13 Summarising qualitative data
So far, you have learnt to ask a RQ, design a study, collect the data, and classify the data. In this chapter, you will learn to:
- summarise qualitative data using the appropriate graphs.
- summarise qualitative data using, for example, medians and odds to summarise qualitative data.
13.1 Introduction
Many quantitative research studies involve qualitative variables Except for very small amounts of data, understanding the data is difficult without a summary. As with quantitative data, quantitative data can be understood by knowing how often values of the variables appear. This is called the distribution of the data (Def. 12.1).
The distribution can be displayed using a frequency table (Sect. 13.2) or a graph (Sect. 13.3). Qualitative data can be summarised by finding modes or, for ordinal qualitative data, using medians (Sect. 13.3.5). The distribution of qualitative data can be summarised numerically by computing proportions, percentages (Sect. 13.4) or odds (Sect. 13.5).
13.2 Frequency tables for qualitative data
Qualitative data are typically collated in a frequency table. The rows (or the columns) should list the levels of the variable (Sect. 11.4), and these should be exhaustive (cover all levels) and exclusive (observations belong to only one level).
For nominal data, the levels of the variables can be displayed alphabetically, by size, by personal preference, or other way: use the order most likely to be useful to readers. For ordinal data, the natural order of the levels should almost always be used.
Example 13.1 (Opinions of AV vehicles) Pyrialakou et al. (2020) surveyed \(400\) residents of Phoenix (Arizona) about their opinions of autonomous vehicles (AVs). Demographic information (Table 13.1) and respondents' opinions of sharing roads with AVs (Table 13.2) were recorded.
The gender of the respondent is nominal (two levels), while the age group is ordinal (six levels); the levels are in the rows. The three questions about safety (Table 13.2) all yield ordinal responses (five levels, in columns).
| Number | Percentage | |
|---|---|---|
| Gender | ||
| Female | \(204\) | \(51\) |
| Male | \(196\) | \(49\) |
| Age group | ||
| \(18\) to \(24\) | \(\phantom{0}52\) | \(13\) |
| \(25\) to \(34\) | \(\phantom{0}76\) | \(19\) |
| \(35\) to \(44\) | \(\phantom{0}76\) | \(19\) |
| \(45\) to \(54\) | \(\phantom{0}72\) | \(18\) |
| \(55\) to \(64\) | \(\phantom{0}56\) | \(14\) |
| \(65+\) | \(\phantom{0}68\) | \(17\) |
| \(n\) | % | \(n\) | % | \(n\) | % | \(n\) | % | \(n\) | % | |
|---|---|---|---|---|---|---|---|---|---|---|
| Driving near an AV | \(58\) | \(14\) | \(\phantom{0}79\) | \(20\) | \(\phantom{0}96\) | \(24\) | \(97\) | \(24\) | \(70\) | \(18\) |
| Cycling near an AV | \(77\) | \(19\) | \(104\) | \(26\) | \(\phantom{0}87\) | \(22\) | \(76\) | \(19\) | \(56\) | \(14\) |
| Walking near an AV | \(63\) | \(16\) | \(\phantom{0}86\) | \(22\) | \(103\) | \(26\) | \(82\) | \(20\) | \(66\) | \(16\) |
13.3 Graphs
Three options for graphing qualitative data are:
- Dot chart (Sect. 13.3.1): Usually a good choice.
- Bar chart (Sect, 13.3.2): Usually a good choice.
- Pie chart (Sect. 13.3.3): Only useful in special circumstances, and can be hard to interpret.
For nominal data, the levels of the variables can be displayed alphabetically, by size, by personal preference, or other way: use the order most likely to be useful to readers. For ordinal data, the natural order of the levels should almost always be used.
Sometimes these graphs are also used for quantitative data (espeiclaly discrete data) with a small number of possible options. Sometimes, graphs used for quantitative data (Sect. 12.3) may be useful for discrete quantitative data if many values are possible.
The purpose of a graph is to display the information in the clearest, simplest possible way, to facilitate understanding the message(s) in the data.
13.3.1 Dot charts (qualitative data)
Dot charts indicate the counts (or corresponding percentages) in each level, using dots on a line starting at zero. The levels can be on the horizontal or vertical axis, and the counts or percentages on the other. Placing the level on the vertical axis often makes for easier reading, and space for long labels.
Example 13.2 (Dot plots) For the AV study in Example 13.1, a dot chart of the age group of respondents is shown in Fig. 13.1 (top left panel).
FIGURE 13.1: The age group of respondents in the AV study. All the graphs present the same data.
For dot charts:
- place the qualitative variable on the horizontal or vertical axis (and label with the levels of the variable).
- use counts or percentages on the other axis.
- for nominal data, think about the most helpful order for the levels.
13.3.2 Bar charts
Bar charts indicate the counts in each category using bars starting from zero. As with dot charts, the levels can be on the horizontal or vertical axis, but placing the level names on the vertical axis often makes for easier reading, and room for long labels.
Example 13.3 (Dot plots) For the AV study in Example 13.1, a bar chart of the age group of respondents is shown in Fig. 13.1 (top right panel).
For bar charts:
- place the qualitative variable on the horizontal or vertical axis (and label with the levels of the variable).
- use counts or percentages on the other axis.
- for nominal data, levels can be ordered any way: Think about the most helpful order.
- bars have gaps between bars, as the bars represent distinct categories.
Bar charts have gaps between all of the bars. In contrast, the bars in histograms are butted together (except when an interval has a count of zero), as the variable-axis represent a continuous numerical scale.
13.3.3 Pie charts
In pie charts, a circle is divided into segments proportional to the number in each level of the qualitative variable.
Example 13.4 (Dot plots) For the AV study in Example 13.1, a pie chart of the age group of respondents is shown in Fig. 13.1 (bottom left panel).
Pie charts may present challenges:
- Pie charts only work when graphing parts of a whole.
- Pie charts only work when all options are present ('exhaustive').
- Pie charts are difficult to use with levels having zero or small counts (see Example 18.2).
- Pie charts are difficult to interpret when many categories are present.
- Pie charts are hard to read: Humans compare lengths (bar and dot charts) better than angles (pie charts) (Friel, Curcio, and Bright 2001).
Example 13.5 Consider studying the percentage of people who use Firefox, Chrome, and Safari as web browsers. A pie chart is not suitable for displaying the data, as people can use more than one of these browsers (i.e., the options are not exclusive) nor exhaustive (i.e., other options exist).
13.3.4 Comparing pie, bar and dot charts
Consider the pie chart in Fig. 13.1 (bottom left panel). Determining which age groups have the most respondents is hard. The equivalent bar chart (or dot chart) makes the comparison easy: clearly the youngest age group has the smallest representation, while the \(25\) to \(34\) and the \(35\) to \(44\) age groups have the most respondents. The tilted pie chart makes this comparison even harder (Fig. 13.1, bottom right panel).
Recall that the purpose of a graph is to display the information in the clearest, simplest possible way, to facilitate understanding the message(s) in the data. A pie chart often makes the message hard to see (Siegrist 1996).
13.3.5 Describing the distribution
Graphs are constructed to help readers understand the data. Hence, after producing a graph, any important features in the graph should be described. One simple way for qualitative data is to identify the level (or levels) with the most observations. This is called the mode.
Definition 13.1 (Mode) A mode is the level (or levels) of a qualitative variable with the most observations.
Example 13.6 (Modes) Consider the data in Tables 13.1 and 13.2. 'Gender' is nominal qualitative; age group is ordinal qualitative. The responses to the three questions are ordinal. The mode can be found for each variable:
- The mode for gender is 'Female' (with \(204\) respondents).
- The mode age groups are \(25\) to \(34\) and \(35\) to \(44\).
- The modal response to the question about driving near AVs is 'Somewhat safe'.
- The modal response to the question about cycling near AVs is 'Somewhat unsafe'.
- The modal response to the question about walking near AVs is 'Neutral'.
Medians can be found for ordinal data (but not nominal data), since ordinal data have levels with a natural order. The median is the location of the middle response, when the levels from all individuals are placed in order.
Medians can be used to summarise qualitative data and ordinal data, but never nominal data.
Example 13.7 (Medians) Consider the data in Tables 13.1 and 13.2. 'Gender' is nominal qualitative, so medians are not appropriate. However, the other variables are ordinal, so medians could be used to describe each variable. Since \(n = 400\), the median response will be halfway between the location of the \(200\)th and \(201\)st response when ordered:
- the median age group is \(35\) to \(44\) (ordered observation numbers \(200\) and \(201\) both fall into this level).
- the median response to the driving question is 'Neutral'.
- the median response to the cycling question is 'Neutral'.
- the median response to the walking question is 'Neutral'.
Qualitative cannot also be numerically summarised using proportions, percentage (Sect. 13.4) or odds (Sect. 13.5). Importantly, these numerical quantities are computed from a sample (i.e., are statistics; Def. 12.3), even though the whole population is of interest (i.e., the parameter; Def. 12.2).
Means are generally not suitable for numerically summarising qualitative data. However, ordinal data may be numerically summarised like quantitative data in rare and very special circumstances: only when
- the levels are considered equally spaced; and
- assigning a number to each level is appropriate (perhaps using a mid-point for numerical groups).
We will not consider means for ordinal data further.
13.4 Numerical summary: proportions and percentages
Qualitative data can be summarised using the proportion or percentage of individuals in each level. These can be given instead of, or with, the counts (as in Tables 13.1 and 13.2).
Definition 13.2 (Proportion) A proportion is a fraction out of a total, and is a number between \(0\) and \(1\).
Definition 13.3 (Percentages) A percentage is a proportion, multiplied by \(100\). In this context, percentages are numbers between \(0\)% and \(100\)%.
Population proportions are almost always unknown. Instead, the population proportion (the parameter), denoted \(p\), is estimated by a sample proportion (a statistic), denoted by \(\hat{p}\).
The symbol \(\hat{p}\) is pronounced 'pee-hat', and refers to a sample proportion.
As always, only one possible samples is studied. Statistics are estimates of parameters, and the value of the statistic is not the same for every possible sample.
Example 13.8 (Proportions and percentages) Consider the AV data in Table 13.1, summarising results from a sample of \(n = 400\) respondents. The sample proportion of respondents aged \(25\) to \(34\) is \(76\div 400\), or \(0.19\). The sample percentage of respondents aged \(25\) to \(34\) is \(0.19 \times 100\), or \(19\)%, as in the table.
13.5 Numerical summary: odds
For the AV data in Table 13.1, the number of females is slightly larger than the number of males. More specifically, the ratio of females to males is \(204\div 196 = 1.04\); that is, there are \(1.04\) times as many females as males. This value of \(1.04\) is the odds that a respondent is female. An alternative interpretation is that there are \(1.04\times 100 = 104\) females for every \(100\) males.
While proportions and percentages are computed as the number of interest divided by the total number, the odds are computed as the number of interest divided by the remaining number.
Definition 13.4 (Odds) The odds are the number (or proportion, or percentage) of times that an event happens, divided by the number (or proportion, or percentage) of times that the event does not happen:
\[
\text{Odds} = \frac{\text{Number of times event happens}}{\text{Number of times event doesn't happen}}
\]
or (equivalently)
\[
\text{Odds}
=
\frac{\text{Proportion of times event happens}}
{\text{Proportion of times event doesn't happen}}.
\]
The odds are how many times an event happens compared to the event not happening.
Example 13.9 (Odds males) The AV data in Table 13.1 includes \(204\) females and \(196\) males. The odds that a respondent is female is \(1.04\), as found above. The odds is greater than one, as the number of females is larger than the number of males.
The odds that a respondent is male is \(196/204 = 0.96\); there are \(0.96\) times as many males as females. The odds is less than one, as the number of males is smaller than the number of females. Alternatively, there are \(96\) males for every \(100\) females.
Example 13.10 (Odds and percentages) Consider the AV data in Table 13.1, summarising results from a sample of \(n = 400\) respondents.
The percentage of respondents that are female is \(204\div400\times 100 = 51\)%. The odds that a respondent is female is \(204\div(400 - 204) = 1.04\).
Similarly, the percentage of respondents aged \(18\) to \(24\) is \(52/400\times 400 = 13\)%. The odds that a respondent is aged \(18\) to \(24\) is \(52/(400 - 52) = 0.15\); that is, the odds that a respondent is aged \(18\) to \(24\) is \(0.15\). This means that the number of respondents aged \(18\) to \(24\) is \(0.15\) times (i.e., less) than the number of respondents aged over \(24\).
The odds that a respondent is aged \(18\) to \(54\) is \((52 + 76 + 76 + 72)/(56 + 68) = 2.23\); that is, the odds that a respondent is aged \(18\) to \(54\) is \(2.23\). This means that the number of respondents aged \(18\) to \(54\) is \(2.23\) times (i.e., greater) than the number of respondents aged over \(54\)
When interpreting odds:
- odds are greater than \(1\) mean the event is more likely to happen than not.
- odds are equal to \(1\) mean the event is equally likely to happen as not.
- odds are less than \(1\) mean the event is less likely to happen than not.
Population odds are almost always unknown.
Instead, the population odds (the parameter) is estimated by a sample odds (a statistic).
No symbol is commonly-used to denote odds.
Take care: proportions and odds are similar, but are different ways of numerically summarising quantitative data (Fig. 13.2).
FIGURE 13.2: Proportions (left) are the number of interest divided by the total number; odds (right) are the number of interest divided by the rest.
13.6 Numerical summary tables
Qualitative variables should be summarised in a table. The table should include, as a minimum, numbers and percentages. While useful in other contexts (see Chap. 14), odds are usually not given in summary table. An example of a summary table is given in the next section.
13.7 Example: water access
López-Serrano et al. (2022) recorded data about access to water for three rural communities in Cameroon (see Sect. 12.9). Numerous qualitative variables are recorded; some are displayed in Fig. 13.3, and a summary table in Table 13.3. Notice that the levels of the two ordinal variables are displayed in the natural order.
The distance to the nearest water source is usually less than \(1\) km, and the wait is often over \(15\) mins. The most common water source is a bore (\(68.6\)%).
| Num. | % | Odds | |
|---|---|---|---|
| Distance to water source | |||
| Under \(100\) m | \(55\) | \(45.5\) | \(0.83\) |
| \(100\) m to \(1000\) m | \(57\) | \(47.1\) | \(0.89\) |
| Over \(1000\) m | \(\phantom{0}9\) | \(\phantom{0}7.4\) | \(0.08\) |
| Wait time at water source | |||
| Under \(5\) min | \(50\) | \(41.7\) | \(0.71\) |
| \(5\) to \(15\) min | \(28\) | \(23.3\) | \(0.30\) |
| Over \(15\) min | \(42\) | \(35.0\) | \(0.54\) |
| Water source | |||
| Tap | \(\phantom{0}7\) | \(\phantom{0}5.8\) | \(0.06\) |
| Bore | \(83\) | \(68.6\) | \(2.18\) |
| Well | \(16\) | \(13.2\) | \(0.15\) |
| River | \(15\) | \(12.4\) | \(0.14\) |
FIGURE 13.3: The distance to the water source (left), the wait time at the water source (centre), and the water sources (right) for the water-access study. (Some data are missing.)
13.8 Chapter summary
Qualitative data can be graphed with a dot chart, bar chart or pie chart (in special circumstances). Qualitative data can be described using the mode or (for ordinal variable only) a median. Qualitative data can be summarised numerically summarized using proportions, percentages or odds.
13.9 Quick review questions
Are the following statements true or false?
- Nominal data can be summarised using a median.
- Ordinal data can be summarised using a mode.
- Odds are the ratio of how often an event occurs, to how often it does not.
- Proportions and percentages are the same.
13.10 Exercises
Answers to odd-numbered exercises are available in App. E.
Exercise 13.1 A study of spider monkeys (C. A. Chapman 1990) examined the types of social groups present (Table 13.4). Construct a suitable plot, and explain what the data reveal about the social groups of spider monkeys.
| Social group | Number |
|---|---|
| Solitary | \(\phantom{0}8\) |
| All males | \(\phantom{0}3\) |
| Female + no young | \(\phantom{0}2\) |
| Mixed young | \(15\) |
| Mixed + no young | \(\phantom{0}1\) |
| One female + offspring | \(23\) |
| Many females + offspring | \(48\) |
Exercise 13.2 Czarniecka-Skubina et al. (2021) studied how Poles prepared and consumed coffee using a sample of \(1\,500\) Poles. Some data are shown in Table 13.5.
- Classify the data.
- Sketch appropriate graphs for the three variables.
- Summarise the three variables.
| Where consumed | |
| Home | 1432 |
| Canteen | 687 |
| Cafe | 922 |
| Others' homes | 994 |
| Work | 1196 |
| Brewing temperature | |
| \(100^\circ\)C | 748 |
| \(98^\circ\)C | 269 |
| \(93^\circ\)C | 453 |
| Unknown | 30 |
| Brewing time | |
| Under \(3\) mins | 226 |
| \(3\) mins | 267 |
| \(4\) mins | 114 |
| \(5\) mins | 82 |
| \(6\) mins | 30 |
| Unknown | 781 |
Exercise 13.3 [Dataset: LungCap]
Tager et al. (1979) studied the lung volume of \(654\) children in East Boston in the 1970s (Table 13.6).
- Construct suitable plots for all variables.
- Compute the mode for each qualitative variable, and the percentage and odds of that level occurin in the data.
| Age | FEV | Height | Gender | Smoking |
|---|---|---|---|---|
| \(3\) | \(1.072\) | \(46\) | F | No |
| \(4\) | \(0.839\) | \(48\) | F | No |
| \(4\) | \(1.102\) | \(48\) | F | No |
| \(4\) | \(1.389\) | \(48\) | F | No |
| \(4\) | \(1.577\) | \(49\) | F | No |
| \(4\) | \(1.418\) | \(49\) | F | No |
Exercise 13.4 Henderson and Velleman (1981) recorded the number of cylinders in many models of cars: eleven had four cylinders, seven had six cylinders, and fourteen had eight cylinders. The number of cylinders is quantitative discrete, but with so few different values, this data could be plotted with some of the graphs used for qualitative data. For these data:
- Produce a dot chart.
- Produce a histogram.
- Produce a bar chart.
- Produce a pie chart.
What graph do you think is best? Why?
Exercise 13.5 A survey of voice assistants (such as Amazon Echo; Google Home; etc.) conducted by Nielsen asked respondents to indicate how they used their voice assistant. The options were:
- Listening to music;
- Listen to news;
- Chat with voice assistant for fun;
- Use alarms, timer.
- Search for real-time information (e.g., traffic; weather);
- Search for factual information (e.g., trivia; history);
What would be the best graph for displaying respondents answers? Would a pie chart be suitable? Explain your answer.
Exercise 13.6 Gębski et al. (2019) studied the taste of bread with varying salt and fibre content, recorded information from \(300\) subjects (including gender, place of residence, and the subjects' responses to the statement 'Rolls with lower salt content taste worse than regular ones', on a five-point ordinal scale from 'Strongly Agree' to 'Strongly Disagree').
- Identify the variables, then classify them as nominal or ordinal.
- For which variables is a mode an appropriate summary (if any)?
- For which variables is a median an appropriate summary (if any)?
- For which variables is a mean an appropriate summary (if any)?
- Compute the above where appropriate.
| Number | Percentage | |
|---|---|---|
| Gender | ||
| Female | \(150\) | \(50\) |
| Male | \(150\) | \(50\) |
| Place of residence | ||
| Rural | \(49\) | \(16\) |
| City up to \(20\, 000\) residents | \(38\) | \(13\) |
| City \(20\, 000\) to \(100\, 000\) residents | \(83\) | \(28\) |
| City more than \(100\, 000\) residents | \(130\) | \(43\) |
| Response to statement | ||
| Strongly agree | \(30\) | \(10\) |
| Agree | \(84\) | \(28\) |
| Neutral | \(78\) | \(26\) |
| Disagree | \(66\) | \(22\) |
| Strongly disagree | \(42\) | \(14\) |
Exercise 13.7 López-Serrano et al. (2022) asked \(231\) farmers what they considered to be the advantages and disadvantages of using reclaimed water. The responses are shown in Table 13.8 (not all farmers responded).
- Produce two bar charts to display the data.
- Produce two dot charts to display the data.
- Produce two pie charts to display the data
- Determine the mode for both the advantages and disadvantages.
- Compute the percentages for the advantages and disadvantages.
- Compute the odds of a farmer stating 'high price' as a disadvantage, among all farmers in the study.
- Compute the odds of a farmer stating 'high price' as a disadvantage, among all farmers who listed a disadvantage.
|
|
Exercise 13.8 Henning, Ferreira Schubert, and Ceccatto Maciel (2020) studied \(284\) university students in Joinville, Brazil, tabulating how students got to campus (Table 13.9; each student could select one option only).
- What is the mode type of active and motorised transport?
- What is the mode type of transport overall?
- Is a median appropriate?
- Compute the percentages for each option.
- Compute the odds that a randomly-chosen student uses motorised transport to get to campus. Explain what this means.
- Compute the odds that a student walks to campus. Explain what this means.
- Construct appropriate plots to display the data.
| x |
|---|
| Active |
| \(\phantom{0}29\) |
| \(\phantom{0}35\) |
| Motorised |
| \(\phantom{0}70\) |
| \(117\) |
| \(\phantom{0}33\) |