DISTRIBUTIONS THAT ARE NOT NORMAL

The normal distribution is often used as a reference for describing other distributions. A distribution is called skewed if it is not symmetric but instead has more cases (more of a "tail") toward one end of the distribution than the other. If the long tail is toward larger values, the distribution is called positively skewed, or skewed to the right. If the tail is toward smaller values, the distribution is negatively skewed, or skewed to the left. A variable such as income has a positively skewed distribution. That is because some incomes are very much above average and make a long tail to the right. Since incomes are rarely less than zero, the tail to the left is not so long.

If a larger proportion of cases falls into the tails of a distribution than into those of a normal distribution, the distribution has positive kurtosis. If fewer cases fall into the tails , the distribution has negative kurtosis. You can compute statistics that measure how much skewness and kurtosis a distribution has, in comparison to a normal distribution. These statistics are zero if the observed distribution is exactly normal. Positive values for kurtosis indicate that the tails of a distribution are heavier than those of a normal distribution. Negative values indicate that a distribution has lighter tails than a normal distribution does. Of course, the measures of skewness and kurtosis for samples from a normal distribution will not be exactly zero. Because of variation from sample to sample, they will fluctuate around zero. To use the computer for the calculations, one only needs to identify the command with skewness and kurtosis , and the computer does the rest.