STATISTICAL PROCEDURES UNIT C: Z SCORES^{1}

In other sections, we have discussed how to describe distributions of (interval level) raw scores graphically and in terms of central tendency, variability, skewness, and kurtosis. Using what we learned from examining these descriptive statistics, we can confirm whether our data are normally distributed or if we must use different procedures.

The normal distribution is very common in social science research, so we need to deepen our understanding of some of the properties of the normal curve. We call the normal distribution a curve, since the histogram forms a curve when the top midpoints of the bars are joined together. Technically, this is called a frequency polygon. If you look at Figure DMUC.22, you will see the SPSS histogram for the school-based math achievement variable. In the figure, SPSS overlaid the normal curve on top of the histogram so you can see the extent to which the data approximate a normal distribution. As you can see from that figure, if you were to connect the top midpoints of the histogram bars with a line, it would not be the smooth line you see, but rather a more jagged line. However, as a database increases its size, the histogram approximates the smooth normal curve in variables that are normally distributed; the jagged line becomes filled in as more cases are added.

When we speak of the normal distribution and how our sample data set is normally distributed, we actually speak about ...

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