Calculating Relative and Cumulative Frequencies

Welcome back. Last lesson focused on turning raw quantitative data into a grouped frequency distribution: equal-width classes, frequencies, and a total that matches the number of observations.

Now we make that table more useful. A count such as “7 scores were between 60 and 69” is helpful, but we often need to know its share of the whole dataset: “about 29%.” We also often need to know how many observations have accumulated up to a point: “13 scores were 69 or below.”

By the end of this lesson, you will be able to calculate and interpret:

• relative frequency: the proportion or percentage in one class;
• cumulative frequency: the running total of counts;
• cumulative relative frequency: the running proportion or percentage.

These are common test-table columns, so the goal is to make the calculation routine rather than something to memorize nervously.

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Three columns, three different questions

Start with an ordinary frequency distribution. Let $f_i$ mean the frequency in row $i$, and let $n$ mean the total number of observations.

Relative frequency: “What share is in this one class?”

The relative frequency of a class is its frequency divided by the total:

$$\text{relative frequency}=\frac{\text{class frequency}}{\text{total frequency}}$$

It can be written as:

• a fraction, such as $\frac{7}{24}$;
• a decimal, such as $0.2917$;
• a percentage, such as $29.17\%$.

The meaning stays the same. A relative frequency of $0.25$, for example, means that one quarter of the observations are in that class.

Cumulative frequency: “How many have accumulated up to here?”

A cumulative frequency is a running total. Begin with the first frequency; then keep adding the next row's frequency as you move down the table.

$$F_i=f_1+f_2+\cdots +f_i$$

For a score class of 60-69, the cumulative frequency means the number of scores at or below 69. It does not mean the number of scores in the 60-69 class alone.

Cumulative relative frequency: “What percentage has accumulated up to here?”

A cumulative relative frequency is the cumulative frequency divided by the total:

$$\text{cumulative relative frequency}=\frac{\text{cumulative frequency}}{\text{total frequency}}$$

Equivalently, it is the running total of the relative-frequency column:

$$R_i=r_1+r_2+\cdots +r_i$$

This is particularly useful for questions asking for the percentage below a cutoff.

For a concise demonstration of the calculations, watch this short segment.

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A complete worked example

Use the score distribution from the previous lesson. There were $24$ test scores total.

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

Step 1: Calculate relative frequencies

Every row uses the same denominator, $24$.

For the class 60-69:

$$\frac{7}{24}=0.2917$$

As a percentage:

$$0.2917(100)=29.17\%$$

So about $29.2\%$ of the scores are from 60 to 69.

Do this for every class. Here, percentages have been rounded to one decimal place.

Test-score class	Frequency	Relative frequency
40-49	2	$2/24=0.0833=8.3\%$
50-59	4	$4/24=0.1667=16.7\%$
60-69	7	$7/24=0.2917=29.2\%$
70-79	6	$6/24=0.2500=25.0\%$
80-89	5	$5/24=0.2083=20.8\%$
Total	24	$100.0\%$

Relative frequency lets you compare groups even when the datasets have different sizes. “7 scores” has a different weight in a class of 24 than in a class of 240; “29.2%” communicates the share immediately.

Step 2: Calculate cumulative frequencies

Now keep a running count.

• First row: $2$
• Second row: $2+4=6$
• Third row: $6+7=13$
• Fourth row: $13+6=19$
• Final row: $19+5=24$

Test-score class	Frequency	Cumulative frequency
40-49	2	2
50-59	4	6
60-69	7	13
70-79	6	19
80-89	5	24

Interpret the row for 60-69 carefully:

• The ordinary frequency is 7: seven scores lie from 60 through 69.
• The cumulative frequency is 13: thirteen scores are 69 or lower.

Cumulative frequency only makes sense when the rows have a meaningful order. Numerical values and ordered class intervals have one. Categories such as favorite colour do not have a natural order, so a cumulative total for them is usually not meaningful.

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Add cumulative relative frequency

You can calculate cumulative relative frequency in either of two valid ways:

1. Add the relative frequencies as you move down.
2. Divide each cumulative frequency by $24$.

The second method is often safer when rounding is involved.

For the 60-69 row:

$$\frac{13}{24}=0.5417$$ $$0.5417(100)=54.17\%$$

So approximately $54.2\%$ of scores are 69 or below.

Test-score class	Frequency	Relative frequency	Cumulative frequency	Cumulative relative frequency
40-49	2	8.3%	2	8.3%
50-59	4	16.7%	6	25.0%
60-69	7	29.2%	13	54.2%
70-79	6	25.0%	19	79.2%
80-89	5	20.8%	24	100.0%

Notice the different interpretations in the same row:

Column	Value for 60-69	Meaning
Frequency	7	Seven scores are from 60 through 69.
Relative frequency	29.2%	About 29.2% of scores are from 60 through 69.
Cumulative frequency	13	Thirteen scores are 69 or below.
Cumulative relative frequency	54.2%	About 54.2% of scores are 69 or below.

The word cumulative always means “include this row and everything before it.”

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Reading grouped tables and cutoff language

The wording of a question tells you which column is useful.

“In this class” language

For questions like “What percentage of scores are from 70 to 79?”, use the relative frequency for that single class.

From the table:

$$25.0\%$$

“At or below” language

For questions like “How many scores are 69 or below?”, use the cumulative frequency in the 60-69 row.

$$13$$

For “What percentage of scores are 69 or below?”, use the cumulative relative frequency.

$$54.2\%$$

“Above” language

For “What percentage of scores are 69 or above?”, it is usually easiest to use the complement of the percentage below 70.

The cumulative relative frequency through 60-69 is $54.2\%$, so:

$$100\%-54.2\%=45.8\%$$

Thus, about $45.8\%$ of the scores are 70 or higher.

Be precise about endpoints. For whole-number classes, the row 60-69 includes values from 60 through 69. For measured data written as “65.95 to less than 67.95,” the cumulative total through that row represents measurements less than 67.95.

This OpenStax reading shows the same ideas first with individual values and then with grouped height intervals.

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Visualizing the difference

A relative-frequency graph has a separate bar for each category or class. A cumulative-relative-frequency graph rises because it keeps all earlier observations.

In the chart:

• Grade 1 has a relative frequency of roughly $26\%$.
• Grade 2 alone has a relative frequency of roughly $23\%$.
• Through Grade 2, the cumulative relative frequency is roughly $49\%$.

That last statement combines the first two grades. It does not say that Grade 2 alone contains $49\%$ of students.

A cumulative relative-frequency line has three predictable features when categories are correctly ordered:

1. It never goes down.
2. It starts at the first class's relative frequency.
3. It ends at $1$, or $100\%$, apart from minor rounding.

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Fast checks before you submit a table

Use this short audit.

Check	What should happen?
Total frequency	Adds to the sample size $n$.
Relative-frequency total	Adds to $1$ or $100\%$, allowing for small rounding differences.
Final cumulative frequency	Equals $n$.
Final cumulative relative frequency	Equals $1$ or $100\%$, allowing for rounding.
Cumulative columns	Stay the same or increase; they can never decrease.
Denominator	Every relative frequency uses the same total $n$.

A very common error is dividing a class frequency by the previous cumulative frequency. Do not do that. For relative frequency, the denominator is always the total number of observations.

Another common error is to add rounded percentages and worry if the answer is $99.9\%$ or $100.1\%$. That is usually just rounding. If exactness matters, calculate the cumulative relative frequency from the cumulative count divided by $n$, then round only the final displayed result.

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Key takeaways

A frequency distribution can show both counts and proportions:

$$\text{relative frequency}=\frac{\text{frequency}}{\text{total}}$$ $$\text{cumulative frequency}=\text{running total of frequencies}$$ $$\text{cumulative relative frequency}=\frac{\text{cumulative frequency}}{\text{total}}$$

Relative frequency answers “what share is in this class?” Cumulative frequency and cumulative relative frequency answer “how many, or what percentage, are at or below this point?”

In the next lesson, you will use grouped frequency distributions to interpret histograms: shape, skewness, gaps, possible outliers, and whether a distribution looks approximately normal.