Chapter 4: Analysing the Data Part II : Descriptive Statistics

# The shape of a distribution

Symmetry. A distribution of scores may be symmetrical or asymmetrical. Imagine constructing a histogram centred on a piece of paper and folding the paper in half the long way. If the distribution is symmetrical, the part of the histogram on the left side of the fold would be the mirror image of the part on the right side of the fold. If the distribution is asymmetrical, the two sides will not be mirror images of each other. True symmetric distributions include what we will later call the normal distribution. Asymmetric distributions are more commonly found.

Skewness. If a distribution is asymmetric it is either positively skewed or negatively skewed. A distribution is said to be positively skewed if the scores tend to cluster toward the lower end of the scale (that is, the smaller numbers) with increasingly fewer scores at the upper end of the scale (that is, the larger numbers). Figure 4.2 is an example of a positively skewed distribution, the majority of people report 0, 1, or 2 sexual partners for the year and increasingly few report more.

A negatively skewed distribution is exactly the opposite. With a negatively skewed distribution, most of the scores tend to occur toward the upper end of the scale while increasingly fewer scores occur toward the lower end. An example of a negatively skewed distribution would be age at retirement. Most people retire in their mid 60s or older, with increasingly fewer retiring at increasingly earlier ages. A graphic example of a negatively skewed distribution can be found in Figure 4.5.

Figure 4.5 An example of a negatively skewed distribution

SPSS example

If we select FrequenciesÉ from SPSS on the "number of sex partners last year" variable and also select the appropriate statistics (as shown in the SPSS Screens and Output booklet), you will find the following output.

Output 4.2 Summarize É. FrequenciesÉwith statistics options.

### Frequencies

Output 4.2 The results of selecting Statistical options within the SPSS FrequenciesÉ procedure.

Output 4.2 shows many numerical descriptive measures for the "number of sex partners" variable. Many we will deal with in the next chapter. For the moment, we will focus on skewness and kurtosis.

If a distribution is not skewed, the numerical value for "Skewness" is zero. The fact that it is positive (3.076) in the output above, shows that the variable is positively skewed.