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1.4 · Organize and graph qualitative data

Learn to organize and graph qualitative data through clear examples and targeted practice.

Athabasca University MATH 215: Introduction to Statistics

Descriptive Statistics

Frequency tables, relative frequencies, and clear displays

Qualitative data describe qualities or group membership rather than measured amounts. Examples include a person’s preferred type of music, a vehicle’s colour, or a survey response such as “yes” or “no.” Organizing these data makes it easier to see which categories are common, which are rare, and how the categories compare. This lesson introduces frequency tables, relative frequencies, percent frequencies, bar charts, and pie charts. The calculations use only addition, division, and multiplication: a frequency is a count, a relative frequency is a count divided by a total, and a percent is a relative frequency multiplied by 100.

What you will learn

Identify qualitative data and distinguish categories from numerical measurements.
Organize category counts in a frequency table and calculate relative and percent frequencies.
Choose and construct a suitable bar chart or pie chart.
Check that a table or graph represents the data fairly and interpret it in context.

Recognize and organize qualitative data

A variable is a characteristic recorded for each person or object. A qualitative variable places each observation into a named category. For example, if a survey records each student’s preferred way to travel to campus, the variable is “preferred travel method,” and possible categories might be bus, bicycle, walking, and car. These category names are not numerical measurements, even if a researcher chooses to assign them number codes.

The population is the entire group a study is interested in, while a sample is the smaller group actually observed. The observations are the recorded responses. A parameter is a numerical description of a population; a statistic is a numerical description calculated from a sample. In a simple frequency table, the category counts summarize the observations. They do not automatically describe a larger population unless the sample and study design support that interpretation.

A frequency is the number of observations in a category. To make a frequency table, list each category once and count how many observations belong to it. The categories should be clear and should not overlap: each observation should fit one category only. Include all observed categories, even if a category has a small count. The total frequency must equal the number of observations.

n=\sum f_i

Qualitative data are names or labels for categories.
A category must be defined so an observation can be assigned consistently.
The sum of the category frequencies equals the total number of observations.

Relative frequency and percent frequency

A frequency table gives counts, but counts alone can be difficult to compare when groups have different totals. Relative frequency gives the share of observations in a category. Divide that category’s frequency by the total frequency. Since a share is between zero and one, relative frequencies across all categories add to one, apart from small differences caused by rounding.

A percent frequency expresses the same share out of 100. Multiply the relative frequency by 100 and attach the percent sign. Percent frequencies should add to 100%, except for a small rounding difference. Use the same number of decimal places across a table when that makes the categories easier to compare.

Before calculating, confirm that the category counts are complete and that the total is correct. Division is appropriate because each relative frequency compares one category’s count with the same overall total. Do not treat a category label as a number or average the labels.

r_i=\frac{f_i}{n},\qquad p_i=100r_i\%

Relative frequency compares a category count with the total count.
Percent frequency is relative frequency expressed out of 100.
Rounding may make a displayed total differ slightly from one or 100%.

Choose and read a graph

A bar chart displays category frequencies or percentages using separate rectangular bars. Put the category names on one axis and a numerical scale on the other. Bars have equal width and a gap between them because the categories are distinct groups, not points along a continuous number line. Give the graph a descriptive title, label the axes, and start a frequency scale at zero so bar lengths represent the counts fairly.

A bar chart can display counts when the group sizes are the same or when the audience needs the actual number of observations. When comparing groups with different totals, percentages may be more informative than counts because they account for the different group sizes. State which quantity the bars represent.

A pie chart represents the categories as parts of one whole. It is most useful when there are only a few categories and each observation belongs to exactly one category. The whole circle represents 100%; each slice represents that category’s share. Use category labels or a legend, and make sure the slices collectively represent the full total. If there are many categories or several small slices, a bar chart is usually easier to read.

A graph should make comparisons possible without exaggeration. Use a consistent scale, avoid decorative effects that distort apparent bar size, and do not omit categories in a way that hides part of the data. Describe what the sample shows; avoid claiming that a graph alone proves why a pattern occurred.

\theta_i=360^\circ r_i

Use separate bars for categories and a clearly labelled scale.
Use a pie chart only when categories form parts of one whole and the display remains readable.
Choose counts or percentages to suit the comparison, and identify which one is shown.

A reliable workflow

Start by naming the population of interest, the observed sample, and the qualitative variable when those details are known. Then check the category definitions: are they understandable, non-overlapping, and complete for the observations? This prevents errors before any graph is made.

Next, tally the observations and verify the total. Calculate relative or percent frequencies only if they help answer the question. Select a display that suits the number of categories and comparison: a bar chart is a flexible choice, while a pie chart emphasizes parts of one whole. Finish by checking titles, labels, scale, and totals.

Interpret the display in context. Identify the category with the greatest or smallest frequency if relevant, and give its count or percentage. A sample result describes the observed sample. It should not be presented as a fact about every member of a population unless the study supports that conclusion.

Check data and category definitions before calculating.
Verify totals and labels after constructing the table or graph.
Interpret only what the displayed sample data support.

Travel method survey summary

Travel method	Frequency	Relative frequency	Percent frequency	Pie-chart angle
Bus	12	0.400000	40.0%	144.0°
Car	8	0.266667	26.7%	96.0°
Walking	6	0.200000	20.0%	72.0°
Bicycle	4	0.133333	13.3%	48.0°
Total	30	1.000000	100.0%	360.0°

Worked example

Travel methods reported by a student group

A class survey asks 30 students which travel method they used to get to campus that day. The recorded responses are 12 bus, 8 car, 6 walking, and 4 bicycle. Organize the responses in a frequency table with relative and percent frequencies, then determine the pie-chart angle for each category. State a suitable graph and interpret the most common response.

Identify the data
The population of interest might be all students at the institution, but the observed sample is the 30 surveyed students. The qualitative variable is travel method used that day. The categories are distinct, and the problem assigns one response to each student. The counts add to 30, so they account for the full sample.
$12+8+6+4=30$
Calculate relative frequencies
Divide each category count by the sample total. For example, the bus share is its count divided by 30. Keep several decimal places while calculating, then round the displayed percentages and angles at the end.
$\frac{12}{30}=0.400000,\quad\frac{8}{30}=0.266667,\quad\frac{6}{30}=0.200000,\quad\frac{4}{30}=0.133333$
Convert shares to percentages and angles
Multiply each relative frequency by 100 for a percentage. For a pie chart, multiply each relative frequency by 360 degrees because the complete circle has 360 degrees. The unrounded angles add to 360 degrees.
$100(0.400000)=40.0000\%,\quad360^\circ(0.400000)=144.0000^\circ$
Select and interpret the display
A bar chart is a clear choice because it makes the four categories easy to compare; its bars can show counts or percentages. A pie chart is also suitable here because the categories divide the 30 responses into parts of one whole. Bus is the most common response in this sample, representing 12 of 30 students, or 40%. This result describes the surveyed class and does not by itself establish the travel habits of all students.
$\frac{12}{30}\times100\%=40\%$

Answer: The frequency table is: bus, 12; car, 8; walking, 6; bicycle, 4. The relative frequencies are 0.400000, 0.266667, 0.200000, and 0.133333, respectively. The rounded percentages are 40.0%, 26.7%, 20.0%, and 13.3%. The pie-chart angles, rounded to one decimal place, are 144.0°, 96.0°, 72.0°, and 48.0°. Bus is the most common reported method in this sample.

Check: The counts sum to 30, the displayed percentages sum to 100.0%, and the pie-chart angles sum to 360.0°. These checks support that the categories cover the sample.

Common mistakes and how to avoid them

Treating category codes as measurements, then averaging them.

Correction: A number used only as a label does not measure an amount. Summarize qualitative data with category counts or shares, not an average of arbitrary codes.

Using touching bars for distinct categories or leaving the axes unclear.

Correction: Leave spaces between category bars, name the categories, label the numerical scale, and state whether bar heights show counts or percentages.

Reporting counts that do not add to the number of observations.

Correction: Recheck the tally and ensure every observation belongs to exactly one listed category.

Claiming a sample graph describes an entire population with certainty.

Correction: Describe what the observed sample shows. A graph alone does not establish that the same pattern holds for everyone in the population.

Lesson summary

Qualitative data place observations into named categories.
A frequency is a category count; relative frequency is that count divided by the total.
Percent frequency is relative frequency multiplied by 100.
Bar charts compare categories; pie charts show categories as parts of one whole.
Check category definitions, totals, labels, and scales, then interpret the display in context.

Check your understanding

Question 1

A survey records 20 responses: 5 prefer tea, 9 prefer coffee, and 6 prefer water. What is the relative frequency for coffee?

0.25
0.30
0.45
9

Show answer and explanation

0.45

Relative frequency is the category count divided by the total. For coffee, 9 divided by 20 is 0.45.

Question 2

Which display is usually clearer for comparing 12 named categories, several of which have small counts?

A pie chart with many small slices
A bar chart with category labels and a clear scale
A pie chart that omits the smallest categories
A bar chart with no numerical scale

Show answer and explanation

A bar chart with category labels and a clear scale

A bar chart makes it easier to compare category sizes, especially when there are many categories. The category labels and scale are needed to interpret it.

Key terms

Qualitative variable: A characteristic that records category membership rather than a measured numerical amount.
Frequency: The number of observations in a category.
Relative frequency: A category’s frequency divided by the total number of observations.
Percent frequency: A relative frequency expressed as a percentage.
Bar chart: A graph that compares categories using separate bars whose heights or lengths show counts or shares.
Pie chart: A circle divided into slices that show categories as parts of a whole.

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Published by DoAssignment. This AI-assisted lesson follows Athabasca University MATH 215: Introduction to Statistics, study topic 1.4. It is a study resource, not an official curriculum publication.

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