What graphical tool is best used to display the relative frequency of grouped, quantitative data?

Frequency Distribution

What is a frequency distribution?

Frequency distributions are visual displays that organise and present frequency counts so that the information can be interpreted more easily.

Table of Contents Show

What is a frequency distribution?
Types of frequency distributions
How to make a frequency table
How to make an ungrouped frequency table
How to make a grouped frequency table
How to make a relative frequency table
How to make a cumulative frequency table
How to graph a frequency distribution
Frequently asked questions about frequency distributions

Frequency distributions can show absolute frequencies or relative frequencies, such as proportions or percentages.

How do we show a frequency distribution?

A frequency distribution of data can be shown in a table or graph. Some common methods of showing frequency distributions include frequency tables, histograms or bar charts.

Frequency Tables

A frequency table is a simple way to display the number of occurrences of a particular value or characteristic.

For example, if we have collected data about height from a sample of 50 children, we could present our findings as:

Height of Children

Height (cm) of children	Absolute frequency	Relative frequency
120 – less than 130	9	18%
130 – less than 140	10	20%
140 – less than 150	13	26%
150 – less than 160	11	22%
160 – less than 170	7	14%
Total	50	100%

From this frequency table we can quickly identify information such as 7 children (14% of all children) are in the 160 to less than 170 cm height range, and that there are more children with heights in the 140 to less than 150 cm range (26% of all children) than any other height range.

Data can also be presented in graphical form.

Frequency Graphs

Histograms and bar charts are both visual displays of frequencies using columns plotted on a graph. The Y-axis (vertical axis) generally represents the frequency count, while the X-axis (horizontal axis) generally represents the variable being measured.

A histogram is a type of graph in which each column represents a numeric variable, in particular that which is continuous and/or grouped.

A histogram shows the distribution of all observations in a quantitative dataset. It is useful for describing the shape, centre and spread to better understand the distribution of the dataset.

Features of a histogram:

The height of the column shows the frequency for a specific range of values.

Columns are usually of equal width, however a histogram may show data using unequal ranges (intervals) and therefore have columns of unequal width.

The values represented by each column must be mutually exclusive and exhaustive. Therefore, there are no spaces between columns and each observation can only ever belong in one column.

It is important that there is no ambiguity in the labelling of the intervals on the x-axis for continuous or grouped data (e.g. 0 to less than 10, 10 to less than 20, 20 to less than 30).

For example:

The histogram below shows the same information as the frequency table.

A bar chart is a type of graph in which each column (plotted either vertically or horizontally) represents a categorical variable or a discrete ungrouped numeric variable.

It is used to compare the frequency (count) for a category or characteristic with another category or characteristic.

Features of a bar chart:

In a bar chart, the bar height (if vertical) or length (if horizontal) shows the frequency for each category or characteristic.

The distribution of the dataset is not important because the columns each represent an individual category or characteristic rather than intervals for a continuous measurement. Therefore, gaps are included between each bar and each bar can be arranged in any order without affecting the data.

For example:

If data had been collected for 'country of birth' from a sample of children, a bar chart could be used to plot the data as 'country of birth' is a categorical variable.

Birthplace of Children

Country of Birth	Absolute frequency	Relative frequency
Australia	16	32%
Fiji	3	6%
India	8	16%
Italy	10	20%
New Zealand	9	18%
United States of America	4	8%
Total	50	100%

The bar chart below shows us that 'Australia' is the most commonly observed country of birth of the 50 children sampled, while 'Fiji' is the least common country of birth.

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What is a frequency distribution?

The frequency of a value is the number of times it occurs in a dataset. A frequency distribution is the pattern of frequencies of a variable. It’s the number of times each possible value of a variable occurs in a dataset.

Types of frequency distributions

There are four types of frequency distributions:

Ungrouped frequency distributions: The number of observations of each value of a variable.
- You can use this type of frequency distribution for categorical variables.
Grouped frequency distributions: The number of observations of each class interval of a variable. Class intervals are ordered groupings of a variable’s values.
- You can use this type of frequency distribution for quantitative variables.
Relative frequency distributions: The proportion of observations of each value or class interval of a variable.
- You can use this type of frequency distribution for any type of variable when you’re more interested in comparing frequencies than the actual number of observations.
Cumulative frequency distributions: The sum of the frequencies less than or equal to each value or class interval of a variable.
- You can use this type of frequency distribution for ordinal or quantitative variables when you want to understand how often observations fall below certain values.

How to make a frequency table

Frequency distributions are often displayed using frequency tables. A frequency table is an effective way to summarize or organize a dataset. It’s usually composed of two columns:

The values or class intervals
Their frequencies

The method for making a frequency table differs between the four types of frequency distributions. You can follow the guides below or use software such as Excel, SPSS, or R to make a frequency table.

How to make an ungrouped frequency table

Create a table with two columns and as many rows as there are values of the variable. Label the first column using the variable name and label the second column “Frequency.” Enter the values in the first column.
- For ordinal variables, the values should be ordered from smallest to largest in the table rows.
- For nominal variables, the values can be in any order in the table. You may wish to order them alphabetically or in some other logical order.
Count the frequencies. The frequencies are the number of times each value occurs. Enter the frequencies in the second column of the table beside their corresponding values.
- Especially if your dataset is large, it may help to count the frequencies by tallying. Add a third column called “Tally.” As you read the observations, make a tick mark in the appropriate row of the tally column for each observation. Count the tally marks to determine the frequency.

Example: Making an ungrouped frequency tableA gardener set up a bird feeder in their backyard. To help them decide how much and what type of birdseed to buy, they decide to record the bird species that visit their feeder. Over the course of one morning, the following birds visit their feeder:

How to make a grouped frequency table

Divide the variable into class intervals. Below is one method to divide a variable into class intervals. Different methods will give different answers, but there’s no agreement on the best method to calculate class intervals.
- Calculate the range. Subtract the lowest value in the dataset from the highest.
- Decide the class interval width. There are no firm rules on how to choose the width, but the following formula is a rule of thumb:
  
  You can round this value to a whole number or a number that’s convenient to add (such as a multiple of 10).
- Calculate the class intervals. Each interval is defined by a lower limit and upper limit. Observations in a class interval are greater than or equal to the lower limit and less than the upper limit:
  
  The lower limit of the first interval is the lowest value in the dataset. Add the class interval width to find the upper limit of the first interval and the lower limit of the second variable. Keep adding the interval width to calculate more class intervals until you exceed the highest value.

Create a table with two columns and as many rows as there are class intervals. Label the first column using the variable name and label the second column “Frequency.” Enter the class intervals in the first column.
Count the frequencies. The frequencies are the number of observations in each class interval. You can count by tallying if you find it helpful. Enter the frequencies in the second column of the table beside their corresponding class intervals.

Example: Grouped frequency distributionA sociologist conducted a survey of 20 adults. She wants to report the frequency distribution of the ages of the survey respondents. The respondents were the following ages in years:

52, 34, 32, 29, 63, 40, 46, 54, 36, 36, 24, 19, 45, 20, 28, 29, 38, 33, 49, 37

Round the class interval width to 10.

The class intervals are 19 ≤ a < 29, 29 ≤ a < 39, 39 ≤ a < 49, 49 ≤ a < 59, and 59 ≤ a < 69.

How to make a relative frequency table

Create an ungrouped or grouped frequency table.
Add a third column to the table for the relative frequencies. To calculate the relative frequencies, divide each frequency by the sample size. The sample size is the sum of the frequencies.

Example: Relative frequency distribution

From this table, the gardener can make observations, such as that 19% of the bird feeder visits were from chickadees and 25% were from finches.

How to make a cumulative frequency table

Create an ungrouped or grouped frequency table for an ordinal or quantitative variable. Cumulative frequencies don’t make sense for nominal variables because the values have no order—one value isn’t more than or less than another value.
Add a third column to the table for the cumulative frequencies. The cumulative frequency is the number of observations less than or equal to a certain value or class interval. To calculate the relative frequencies, add each frequency to the frequencies in the previous rows.
Optional: If you want to calculate the cumulative relative frequency, add another column and divide each cumulative frequency by the sample size.

Example: Cumulative frequency distribution

From this table, the sociologist can make observations such as 13 respondents (65%) were under 39 years old, and 16 respondents (80%) were under 49 years old.

How to graph a frequency distribution

Pie charts, bar charts, and histograms are all ways of graphing frequency distributions. The best choice depends on the type of variable and what you’re trying to communicate.

Pie chart

A pie chart is a graph that shows the relative frequency distribution of a nominal variable.

A pie chart is a circle that’s divided into one slice for each value. The size of the slices shows their relative frequency.

This type of graph can be a good choice when you want to emphasize that one variable is especially frequent or infrequent, or you want to present the overall composition of a variable.

A disadvantage of pie charts is that it’s difficult to see small differences between frequencies. As a result, it’s also not a good option if you want to compare the frequencies of different values.

Bar chart

A bar chart is a graph that shows the frequency or relative frequency distribution of a categorical variable (nominal or ordinal).

The y-axis of the bars shows the frequencies or relative frequencies, and the x-axis shows the values. Each value is represented by a bar, and the length or height of the bar shows the frequency of the value.

A bar chart is a good choice when you want to compare the frequencies of different values. It’s much easier to compare the heights of bars than the angles of pie chart slices.

Histogram

A histogram is a graph that shows the frequency or relative frequency distribution of a quantitative variable. It looks similar to a bar chart.

The continuous variable is grouped into interval classes, just like a grouped frequency table. The y-axis of the bars shows the frequencies or relative frequencies, and the x-axis shows the interval classes. Each interval class is represented by a bar, and the height of the bar shows the frequency or relative frequency of the interval class.

Although bar charts and histograms are similar, there are important differences:

	Bar chart	Histogram
Type of variable	Categorical	Quantitative
Value grouping	Ungrouped (values)	Grouped (interval classes)
Bar spacing	Can be a space between bars	Never a space between bars
Bar order	Can be in any order	Can only be ordered from lowest to highest

A histogram is an effective visual summary of several important characteristics of a variable. At a glance, you can see a variable’s central tendency and variability, as well as what probability distribution it appears to follow, such as a normal, Poisson, or uniform distribution.

Frequently asked questions about frequency distributions

What’s the difference between relative frequency and probability?

Probability is the relative frequency over an infinite number of trials.

For example, the probability of a coin landing on heads is .5, meaning that if you flip the coin an infinite number of times, it will land on heads half the time.

Since doing something an infinite number of times is impossible, relative frequency is often used as an estimate of probability. If you flip a coin 1000 times and get 507 heads, the relative frequency, .507, is a good estimate of the probability.