Frequency distributions are critical to understanding our use of measurement terms. We begin this chapter with a discussion of frequency distributions and two examples. Frequency tables and distributions can be used whether the variable involved has ordered or unordered levels or values. In this section, we only consider variables with many ordered values.
A frequency distribution is a tally or count of the number of times each score on a single variable occurs. For example, the frequency distribution of final grades in a class of 50 students might be 7 As, 20 Bs, 18 Cs, and 5 Ds. Note that in this frequency distribution most students have Bs or Cs (grades in the middle) and similar smaller numbers have As and Ds (high and low grades). When there are a small number of scores for the low and high values and most scores are for the middle values, the distribution is said to be approximately normally distributed. We discuss this distribution and the normal curve later in this chapter.
When the variable is continuous or has many ordered levels (or values), the frequency distribution usually is based on ranges of values for the variable. For example, the frequencies (number of students), shown by the bars in Fig 3.1, are for a range of points. (In this case the program selected a range of 50: 250-299, 300-349, 350-399, etc.) Notice that the largest number of students (about 20) had scores in the middle two bars of the range (450-499 and 500-549). Similar small numbers of students have very low and very high scores. The bars in the histogram form a distribution (pattern or curve) that is similar to the normal, bell-shaped curve. Thus, the frequency distribution of the SAT math scores is said to be approximately normal.
Figure 3.2: A grouped frequency distribution for SAT math scores
Figure 3.2 shows the frequency distribution for the competence scale. Notice that the bars form a pattern very different from the normal curve in Figure 3.1. This distribution can be said to be not normally distributed. As we see later in the chapter, the distribution is negatively skewed. That extreme scores or the tail of the curve are on the low end or left side. As you will see in the Levels of Measurement section, we call the competence scale variable ordinal.
You can create these figures yourself using the hsbdata.sav file. Select:
- Graphs → Legacy Dialogs → Histogram…
- Then move scholastic aptitude test – math (or competence scale) into the Variable box
- Click OK. The program can superimpose a normal curve on the histogram if you request it, but we have found this curve more confusing than helpful to our students.
Fig 3.2. A grouped frequency distribution for the competence scale
Source: Morgan George A, Leech Nancy L., Gloeckner Gene W., Barrett Karen C.
(2012), IBM SPSS for Introductory Statistics: Use and Interpretation, Routledge; 5th edition; download Datasets and Materials.
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