Constructing a Grouped Frequency Distribution

Hello again. In the previous lesson, you learned to judge how a sample was chosen and whether it could be biased. Once data have been collected, the next challenge is making a long list of numbers readable.

This lesson introduces the grouped frequency distribution: a table that places quantitative values into sensible intervals and counts how many observations fall in each interval. By the end, you will be able to choose a class width, write non-overlapping class limits, and check that the table includes every value exactly once.

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Why group numerical data?

A regular frequency table can list each individual value. That works for a small set such as the number of books bought by students: 1 book, 2 books, 3 books, and so on.

But suppose you have 75 test scores ranging from 43 to 89. Listing every score separately produces a long table with many frequencies of 0, 1, or 2. It is much more useful to combine scores into intervals such as 40–49, 50–59, and 60–69.

A grouped frequency distribution has two main pieces:

• Classes: non-overlapping intervals of values
• Frequency: the number of observations in each class

The goals are simple but important:

1. Every data value must fit in a class.
2. No data value can fit in two classes.
3. Classes should have a consistent width.
4. The grouping should make the distribution easier—not harder—to see.

The grouped table is the foundation for a histogram, which you will interpret later in this module.

To see the full construction process once before breaking it into rules, watch Math with Mr. J’s worked example.

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The vocabulary: classes, limits, and width

Consider the grouped distribution below. It summarizes a variable called “Wins.”

For the first class, 26–30:

• Lower class limit: 26
• Upper class limit: 30
• Frequency: 2
• Class width: 5

Why is the width 5 rather than $30-26=4$? Because this is a whole-number class that includes five possible values:

$$26,27,28,29,30$$

For whole-number data written with inclusive limits, calculate the class width by counting the values:

$$\text{class width}=\text{upper class limit}-\text{lower class limit}+1$$

You can also find it by subtracting consecutive lower class limits:

$$31-26=5$$

That second method is often faster. In the image, the lower limits are 26, 31, 36, 41, and so on; each increases by 5.

A key distinction: whole-number versus measured data

For counts recorded as whole numbers, such as wins, customers, or books, classes are often written inclusively:

Class	Values included
20–29	20 through 29
30–39	30 through 39
40–49	40 through 49

Each class has width 10.

For measured data that may include decimals, such as height or time, it is clearer to write classes using “less than”:

Class	Meaning
60 to less than 62	$60\le x<62$
62 to less than 64	$62\le x<64$
64 to less than 66	$64\le x<66$

Here each class has width 2. A height of 62 belongs in the second class, not both classes. The phrase “less than” prevents overlap.

For this lesson, focus first on the common test format using whole-number class limits.

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A dependable method for constructing classes

Often a question tells you how many classes to use. If it does not, a common guideline is roughly 5 to 15 classes: enough to show a pattern, but not so many that the table becomes cluttered.

Suppose you know:

• the minimum value, $x_{\min }$;
• the maximum value, $x_{\max }$;
• the desired number of classes, $k$.

Step 1: Find the range

$$\text{range}=x_{\max }-x_{\min }$$

The range tells you how far the data extend from smallest to largest.

Step 2: Estimate a class width

A standard starting point is:

$$\text{approximate width}=\frac{x_{\max }-x_{\min }}{k}$$

For whole-number data, choose a convenient whole-number width that is at least this large—usually by rounding up, not by ordinary rounding.

Rounding upward prevents the last class from ending before the maximum value.

A useful safeguard for closed whole-number classes is:

$$\text{width}=\lceil \frac{x_{\max }-x_{\min }+1}{k}\rceil$$

The $+1$ accounts for the fact that a class such as 20–29 includes both endpoints. You do not need to memorize this version if your course uses the approximate-width rule, but it explains why you should always check that your final class includes the maximum.

Step 3: Choose the first lower class limit

You may begin at the minimum value, or at a slightly smaller, convenient number.

For example, if the smallest age is 23 and your width is 10, beginning at 20 gives clean classes:

• 20–29
• 30–39
• 40–49

Starting at 20 is valid because it includes the minimum value of 23. It also makes the table easy to read.

Step 4: Write all lower class limits

Add the class width repeatedly.

If the first lower class limit is 20 and the width is 10, the lower limits are:

$$20,30,40,50,60$$

Step 5: Write the upper class limits

For inclusive whole-number classes:

$$\text{upper class limit}=\text{lower class limit}+\text{width}-1$$

So a class beginning at 20 with width 10 ends at:

$$20+10-1=29$$

The next class begins at 30, so there is no gap and no overlap.

This short reading gives the formal requirement that every observation must fall into one—and only one—class, then applies the width calculation to a height dataset.

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Worked example: constructing the complete table

Suppose these are test scores for 24 students:

$$43,47,52,54,56,58,60,61,63,64,65,67,$$ $$68,70,72,73,74,76,77,80,81,83,86,89$$

Construct a grouped frequency distribution using 5 classes.

1. Identify the minimum and maximum

$$x_{\min }=43$$ $$x_{\max }=89$$ $$\text{range}=89-43=46$$

2. Estimate the width

$$\frac{46}{5}=9.2$$

Round up to a convenient width of 10.

3. Select a starting lower limit

The minimum is 43. Starting at 40 is slightly lower, clean, and still includes every score.

With width 10, the five classes are:

• 40–49
• 50–59
• 60–69
• 70–79
• 80–89

Notice the last class includes 89, the maximum score. That is an essential check.

4. Tally and count the observations

Test score class	Frequency
40–49	2
50–59	4
60–69	7
70–79	6
80–89	5
Total	24

For example, the class 60–69 contains:

$$60,61,63,64,65,67,68$$

so its frequency is 7.

Finally, check the total:

$$2+4+7+6+5=24$$

The total frequency must equal the number of original observations. If it does not, a value was missed, counted twice, or placed in the wrong interval.

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The three checks that prevent most mistakes

Before accepting a grouped frequency distribution, run this quick audit.

1. Equal width

The lower class limits should increase by the same amount each time.

For the score example:

$$40,50,60,70,80$$

Each lower limit increases by 10, so the widths are consistent.

2. No gaps or overlaps

For whole-number classes with width 10:

• 40–49 is followed by 50–59: no whole-number score is missing.
• A score of 50 belongs only to 50–59: no overlap occurs.

Bad class limits might look like this:

Problem	Example	Why it fails
Gap	40–49, 51–60	A score of 50 fits nowhere.
Overlap	40–50, 50–60	A score of 50 fits twice.
Unequal width	40–49, 50–64	The second interval is much wider.
Maximum excluded	40–49, 50–59, 60–69 when the maximum is 72	The value 72 has no class.

3. Frequencies total correctly

Add the frequencies. The result must equal the sample size.

This does not prove that every value was classified correctly, but it catches many basic tallying errors.

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Choosing sensible limits is partly a judgment call

More than one grouped frequency distribution can be correct for the same data.

For the test scores, a width of 10 beginning at 40 is especially readable. But if a question specified a different number of classes or a particular width, you would follow those directions.

When you have freedom to choose, prefer classes that are:

• easy to interpret;
• equal in width;
• broad enough to show a general pattern;
• narrow enough that important differences are not hidden;
• guaranteed to include the full data range.

For instance, grouping ages as 20–29, 30–39, and 40–49 is generally more readable than using awkward limits such as 23–31, 32–40, and 41–49—unless there is a specific reason to do so.

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Key takeaways

A grouped frequency distribution organizes quantitative data into class intervals and records the number of observations in each class.

The core procedure is:

1. Find the minimum and maximum.
2. Decide on the number of classes, if it is not given.
3. Estimate a class width using the range divided by the class count.
4. Round the width upward to a convenient value.
5. Choose a sensible first lower class limit.
6. Build equal-width, non-overlapping classes that include the maximum.
7. Tally the data and verify that frequencies add to the total number of observations.

For whole-number classes, remember that 20–29 has width 10 because it contains ten possible values. For measured data, class labels such as “60 to less than 62” make the boundary rule clear.

Next, you will extend this same table by calculating relative frequencies and cumulative frequencies.