Whereas categorical data involve variables which refer to different categories or classes, continuous data involve variables which vary on a continuous dimension. Occupation is a categorical variable because people are either in one occupational type or another. Age, on the other hand, is a continuous variable because we are not simply in one age category or another, we all exist at some point along a continuum of age, from the moment we are born to the moment we die. So, someone who is 26 years old lies on an age dimension at point 26. Another critical difference between categorical and continuous data is that it is possible to get a ‘score’ on a continuous dimension which is greater than, or less than, that scored by someone else. A person who is 36 years old has a greater age (and therefore a greater ‘score’) than someone who is 26. Similarly, John may have higher-rated job perform-ance than Jim, and Mary may have greater well-being than Sally. Here, job performperform-ance and well-being, like age, are regarded as continuous rather than categorical variables, and each case has a score on a particular variable or dimension.
Sometimes continuous scales (like categorical ones) are present before research begins.
For example, we may wish to measure how much people are paid and how old they are, and in these cases convenient continuous measurement scales are probably already available in the form of annual salary, and age expressed in years or months. However, at other times it is necessary to develop a scale specifically to measure a continuous dimension of research interest. For example, a common way of obtaining continuous data in organizational research is with a Likert scale (named after the psychologist who proposed it). A Likert scale consists of a series of written statements that express a clearly favourable, or unfavourable, attitude towards something. Respondents have to indicate how much they agree or disagree with each statement, usually in terms of whether they strongly disagree, disagree, are uncertain, agree, or strongly agree. For example, you might measure well-being very crudely with the following scale which is based on a single statement.
■ I am very happy with my life:
You can then allocate scores to this scale as follows:
5 Strongly agree 4 Agree 3 Uncertain 2 Disagree
1 Strongly disagree
If 10 people complete the scale they might obtain scores like those shown in Table 1.7.
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Table 1.7 Ten well-being scores
Other common methods of constructing measure-ment scales in organizational research are the graphic rating scale where someone is asked to mark a point on a line which corresponds to their view (see Figure 1.3), and the itemized rating scale in which the respondent is asked to indicate which of several response categories they agree with (see Figure 1.4).
As with Likert scales, answers to these can be allo-cated a score. So, in Figure 1.3 a respondent would probably receive a score of 1 if they indicated that they felt ‘extremely negative’ about their manager, 5 if they indicated that they felt ‘in between’, 8 if they felt one point less than ‘extremely positive’, etc.
Similarly, in Figure 1.4 they might receive a score of 1 if they indicated that the marketing programme was not at all effective, 2 if it was quite effective, etc.
An interesting feature of the Likert scale, the graphic scale and the itemized rating scale is that while people can only obtain a limited range of scores on each (e.g. 1, 2, 3, 4 or 5) the data obtained from them are treated as continuous. The reason is that it is assumed that the underlying dimension being meas-ured is continuous. For example, in Figure 1.2 the underlying dimension measured by the Likert scale, well-being, is assumed to be continuous because, in principle, someone can have anything from zero well-being to the maximum possible level of well-well-being.
This illustrates that it is important to distinguish between the nature of the variable being measured and the scale used to measure it. For example, weight (a variable) may be measured in kilograms or pounds (two different measurement scales). In the same way, the extent to which a manager is committed might be measured by counting the number of times she works late over a six-month period, or asking her to indicate on a question-naire whether she is uncommitted, slightly committed, moderately committed or highly committed to the organization (again, two different measurement scales).
Stevens (1946) made a widely cited distinction between four different types of measure-ment scale: nominal, ordinal, interval and ratio. A nominal scale is used to measure categorical variables. So, if the categorical variable is type of manager, and the scale used to measure it is whether someone is a junior manager, a middle manager, or a senior manager, we would refer to this as a nominal scale.
Ordinal, interval and ratio scales are used for continuous variables. An ordinal scale is one in which the points on the scale are not equal distances apart. For example, imagine that the variable is job performance, and it is measured with a scale with the following
9
5
1
Extremely positive On a scale of 1 to 9, how positive do you feel about your manager?
In between
Extremely negative
Figure 1.3 An example of a graphic rating scale
o Not at all effective o Not very effective o In between o Quite effective o Very effective
In your view, how effective is the new marketing programme?
Figure 1.4 An example of an itemized rating scale
points: poor job performance, very good job performance and excellent job perform-ance. Here, the interval between poor job performance and very good job performance is quite large whereas the interval between very good and excellent performance is relatively small. The intervals between each pair of points are therefore not equal in this case, and for this reason it is referred to as an ordinal scale. Now consider a quite different example, the centigrade scale used to measure the variable of temperature. Here, the interval between each pair of points on the scale is equal: the difference in temperature between 10 degrees centigrade and 11 degrees centigrade is the same as the difference between 11 degrees and 12 degrees. When the intervals between the points on a scale are equal, we refer to it as an interval scale.
Finally, there is the ratio scale. The ratio scale has the same properties as the interval scale, but it also has an additional property: an absolute zero. The centigrade measure of temperature is not a ratio scale because zero degrees centigrade does not mean that there is zero temperature; it is just the point at which water freezes. With a ratio scale the points have equal intervals and zero really does mean zero. In an organizational context it is not uncommon to find ratio scales. For example, when annual salary is measured in the local currency this is a ratio scale because having a salary of, say, zero dollars a year would actu-ally mean earning nothing at all.
The distinction between nominal, ordinal, interval and ratio scales is often presented as being very important for applied statistics. In particular, it is suggested that when choosing a statistical test, the researcher should carefully consider which of Stevens’ four measure-ment scales has been used to collect the data (Stevens 1951). For example, texts may recommend that the ubiquitous t-test (to be discussed in Chapter 4), should only be used with data obtained from interval or ratio measurement scales. However, this was a contro-versial issue among statisticians from the outset, with Stevens’ argument criticized almost as soon as he had made it by Lord (1953). Velleman and Wilkinson (1993) provide an inter-esting review of the issue, and conclude that while the distinction between categorical and continuous variables is vital, the rigid application of Stevens’ four measurement scales is not. This position, that measurement scales are not of critical importance when choosing statistical tests, also endorsed by Howell (1997), will be adopted in this book as well.
However, there are three qualifications to this. First, ignoring the distinction between Stevens’ scales of measurement does not imply that we can also ignore the distinction between categorical and continuous data. On the contrary, being able to differentiate between categorical and continuous data is vital. Second, playing down the importance of measurement scales in the process of deciding which type of statistical technique to use does not imply that measurement scales should be dismissed when research is planned, or when research findings are interpreted. When designing research it is certainly good practice to try to measure continuous variables with the best scale possible, with ratio scales being better than interval ones, interval scales better than ordinal, and ordinal scales with reason-ably equal distances between points better than ordinal scales with gross variations in the differences between points, such as a 3-point rating scale of ‘absolutely disagree’,
‘strongly disagree’ and ‘absolutely agree’. And when interpreting research findings a researcher should always actively interpret the results of the analysis, and critically examine them, in the light of both what the variables are (e.g. well-being or job satisfaction) and 1111
the way they have been measured (e.g. with a set of questions which require responses which are measured on a Likert scale). Without such active and thoughtful interpretation, meaningless and potentially harmful conclusions can be drawn. For example, if the job performance of two people in a department is rated as excellent and the other two as extremely poor, it is not sensible to use this information to grade the performance of the department as mediocre.
Having discussed measurement scales in a little detail, let’s turn to ways in which contin-uous data can be summarized.
Summarizing continuous data
When a researcher has collected data on a continuous variable in an organization (e.g. the age of employees in years) he or she will want to get an impression of the overall nature of this data; to summarize the data in a way that will make it easy to digest and understand.
The two pieces of information that are most often used to achieve this are central tendency and dispersion.
Measures of central tendency
Central tendency is concerned with expressing information not about the extremes of continuous data but about the central part of it. So, if information is collected about the age of all the people who work in a large organization, measures of central tendency are designed to tell you not about the very oldest or the very youngest employees, but about where the ages in the centre of this range tend to be. There are three ways of measuring central tendency:
■ the mode;
■ the median;
■ the mean.
THE MODE
The mode is simply the number that occurs most frequently. So, in Table 1.7 the mode is 3 because there are more 3s than any other single number.
THE MEDIAN
The median is the figure obtained by placing all of the numbers in rank order, and finding the one that falls half-way from the smallest number to the largest. In Table 1.8 the data points relating to well-being have been rank ordered.
Now that, in Table 1.8, the numbers are sorted from the lowest to the highest, you can identify the one that comes half-way up. It is case number 5, because there are five cases below this case, and another five above it. So the median is the score for case number five, which is 3.
In the above situation, working out the median is very straightforward because, as there is an odd number of cases (11), one case will always appear half-way up the rank order.
However, if there is an even number of cases this does not apply. For example, if there had been 10 cases here instead of 11, there would not be a case half-way up the rank order.
To get round this problem, and obtain the median with an even number of cases, add together the two numbers closest to the middle of the rank order, and divide by two.
So, if there are 10 rank-ordered cases, the fifth highest has a score of 3, and the sixth highest has a score of 4, the median is:
3⫹4 divided by 2 ⫽3.5
THE MEAN
The last way of presenting information about central tendency is the mean. The mean is the figure that people usually refer to as ‘the average’, and you are almost certainly familiar with it already. If you have a 10-year-old, an 11-year-old, and a 12-year-old child, what is their average age? It is the sort of thing you probably learned early on at school, and the answer, as of course you know, is obtained by adding all the scores together and dividing by the number of scores:
10⫹11⫹12⫽33, and
33 divided by 3 is 11
So, 11 is the mean age of the three children.
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Table 1.8 Rank-ordered well-being scores
Rank Case Well-being
order score
Table 1.9 Ten well-being scores
Case Well-being
How measures of central tendency are used
The most widely used ways of presenting information about central tendency are the median and the mean. The mode is not used very much because as it is only based on one figure – the one which occurs most frequently – it often gives a misleading picture of the central tendency of the data. For example, imagine that you obtained the information shown in Table 1.9, on the well-being of 10 employees, using a Likert scale. Here, the mode would be 5 because there are more 5s than any other score. But to say that the central tendency of the data is 5 gives a misleading impression. In fact, 5 is an extreme score, and if you consider this data you will probably agree that the central tendency actually seems to be closer to 3 than to 5.
The median is normally better than the mode at capturing the central tendency of the data. Here the median would be 3 (i.e. the fifth highest score is 3, the sixth highest is also 3 and 3 ⫹3 divided by 2 ⫽3), and this clearly does represent the central tendency of the data better than the mode of 5.
The mean has an advantage over the median and the mode in that the values of all the data points contribute to it directly. However, there are circumstances in which the median can nevertheless be a better measure of central tendency than the mean. This occurs when the data contain some extreme scores, or ‘outliers’. Consider the data set in Table 1.10, which shows the ages of 10 staff.
The median age of the staff in Table 1.10 is 22.5, and this fairly represents the central tendency of the staff who are mostly in their early twenties. But the mean is 29.2 – inflated by the two older people in their sixties. So in cases like this, where there are some extreme scores, the median gives a better impression of central tendency than the mean.
One of the reasons why measures of central tendency are helpful is that they can be used to summarize information about a great deal of data in a highly concentrated and efficient way. Imagine that you collect information about the ages of 2,000 employees. In this case scrutinizing so many numbers would be virtually impossible. But by indicating that the mean age is 43.7 you immediately get an impression of the central tendency of this data. Even when you only have information about a small number of people, measures of central tendency can be very helpful.
Let’s say you were to run two training programmes, one using training method A, and one using training method B. You are interested in which of these programmes is more effective, so you train 30 people using one programme, and another 30 people using the other. You then measure how much each person has learnt using a continuous scale, with 0 indicating that they have learnt nothing, and 100 indicating that they have learnt everything perfectly.
You could then use the median or mean amount learnt by the people trained with each method to
Table 1.10 The ages of 10 staff
Case Age
1 23
2 22
3 62
4 18
5 64
6 22
7 19
8 20
9 21
10 21
make a direct comparison between the amount learnt in the two training programmes.
Comparing the amount learnt with the two methods would be far more difficult if you did not use a measure of central tendency, and instead tried to compare the 30 individual scores obtained with one training method with the 30 obtained from the other one.
Measures of dispersion
As well as summarizing continuous data with measures of central tendency, it is also helpful to provide summary information about the dispersion of the scores – how spread out they are. How can you measure dispersion? In fact, there are several ways to do it.
THE RANGE
The first, and simplest method, is known as the range. The range is simply obtained by subtracting the smallest score from the largest one. Thus:
– the range for Department A is 46 ⫺44⫽2;
– the range for Department B is 65 ⫺25⫽40.
This gives a straightforward and clearly understandable measure of the extent to which data on a continuous dimension are dispersed. However, the range is not always a good measure of dispersion. If you collect information about the ages of people in Departments C and D you might find the following:
– Department C: 25, 26, 27, 45, 63, 64, 65;
– Department D: 25, 43, 44, 45, 46, 47, 65.
Here the range is the same in both cases: 40 years, but the ages in Department C are gener-ally more spread out than the ages in Department D. What is needed to get round this problem is a measure of dispersion that does more than simply reflect how spread out the largest and smallest scores are and which prevents extreme scores from having a dispro-portionate influence. In fact, there is a measure of dispersion which has these properties:
the inter-quartile range.
THE INTER-QUARTILE RANGE
The inter-quartile range is actually quite closely related to the median. Imagine that you collect data about the salaries of all 1,001 people who work in a large organization. If you rank-order these salaries from the lowest to the highest, and then choose the salary that is half-way between the lowest and the highest, what would this be called? You will no doubt remember that this would be the median salary.
Another way to think about the median is that it occurs 50 per cent of the way up the range of data when they are ordered from the lowest to the highest. It is for this reason that the median can also be called the 50th percentile. To understand what percentiles are, imagine that you have collected information about the salary of all 7,000 people who work in a financial organization. George Brown, an employee at the company, earns
$37,000 a year, and you want to know how this salary compares to everyone else who 1111
works there. To create percentiles, all 7,000 salaries are rank-ordered and then divided into 100 equal portions. The highest salary in each portion is known as a percentile. So the 1st percentile is the value that only 1 per cent of the salaries fall at or below, and the 61st
works there. To create percentiles, all 7,000 salaries are rank-ordered and then divided into 100 equal portions. The highest salary in each portion is known as a percentile. So the 1st percentile is the value that only 1 per cent of the salaries fall at or below, and the 61st