Frequency Distribution — Tables, Graphs & Class Intervals

A frequency distribution is a table or graph that shows how often each value, category, or interval appears in a data set. If the data is large, nearby values are often grouped into class intervals, and the count in each interval is the frequency.

It helps because raw data is often hard to scan. A frequency distribution makes the pattern visible quickly: where values cluster, where they thin out, and which results are most common.

Frequency Distribution Table Vs. Grouped Table

For ungrouped data, the table can list each value separately. If the scores are $4, 5, 5, 6, 6, 6$, the frequency of $6$ is $3$ because it appears three times.

For larger numerical data sets, exact values are often grouped into intervals such as $40$-$49$, $50$-$59$, and $60$-$69$. That version is called a grouped frequency distribution.

How Class Intervals Work

A class interval is a range used to collect nearby values into one group. In a good grouped table, every observation fits into exactly one class, and the classes do not overlap.

If you use intervals like $10$-$19$, $20$-$29$, and $30$-$39$, a value such as $24$ belongs in exactly one class. That clear rule matters. If class limits overlap, the table becomes ambiguous.

The class width is the size of each interval. If the classes are $10$-$19$, $20$-$29$, and $30$-$39$, the width is constant. This matters when you draw a histogram: direct bar-height comparisons are safe only when the class widths are equal.

Worked Example: Reading A Frequency Distribution

Suppose a teacher records the quiz scores of $20$ students and groups them into intervals:

Score interval	Frequency
$0$-$9$	$2$
$10$-$19$	$5$
$20$-$29$	$8$
$30$-$39$	$4$
$40$-$49$	$1$

The interval $20$-$29$ has the highest frequency, so it is the most common score range. This does not mean every student scored the same number. It means $8$ students scored somewhere inside that interval.

The frequencies also add up to the total number of students:

$$2 + 5 + 8 + 4 + 1 = 20$$

If you want a proportion instead of a count, use relative frequency:

$$\text{relative frequency} = \frac{\text{frequency}}{\text{total number of observations}}$$

For the interval $20$-$29$, the relative frequency is:

$$\frac{8}{20} = 0.4$$

So $40\%$ of the students scored between $20$ and $29$.

Frequency Distribution Graphs: Bar Graph Or Histogram?

A frequency distribution can be shown as a table, a bar graph, or a histogram. The right graph depends on the kind of data you have.

Use a bar graph when you are counting separate categories, such as favorite fruits or types of transport. The bars are separated because the categories are distinct.

Use a histogram when you are grouping numerical data into intervals. The bars touch because the intervals represent a continuous scale.

If all class intervals have the same width, taller histogram bars mean larger frequencies. If the class widths are different, raw height alone can be misleading. In that case, the histogram should use frequency density so that bar area, not just height, represents frequency.

Common Mistakes In Frequency Distribution Tables

Mixing Up Categories And Intervals

A bar graph for categories and a histogram for grouped numerical data do not mean the same thing. Using the wrong graph can hide the structure of the data.

Using Overlapping Classes

Intervals need a clear rule. A setup like $0$-$10$ and $10$-$20$ is a problem unless you say exactly where the value $10$ belongs.

Forgetting That Grouping Hides Detail

A grouped frequency distribution summarizes the data, but it does not preserve every original value. Once scores are bundled into intervals, you can see the pattern more easily, but you lose some precision.

Comparing Unequal Bars As If They Were Equal

If one class interval is twice as wide as another, a histogram should not be read the same way as an equal-width histogram. The condition matters: equal-width classes support direct height comparison; unequal-width classes do not.

When Frequency Distributions Are Used

Frequency distributions are common in statistics, classrooms, surveys, quality control, and lab work. They are most useful when the raw list is long enough that a quick scan no longer shows the pattern clearly.

They are also the starting point for related ideas such as histograms, cumulative frequency, grouped means, and estimates of spread.

Try A Similar Problem

Take $15$ to $20$ numbers from a worksheet, experiment, or score list. First make an ungrouped frequency table, then regroup the same data into class intervals. Comparing the two versions is one of the fastest ways to see what a frequency distribution helps you notice and what detail grouping hides.