12 Summarising qualitative data
So far, you have learnt to ask a RQ, design a study, collect the data, classify the data, and summarise quantitative data In this chapter, you will learn to:
- summarise qualitative data using the appropriate graphs.
- summarise qualitative data using, for example, medians, proportions and odds.
12.1 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, qualitative data can be understood by knowing how often values of the variables appear. This is called the distribution of the data (Def. 11.1).
The distribution can be displayed using a frequency table (Sect. 12.2) or a graph (Sect. 12.3). Qualitative data can be summarised by finding modes or, for ordinal qualitative data, using medians (Sect. 12.3.5). The distribution of qualitative data can be summarised numerically by computing proportions, percentages (Sect. 12.4) or odds (Sect. 12.5).
12.2 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. 10.4), and these should be exhaustive (cover all levels) and exclusive (observations belong to only one level). The number of observations or the percentage of observations (or both) are then used to describe the levels.
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 12.1 (Opinions of AV vehicles) Pyrialakou et al. (2020) surveyed \(400\) residents of Phoenix (Arizona) about their opinions of autonomous vehicles (AVs). Demographic information (Table 12.1) and respondents' opinions of sharing roads with AVs (Table 12.2) 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 12.2) 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 | \(63\) | \(16\) | \(\phantom{0}86\) | \(22\) | \(103\) | \(26\) | \(82\) | \(20\) | \(66\) | \(16\) |
12.3 Graphs
Three options for graphing qualitative data include:
- Dot charts (Sect. 12.3.1): usually a good choice.
- Bar charts (Sect, 12.3.2): usually a good choice.
- Pie charts (Sect. 12.3.3): 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 any 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, an 'Other' (or similar) level may be used, and may be placed at the end of the list of levels.)
Sometimes these graphs are also used for discrete quantitative data with a small number of possible options. Sometimes, graphs used for quantitative data (Sect. 11.3) may be useful for qualitative 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.
12.3.1 Dot charts (qualitative data)
Dot charts indicate the counts (or corresponding percentages) in each level. Dots indicate the number (or percentage) in each level. 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.
The axis displaying the counts (or percentages) should start from zero, since the distance of the dots from the axis visually implies the frequency of those observations (see Example 17.3).
Example 12.2 (Dot plots) For the AV study in Example 12.1, a dot chart of the age group of respondents is shown in Fig. 12.1 (top left panel).
FIGURE 12.1: 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.
12.3.2 Bar charts
Bar charts indicate the counts in each category. Bars indicate the counts (or percentages) in each category. 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.
The axis displaying the counts (or percentages) should start from zero, since the height of the bars visually implies the frequency of those observations (see Example 17.3).
Example 12.3 (Bar plots) For the AV study in Example 12.1, a bar chart of the age group of respondents is shown in Fig. 12.1 (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 usually represent a continuous numerical scale.
12.3.3 Pie charts
In pie charts, a circle is divided into segments proportional to the number in each level of the qualitative variable.
Example 12.4 (Pie charts) For the AV study in Example 12.1, a pie chart of the age group of respondents is shown in Fig. 12.1 (bottom left panel).
Using 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 17.3).
- 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 12.5 (Pie chart unsuitable) 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).
12.3.4 Comparing pie, bar and dot charts
Consider the pie chart in Fig. 12.1 (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. 12.1, 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).
12.3.5 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 12.1 (Mode) A mode is the level (or levels) of a qualitative variable with the most observations.
Example 12.6 (Modes) Consider the data in Tables 12.1 and 12.2. '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, or \(51\)%).
- The mode age groups are \(25\) to \(34\) and \(35\) to \(44\) (each with \(19\) respondents, or \(4.8\)%).
- 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. The sample median estimates the unknown population median.
Medians can be used to summarise qualitative data and ordinal data, but never nominal data.
Example 12.7 (Medians) Consider the data in Tables 12.1 and 12.2. '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 data can also be numerically summarised using proportions, percentages (Sect. 12.4) or odds (Sect. 12.5). Importantly, these numerical quantities are computed from a sample (i.e., are statistics; Def. 11.3), even though the whole population is of interest (i.e., the parameter; Def. 11.2).
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.
12.4 Numerical summary: proportions and percentages
Qualitative data can be summarised numerically by using the proportion or percentage of individuals in each level. These can be given instead of, or with, the counts (as in Tables 12.1 and 12.2).
Definition 12.2 (Proportion) A proportion is a fraction out of a total, and is a number between \(0\) and \(1\).
Definition 12.3 (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. The caret above the \(p\) is called a 'hat'.
As always, only one possible sample is studied. Statistics are estimates of parameters, and the value of the statistic is not the same for every possible sample.
Example 12.8 (Proportions and percentages) Consider the AV data in Table 12.1, 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.
12.5 Numerical summary: odds
For the AV data in Table 12.1, 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 results of interest divided by the total number, the odds are computed as the number of results of interest divided by the remaining number.
Definition 12.4 (Odds) The odds are the number (or proportion, or percentage) of results of interest, divided by the remaining number (or proportion, or percentage) of results: \[ \text{Odds} = \frac{\text{Number of results of interest}}{\text{Remaining number of results}} \] or (equivalently) \[ \text{Odds} = \frac{\text{Proportion of results of interest}} {\text{Remaining proportion of results}}. \] The odds are how many times the result of interest occurs compared to the number of times the results of interest does not occur.
Example 12.9 (Interpreting odds) The AV data in Table 12.1 includes \(204\) females and \(196\) males. The odds that a respondent is female are \(1.04\), as found above. The odds are greater than one, as the number of females is larger than the number of males. Alternatively, there are \(104\) females for every \(100\) males.
The odds that a respondent is male are \(196/204 = 0.96\); there are \(0.96\) times as many males as females. The odds are less than one, as the number of males is smaller than the number of females. Alternatively, there are \(96\) males for every \(100\) females.
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.
Example 12.10 (Odds and percentages) Consider the AV data in Table 12.1, 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 \(55\) or over.
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. 12.2).
FIGURE 12.2: Proportions (left) are the number of interest divided by the total number. Odds (right) are the number of interest divided by the rest.
12.6 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. 13), odds are usually not given in summary table. An example of a summary table is given in the next section.
12.7 Example: water access
López-Serrano et al. (2022) recorded data about access to water for three rural communities in Cameroon (see Sect. 11.10). Numerous qualitative variables are recorded; some are displayed in Fig. 12.3, and summarised in Table 12.3. Notice that the levels of the two ordinal variables are displayed in their natural order.
The distance to the nearest water source is usually less than \(1\), and the wait is often over \(15\). The most common water source is a bore (\(68.6\)%).
| Num. | % | Odds | |
|---|---|---|---|
| Distance to water source | |||
| Under \(100\) | \(55\) | \(45.5\) | \(0.83\) |
| \(100\)to \(1000\) | \(57\) | \(47.1\) | \(0.89\) |
| Over \(1000\) | \(\phantom{0}9\) | \(\phantom{0}7.4\) | \(0.08\) |
| Wait time at water source | |||
| Under \(5\) | \(50\) | \(41.7\) | \(0.71\) |
| \(5\) to \(15\) | \(28\) | \(23.3\) | \(0.30\) |
| Over \(15\) | \(42\) | \(35.0\) | \(0.54\) |
| 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 12.3: 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.)
12.8 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 numerically summarised using proportions, percentages or odds.
12.9 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 something of interest occurs, to how often it does not occur.
- Proportions and percentages are the same.
12.10 Exercises
Answers to odd-numbered exercises are available in App. E.
Exercise 12.1 A study of spider monkeys (C. A. Chapman 1990) examined the types of social groups present (Table 12.4). 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 12.2 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 12.5.
- 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 12.3 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, the 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 12.4 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 12.5 Gębski et al. (2019) studied the taste of bread with varying salt and fibre content, and 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'); see Table 12.6.
- 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)?
- Compute the above statistics where appropriate.
- Compute and interpret the odds of a respondent coming from a city background.
- Compute and interpret the odds of a respondent agreeing or strongly agreeing with the statement.
- Compute and interpret the odds of a respondent being male.
| 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 12.6 López-Serrano et al. (2022) asked \(231\) farmers what they considered to be the advantages and disadvantages of using reclaimed water on the farm. The responses are shown in Table 12.7 (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 farmers in the study.
- Compute the odds of a farmer stating 'high price' as a disadvantage, among farmers who listed a disadvantage.
|
|
Exercise 12.7 Henning, Ferreira Schubert, and Ceccatto Maciel (2020) studied \(284\) university students in Joinville, Brazil, tabulating how students got to campus (Table 12.8; each student could select one option only).
- What is the mode type of active transport? What about motorised transport?
- What is the mode type of transport overall?
- Are medians appropriate? If so, compute the median for active transport types, and motorised transport types.
- Compute the percentages for each option, out of the total sample.
- 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 |
| \(\phantom{0}70\) |
| \(117\) |
| \(\phantom{0}33\) |
Exercise 12.8 The data in Fig. 11.2 give the ender of \(44\) babies born in a hospital on one day (P. K. Dunn 1999; Steele 1997). The data are given in the order in which the births occurred.
- What is the mode sex?
- If appropriate, compute the median sex.
- Compute the percentages 8for each sex.
- Compute the odds that a randomly-chosen baby is female. Explain what this means.
- Construct appropriate plots to display sex of the baby.
Exercise 12.9 [Dataset: LungCap]
Tager et al. (1979) studied the lung volume of \(654\) children in East Boston in the 1970s (Table 12.9).
- Construct suitable plots for all variables.
- For each qualitative variable, determine the mode.
- For each qualitative variable, compute the percentage and odds of one of the levels occurring in the data.
- Compute appropriate statistics for each quantitative variable.
| 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 12.10 Swinnen et al. (2018) studied the influence of using ankle-foot orthoses in children with cerebral palsy. The data in Table 10.3 give the data for the \(15\) subjects. (GMFCS is the Gross Motor Function Classification System) used to describe the impact of cerebral palsy on their motor function; where lower levels mean better functionality.)
- Construct suitable plots for all variables.
- For each qualitative variable, determine the mode.
- For each qualitative variable, compute the percentage and odds of one of the levels occurring in the data.
- Compute appropriate statistics for each quantitative variable.
Exercise 12.11 [Dataset: PremierL]
In the 2019/2020 Premier League season, Chelsea had \(4\) wins from \(10\) games at home, and \(7\) wins from \(11\) wins away from home.
What is the odds ratio of a win (comparing home games and away games)?