34% of the scores above and 34% of the scores below the mean to total 68%;
two standard deviations contain 34% plus 13.5% above and below the mean to
equal 95% of the scores. On a standard deviation graph, the mean has a z score
of zero. Other z-score percentages may be determined by using a table showing
the area under the curve data.
Figure 13-9 shows that normal distributions may also be used to approxi-
mate percentile ranks. Actual percentile values must always be determined by
using a normal distribution table to look up a given z score.
The age data from previous examples do not approximate a normal distribution
because the mean, median, and mode are not equal. If, however, we used data
that were standardized with a mean of 23 and a standard deviation of 1.5 years,
the normal curve for these data would look like the graph in Figure 13-10.
Sixty-eight percent of these people would be between 21.5 and 24.5 years of
age, and someone 26 years old would be in the 95th percentile.
0.15% 0.15%
2.35%13.5%34% 34%13.5%2.35%68%95%99.7%–3 SD –2 SD –1 SD x– +1 SD +2 SD +3 SDFIGURE 13-8 Standard Deviations and Percentage Distribution
13.5 Measures of Variability 349