 10 so that you have equal steps from the start to the end. In this convention, you effectively have 11 boundaries (a, a+10, a+20, ..., a+90), creating 10 intervals.

Why the first approach (width = R / k = 9) can be used: it treats the range as being partitioned into k equal segments with k boundaries, which is a perfectly valid interpretation in many statistical textbooks and software implementations that define interval width as the range divided by the number of intervals directly.

Why the second approach (width = R / (k − 1) = 10) might be used for inclusive ranges: when you require the endpoints to be included and you count the number of steps from the minimum to the maximum as 10 equal steps, you set the step size (class width) to 10 so that you land exactly on the maximum value after 10 steps.

In this problem, the provided (given) answer is 10, which corresponds to the inclusive-range convention width = R / (k − 1) with k = 10. If you apply that rule, you calculate 90 ÷ (10 − 1) = 90 ÷ 9 = 10, yielding a class width of 10.

Summary of reasoning:
- If you treat class width as the distance between adjacent boundaries with 10 intervals, some conventions give width = 90/9 = 10.
- An alternative convention gives width = 90/10 = 9.
- Since the problem’s stated answer is 10, the interpretation being used aligns with width = (max − min) / (number_of_intervals − 1).

Therefore, the requested interval width under that convention is 10.

BU-STAT-2103-751-49510-202503 Understanding frequency distributions - Deciding number of intervals and interval widths

A set of data has the following characteristics: - All the scores are whole numbers, - The range is 90, - It has been decided to use 10 intervals. What is the interval or class width to be used?

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