This chapter:
stresses the advantages of using a software package when analysing quantitative data
and your likely need for help and advice when doing this;
shows how to create a data set for entry into a computer;
distinguishes between exploratory and confirmatory data analysis;
explains statistical significance and discusses its controversial status;
advocates greater reliance on measures of effect sizes;
suggests how to explore, display, and summarize the data;
discusses ways of analysing relationships between various types of data and a range of
statistical tests that might be used;
does the same thing for analysing differences between data; and
considers issues specific to the analysis of quasi‐experiments, single‐case experiments,
and non‐experimental fixed designs.
Introduction
You would have to work quite hard in a research project not to have at least some data in the
form of numbers, or which could be sensibly turned into numbers of some kind. Hence, techniques
for dealing with such quantitative data are an essential feature of your professional tool‐
kit. Their analysis covers a wide range of things, from simple organization of the data to complex
statistical analysis. This chapter does not attempt a comprehensive treatment of all aspects of
quantitative data analysis. Its main aim is to help you appreciate some of the issues involved so
that you have a feeling for the questions you need to ask when deciding on an appropriate kind
of analysis.
The analysis and interpretation
of quantitative data
CHAPTER 17
REAL WORLD RESEARCH410
Some assumptions
1. Everyone doing real world research needs to understand how to summarize and display quantitative
data. This applies not only to those using fixed and multi‐strategy designs, but also to users
of flexible designs where their data are essentially qualitative. Even die‐hard qualitative
researchers will often collect small amounts of numerical data or find advantage in turning
some qualitative data into numbers for summary or display purposes. This does not necessarily
call for the use of statistical tests. Simple techniques may be all you need to interpret
your data.
2. For relatively simple statistical tests specialist statistical software is not essential. If you only have a
very small amount of quantitative data, it may be appropriate for you to carry out analyses
by ‘hand’ (or with the help of an electronic calculator). However, the drudgery and potential
for error in such calculation, and the ease with which the computer can perform such mundane
chores for you, suggest strongly that you make use of the new technology if at all possible.
For such tasks, and for simple statistical tests, spreadsheet software such as Excel may
be all that you need. ‘Analyse‐it’ (http://www.analyse‐it.com) is a straightforward package
which can be used with Excel to produce most of the commonly used statistics and charts.
Appendix A gives details. It has been used for several of the figures showing the results of
different statistical analyses in this chapter.
3. If you need to carry out complex statistical tests you will need to use a specialist statistical computer
package. A range of commonly used statistical packages is discussed in Appendix A. SPSS (the
Statistical Package for the Social Sciences) is the market leader by some margin but other
packages are well worth considering, particularly if you wish to follow the exploratory data
analysis (EDA) approach highlighted in the chapter. Facility in the use of at least one specialist
statistical package is a useful transferable skill for the real world researcher.
4. You have some prior acquaintance with the basic concepts and language of statistical analysis. If not,
you are recommended to spend some time with one of the many texts covering this at an
introductory level (e.g. Graham, 2013; Robson, 1994; Rowntree, 2000).
5. You will seek help and advice in carrying out statistical analyses. The field of statistical analysis is
complex and specialized and it is unreasonable to expect everyone carrying out real world
research to be a statistical specialist. It is, unfortunately, a field where it is not at all difficult to
carry out an analysis which is simply wrong, or inappropriate, for your data or your purposes.
And the negative side of readily available specialist statistical software is that it becomes
that  much easier to generate elegantly presented rubbish (remember GIGO – Garbage In,
Garbage Out).
Preferably, such advice should come from an experienced statistician sympathetic to the particular
difficulties involved in applied social research. Repeating the advice once more – it should
be sought at the earliest possible stage in the design of your project. Inexperienced non‐numerate
researchers often have a touching faith that research is a linear process in which they first collect
the data and then the statistician shows them the analysis to carry out. It is, however, all too easy
to end up with unanalysable data, which, if they had been collected in a somewhat different
way, would have been readily analysable. In the absence of personal statistical support, you
should be able to use this chapter to get an introduction to the kind of approach you might take.
The references provided should then help with more detailed coverage.
THE ANALYSIS AND INTERPRETATION OF QUANTITATIVE DATA 411
Organization of the chapter
The chapter first covers the creation of a ‘data set’ as a necessary precursor to data analysis.
Suggestions are then made about how you might carry out various types of data analysis appropriate
for different research designs and tasks.
Creating a data set
If you are to make use of a computer to help with analysis, then the data must be entered into the
computer in the form required by the software you are using. This may be done in different ways:
1. Direct automatic entry. It may be feasible for the data to be generated in such a way that entry
is automatic. For example, you may be using a structured observation schedule with some
data collection device (either a specialized instrument or a laptop computer) so that the data
as collected can be directly usable by the analysis software.
2. Creation of a computer file which is then ‘imported’ to the analysis software. It may be easier for your
data to be entered into a computer after collection. For example, a survey might use questionnaires
which are ‘optically readable’. Respondents, or the person carrying out the survey, fill in
boxes on the form corresponding to particular answers. The computer can directly transform
thisresponseintodatawhichitcanuse.Suchdataformacomputer‘file’whichisthen‘imported’
into the particular analysis software being used. This is feasible with most statistical packages
although you may need assistance to ensure that the transfer takes place satisfactorily.
3. Direct ‘keying’ of data into analysis software. For much small‐scale research, automatic reading
or conversion of the data into a computer file will either not be possible or not be economically
justifiable. There is then the requirement for manual entry of data into the analysis software.
The discussion below assumes that you will be entering the data in this way.
Whichever approach is used, the same principle applies. Try at the design stage to capture
your data in a form which is going to simplify this entry process. Avoid intermediate systems
where the original response has to be further categorized. The more times that data are transferred
between coding systems, the greater the chance of error. Single‐transfer coding (i.e. where
the response is already in the form which has to be entered into the computer) is often possible
with attitude and other scales, multiple‐choice tests, inventories, checklists, and many questionnaires.
In a postal or similar survey questionnaire, you will have to weigh up whether it is more
important to simplify the task of the respondent or the task of the person transferring the code
to the computer. Box 17.1 shows possible alternatives.
The conventions on coding are essentially common sense. Suggestions were made in
Chapter 11 (p. 272) about how this might be dealt with in relation to questionnaires. Note that it
is helpful to include the coding boxes on the questionnaire itself, conventionally in a column on
the right‐hand side of each page.
The data sets obtained from other types of project will be very various. However, it is almost
always possible to have some sensible arrangement of the data into rows and columns. Typically
each row corresponds to a record or case. This might be all of the data obtained from a particular
REAL WORLD RESEARCH412
respondent. A record consists of cells which contain data. The cells in a column contain the data
for a particular variable. Figure 17.1 presents a simple example derived from a survey‐type study.
BOX 17.1
Question formats requiring (a) single‐transfer coding and
(b) double‐transfer coding
(a) How many children are there in your school?
under 40 40–49 50–59 60–69 70–79 80–89 90–100 over 100
code 1 2 3 4 5 6 7 8
enter code ( )
(b) How many children are there in your school?
(please circle)
under 40 40–49 50–59 60–69 70–79 80–89 90–100 over 100
(response has then to be translated into appropriate code)
Student Faculty Sex Entry points Degree class Income
1 A F 14 2.1 14,120
2 EN M 6 2.2 15,900
3 EN M 5 Fail 11,200
4 ED F 10 2.2 21,640
5 S M 4 2.1 25,000
6 B F 13 2.1 11,180
7 A F 16 2.1 12,600
8 EN M 6 3 9,300
9 ED M 5 3 2,200
10 EN M * 2.2 17,880
Key: A = Arts; B = Business; Ed = Education; EN = Engineering; S = Sciences;
M = Male; F = Female; * = missing data
Note: data are fictitious, but modelled on those in Linsell and Robson, 1987
Figure 17.1 Faculty, entry points, degree classification, and
income two years after graduating of a sample of students.
THE ANALYSIS AND INTERPRETATION OF QUANTITATIVE DATA 413
A similar matrix would be obtained from a simple experiment where, say, the columns represent
scores obtained under different experimental conditions.
Entering the data into the computer
The details of the procedure for entering this data set into the computer vary according to the
particular software you are using. With early versions of software, this was quite complex but
later versions are straightforward to use, particularly if you are familiar with the operation of
spreadsheets.
Missing data
‘The most acceptable solution to the problem of missing information is not to have any’
(Youngman, 1979, p. 21). While this is obviously a counsel of perfection, it highlights the problem
that there is no really satisfactory way of dealing with missing data. It may well be that the reason
why data are missing is in some way related to the question being investigated. Those who avoid
filling in the evaluation questionnaire, or who are not present at a session, may well have different
views from those who responded. So it is well worth spending considerable time, effort and
ingenuity in seeking to ensure a full response. Software normally has one or more ways of dealing
with missing data when performing analyses and it may be necessary to investigate this further
as different approaches can have substantially different effects on the results obtained.
Technically there is no particular problem in coding data as missing. There simply needs to
be a signal code which is used for missing data, and only for missing data. Don’t get in the habit
of using 0 (zero) to code for missing data as this can cause confusion if the variable in question
could have a zero value or if any analytic procedure treats it as a value of zero (99 or –1 are frequently
used). Software packages should show the value that you have specified as missing data
and deal with it intelligently (e.g. by computing averages based only on the data present).
It is worth noting that a distinction may need to be made between missing data where there
is no response from someone, and a ‘don’t know’ or ‘not applicable’ response, particularly if you
have catered for possible responses of this type by including them as one of the alternatives.
Cleaning the data set after entry
Just as one needs to proof‐read text for errors, so a computer data set needs to be checked for
errors made while ‘keying in’ the data. One of the best ways of doing this is for the data to be
entered twice, independently, by two people. Any discrepancies can then be resolved. This is
time‐consuming but may well be worthwhile, particularly if substantial data analysis is likely.
A valuable tip is to make use of ‘categorical’ variables whenever feasible. So, in the data set
of Box 17.1 ‘degree class’ has the categories ’first, ‘upper second’, etc. The advantage is that the
software will clearly show where you have entered an invalid value.
While this eliminates several potential mistakes, it is, of course, still possible to enter the
wrong class for an individual. The direct equivalent of proof‐reading can be carried out by checking
the computer data set carefully against the original set. Simple frequency analyses (see below)
on each of the columns are helpful. This will throw up whether ‘illegal’, or highly unlikely, codes
have been entered. For continuous variables box plots can be drawn, and potential ‘outliers’
highlighted (see p. 420).
REAL WORLD RESEARCH414
Cross‐tabulation
This involves counting the codes from one variable that occur for each code in a second variable.
It can show up more subtle errors. Suppose that the two variables are ‘withdrew before completing
degree’ and ‘class of final degree’. Cross‐tabulation might throw up one or two students who
appeared to have withdrawn before completion but were nevertheless awarded a classified
degree. These should then be checked, as while this might be legitimate (perhaps they returned),
it could well be a miscoding. Cross‐tabulation is easy when the variables have only a few values,
as is the case with most categorical variables. However, it becomes very tedious when continuous
variables such as age or income, which can take on many values, are involved. In this circumstance,
scattergrams/scatter plots (see below) provide a useful tool. These are graphs in which
corresponding codes from two variables give the horizontal and vertical scale values of points
representing each record. ‘Deviant’ points which stand out from the general pattern can be followed
up to see whether they are genuine or miscoded.
The ‘cleaned’ data set is an important resource for your subsequent analyses. It is prudent to keep a
couple of copies, with one of the copies being at a separate physical location from the others. You will
be likely to modify the set in various ways during analysis (e.g. by combining codes); however, you
should always retain copies of the original data set.
Starting data analysis
Now that you have a data set entered into the computer you are no doubt itching to do something
with it. Data analysis is commonly divided into two broad types: exploratory and confirmatory.
As the terms suggest, exploratory analysis explores the data trying to find out what
they tell you. Confirmatory analysis seeks to establish whether you have actually got what you
expected to find (for example on the basis of theory, such as predicting the operation of particular
mechanisms).
With all data sets, and whatever type of research design, there is much to be said for having
an initial exploration of the data. Try to get a feeling for what you have got and what it is trying
to tell you. Play about with it. Draw up tables. Simple graphical displays help: charts, histograms,
graphs, pie‐charts, etc. Get summaries in the form of means and measures of the amount
of variability, etc. (Details on what is meant by these terms, and how to do it, are presented later
in the chapter.) Acquiring this working knowledge is particularly useful when you are going on
to use various statistical tests with a software package. Packages will cheerfully and quickly
produce complex nonsense if you ask them the wrong question or misunderstand how you enter
the data. A good common‐sense understanding of the data set will sensitize you against this.
Exploratory approaches of various kinds have been advocated at several points during this
book. They are central to much flexible design research. While these designs mainly generate
qualitative data, strategies such as case study commonly also result in quantitative data which
we need to explore to see what has been found and to help direct later stages of data collection.
Much fixed design research is exclusively quantitative. The degree of pre‐specification of
design and of pre‐thought about possible analyses called for in fixed design research means
that the major task in data analysis is confirmatory; i.e. we are seeking to establish whether our
THE ANALYSIS AND INTERPRETATION OF QUANTITATIVE DATA 415
predictions or hypotheses have been confirmed by the data. Such confirmatory data analysis
(CDA) is the mainstream approach in statistical analysis.
However, there is an influential approach to quantitative analysis known as exploratory data
analysis (EDA) advocated by Tukey (1977) – see also Myatt and Johnson (2014). Tukey’s approach
and influence come in at two levels. First, he has proposed several ingenious ways of displaying
data diagrammatically. These devices, such as ‘box plots’, are non‐controversial, deserve wider
recognition, and are discussed below. The more revolutionary aspect of the EDA movement is
the centrality it places on an informal, pictorial approach to data. EDA is criticized for implying
that the pictures are all that you need; that the usual formal statistical procedures involving
tests, significance levels, etc. are unnecessary. Tukey (1977) does acknowledge the need for CDA;
in his view it complements EDA and provides a way of formally testing the relatively risky
inductions made through EDA.
To a large extent, EDA simply regularizes the very common process whereby researchers
make inferences about relationships between variables after data collection which their study
was not designed to test formally – or which they had not expected prior to the research – and
provides helpful tools for that task. It mirrors the suggestion made in Chapter 6 that while in
fixed design research strong pre‐specification is essential and you have clear expectations of
what the results will show (i.e. the task of analysis is primarily confirmatory), this does not preclude
additional exploration. Using EDA approaches, with a particular focus on graphical display,
has been advocated by Connolly (2006) as a means of avoiding the ecological fallacy of
making inferences about individuals from the group data provided from summary statistics.
In practice the EDA/CDA distinction isn’t clear cut. As de Leeuw puts it (in Van de Geer,
1993), the view that:
The scientist does all kinds of dirty things to his or her data . . . and at the end of this thoroughly
unrespectable phase he or she comes up (miraculously) with a theory, model, or
hypothesis. This hypothesis is then tested with the proper confirmatory statistical methods.
[This] is a complete travesty of what actually goes on in all sciences some of the time
and in some sciences all of the time. There are no two phases that can easily be distinguished
(emphasis in original).
The treatment in this chapter is influenced by EDA and seeks to follow its spirit. However, there is
no attempt to make a rigid demarcation between ‘exploring’ and ‘confirming’ aspects.
A note on ‘levels’ of measurement
A classic paper by Stevens (1946) suggested that there were four ‘levels’ of measurement (‘nominal’,
‘ordinal’, ‘interval’ and ‘ratio’). Nominal refers to a set of categories used for classification
purposes (e.g. marital status); ordinal also refers to a set of categories where they can be ordered
in some meaningful way (e.g. social class); interval refers to a set of categories which are not only
ordered but also have equal intervals on some measurement scale (e.g. calendar time); ratio is the
same as interval level, but with a real or true zero (e.g. income).
Although very widely referred to in texts dealing with the analysis of quantitative data, the
value of this typology has been queried by statisticians (e.g. Velleman and Wilkinson, 1993).
Gorard (2006) considers it unnecessary and confusing. He claims that there is little practical
REAL WORLD RESEARCH416
difference between interval and ratio scales and points out that the same statistical procedures
are traditionally suggested for both. Also that:
So‐called ‘nominal’ measures are, in fact, not numbers at all but categories of things that can
be counted. The sex of an individual would, in traditional texts, be a nominal measure. But
sex is clearly not a number . . . The only measure involved here is the frequency of individuals
in each category of the variable ‘sex’ – i.e. how many females and how many males (p. 61).
Such frequencies are, of course, ‘real numbers’ and can be added, subtracted, multiplied and
divided like other numbers. ‘Ordinal’ measures are also categories of things that can be counted
and can be treated in exactly the same way. The only difference is in the possibility of ordering
which can be used when describing and displaying frequencies.
He highlights the fact that a major problem arises when ordinal categories are treated as real
numbers. For example examination grades, A, B, C, D and E may be given points scores, say that
A is 10 points, B is 8 points, etc. As such points scores are essentially arbitrary; attempts to treat
them as real numbers, for example by working out average points scores, lead to arbitrary
results. Gorard’s advice is to:
. . . use your common sense but ignore the idea of ‘levels’ of measurement. If something is
a real number then you can add it. If it is not a real number then it is not really any kind of
number at all (p. 63).
Our advice is to take note of this advice but not to let it inhibit you from carrying out any of
the statistical analyses (particularly the simple ones) covered in the chapter – providing you
understand what you are doing, and it seems likely to shed light on what the data are trying to
tell you. The notion that specific measurement scales are requirements for the use of particular
statistical procedures, put forward by Stevens (1946), followed up in influential statistics textbooks
(e.g. Siegel, 1959), and still commonly found, is rejected by many mathematical statisticians
(see Gaito, 1980; Binder, 1984). There is nothing to stop you carrying out any analysis on
quantitative data on statistical grounds. As Lord (1953) trenchantly put it in an early response to
Stevens, ‘the numbers do not know where they came from’ (p. 751). The important thing is the
interpretation of the results of the statistical analysis. It is here that the provenance of the numbers
has to be considered, as well as other matters including the design of the study.
Exploring the data set
Frequency distributions and graphical displays
A simple means of exploring many data sets is to recast them in a way which counts the frequency
(i.e. the number of times) that certain things happen and to find ways of displaying that
information. For example, we could look at the number of students achieving different degree
classifications. Some progress can be made by drawing up a frequency distribution as in Figure 17.2.
This table can, alternatively, be presented as a bar chart (Figure 17.3).
THE ANALYSIS AND INTERPRETATION OF QUANTITATIVE DATA 417
Bar chart of degree class
0
10
20
30
40
50
60
70
first upper second lower second third pass fail
degree class
Figure 17.3 Bar chart showing distribution of students across ‘degree class’.
The chart can be shown with either frequencies or percentages on the vertical axis; be sure
to indicate which you have used. The classes of degree are ordered (here shown from first
class ‘downward’ going from left to right). For some other variables (e.g. for faculties) the
ordering is arbitrary. A distinction is sometimes made between histograms and bar charts.
A bar chart is a histogram where the bars are separated from each other, rather than being
joined together. The convention has been that histograms are only used for continuous variables
(i.e. where the bar can take on any numerical value and is not, for example, limited to whole
number values).
Pie charts provide an alternative way of displaying this kind of information (see Figure 17.4).
Bar charts, histograms and pie charts are probably preferable ways of summarizing data to the
corresponding tables of frequency distributions. It is claimed they are more quickly and easily
understood by a variety of audiences – see Spence and Lewandowsky (1990) for a review of
relevant empirical studies. Note, however, that with continuous variables (i.e. ones which can
take on any numerical value, not simply whole numbers) both frequency tables and histograms
may lose considerable detailed information. This is because of the need to group together a
Degree class First Upper second Lower second Third Pass Fail Total
Frequency 9 64 37 30 7 3 150
Percentage 6 42.7 24.7 20 4.7 2 100
Note: ‘Frequency’ is the number of students with that degree class.
Figure 17.2 Frequency distribution of students across ‘degree class’.
REAL WORLD RESEARCH418
Pie chart for faculty membership
education
12%
arts
25%
engineering
30%
business
17%
science
16%
Figure 17.4 Pie chart showing relative numbers of students in different faculties.
range of values for a particular row of the frequency table or bar of the histogram. In all cases
there will be a trade‐off between decreasing the complexity of the display and losing information.
An alternative EDA approach to displaying the data is the box plot (see p. 420).
Graphs (line charts) are well‐known ways of displaying data. Excel, and statistical packages,
provide ways of generating and displaying them although the quality of output many not be
high enough for some needs. Specialized graphics packages (e.g. DeltaGraph, available from
http://www.redrocksw.com) have a range of such displays available. Increasingly, professional
standard displays are expected in presenting the results of projects, and apart from assisting
communication, can help in getting over messages about the quality of the work. It is a matter
of judgement whether or not any package to which you have access provides output of a quality
adequate for presentation to a particular audience.
Marsh and Elliott (2008) give detailed, helpful and down‐to‐earth suggestions for producing
numerical material clearly, in a section on ‘Good Table Manners’ (pp. 126–9). Tufte (2001) provides
a fascinating compendium for anyone who needs to take graphical display seriously.
Summary or descriptive statistics
Summary statistics (also commonly known as descriptive statistics) are ways of representing
some important aspect of a set of data by a single number. The two aspects most commonly dealt
with in this way are the level of the distribution and its spread (otherwise known as dispersion).
Statistics summarizing the level are known as measures of central tendency. Those summarizing
the spread are called measures of variability. The skewness (asymmetricality), and other aspects of
the shape of the distribution which are also sometimes summarized, are considered below in the
context of the normal distribution (see p. 424).
THE ANALYSIS AND INTERPRETATION OF QUANTITATIVE DATA 419
Measures of central tendency
The notion here is to get a single figure which best represents the level of the distribution. The
most common such measure to the layperson is the ‘average’, calculated by adding all of the
scores together and then dividing by the number of scores. In statistical parlance, the figure
obtained by carrying out this procedure is referred to as the arithmetic mean. This is because
average, as a term in common use, suffers from being imprecise – some other more‐or‐less mid‐
value might also be referred to as average. There are, however, several other measures of central
tendency in use, some appropriate for special purposes. Box 17.2 covers some of them.
Measures of variability
The extent to which the data values in a set of scores are tightly clustered or relatively widely
spread out is a second important feature of a distribution for which several indices are in use.
Box 17.3 gives details of the most commonly used measures. Several of them involve calculating
deviations which are simply the difference between an individual score and the mean. Some individual
scores are above the mean (positive deviations) and others below (negative deviations).
It is an arithmetical feature of the mean that the sum of positive deviations is the same as the
sum of negative deviations. Hence the mean deviation is calculated by ignoring the sign of the
deviations, so that a non‐zero total is obtained. The standard deviation and variance are probably
the most widely used measures of variability, mainly because of their relationship to popular
statistical tests such as the t‐test and analysis of variance (discussed later in the chapter).
However, Gorard (2006, pp. 17–19 and 63–73) makes a strong case for using the mean absolute
deviation (i.e. ignoring the sign of the difference) rather than standard deviation, as it is simpler
to compute, has a clear everyday meaning, and does not overemphasize extreme scores. This is
part of his campaign in favour of ‘using everyday numbers effectively in research’.
BOX 17.2
Measures of ‘central tendency’
The most commonly used are:
Mean (strictly speaking this should be referred to as the arithmetic mean as there are other,
rarely used, kinds of mean) – this is the average, obtained by adding all the scores
together and dividing by the number of scores.
Median – this is the central value when all the scores are arranged in order of size (i.e. for
11 scores it is the sixth). It is also referred to as the ‘50th percentile’ (i.e. it has 50 per cent
of the scores below it, and 50 per cent above it).
Mode – the most frequently occurring value.
Note: Statistics texts give formulae and further explanation.
REAL WORLD RESEARCH420
BOX 17.3
Measures of variability
Some commonly used measures are:
Range – difference between the highest and the lowest score.
Midspread or inter‐quartile range – difference between the score which has one quarter of
the scores below it (known as the ‘first quartile’, or ‘25th percentile’) and that which has
three‐quarters of the scores below it (known as the ‘third quartile’, or ‘75th percentile’).
Mean deviation – the average of the deviations of individual scores from the mean (ignoring
the sign or direction of the deviation).
Variance – the average of the squared deviations of individual scores from the mean.
Standard deviation – square root of the variance.
Standard error (SE) – the standard deviation of the mean score.
Note: Statistics texts give formulae and further explanation.
Statistics packages provide a very wide range of summary statistics, usually in the form of
an optional menu of ways of summarizing any column within your data table.
Further graphical displays for single variables
It is possible to incorporate summary statistics into graphical displays in various ways.
Standard deviation error bars
A standard deviation error bar is a display showing the mean value as a dot, which has extending
above and below it an ‘error bar’. This represents one standard deviation unit above and
below the mean. Typically, about two‐thirds of the observed values will fall between these two
limits (see the discussion of the normal distribution below).
This is often a useful way of displaying the relative performance of sub‐groups, and more
generally of making comparisons. A similar‐looking display is used to show the confidence intervals
for the mean. These are limits within which we can be (probabilistically) sure that the mean
value of the population from which our sample is drawn lies: 95 per cent limits (i.e. limits within
which we can be 95 per cent sure) are commonly used, but others can be obtained. Figure 17.5
shows both error bar charts for one standard deviation and 95 per cent confidence intervals.
Box plots and whiskers
Figure 17.6 shows the general meaning of the box and its upper and lower ‘whiskers’. Note that the
plot is based on medians and other percentiles, rather than on means and standard deviations.