The bell curve, also known as the normal distribution, provides a foundation for the majority of statistical procedures currently used in sociology. It can be thought of as a histogram of a continuous variable, but with such fine distinctions between outcomes that it is not possible to differentiate individual bars, so that the histogram appears to be a smooth line in the shape of a bell. Beneath this line is 100 percent of the possible outcomes, with the x axis describing the range of possible outcomes and the y axis describing the proportion or probability for each outcome.

The shape of the distribution is symmetrical, so that if it is divided in two, one half is the mirror image of the other. It is also unimodal, meaning that there is only one mode (most frequent value in the distribution). Because the bell curve is unimodal and symmetrical, the distribution’s mean, median, and mode are identical and in the exact center of the distribution. Additionally, the ‘‘tails’’ of the curve extend indefinitely, without ever actually reaching the x axis.

The bell curve has a specific distribution of scores. One standard deviation from the mean will always take up 34.13 percent of the area under the curve, or 34.13 percent of scores for the variable. Two standard deviations from the mean will always take up 47.72 percent of the area under the curve. Three standard deviations will always take up 49.87 percent of the area under the curve. Since the distribution is symmetrical, the distance from the mean will be the same regardless of whether the standard deviations are above or below the mean. Each additional standard deviation from the mean adds progressively less area under the curve because scores are less likely the farther they are from the mean.

The bell curve is especially useful for hypothesis testing because of the central limit theorem. This theorem states that, even when individual scores are not normally distributed, in random samples of a sufficient size, the distribution of sample means will be approximately normally distributed around the population mean. This facilitates hypothesis testing by allowing a sociologist to examine the probability of producing a specific sample mean, based on a hypothesized population mean. If this sample mean is unlikely to occur simply through chance, the sociologist can reject the hypothesized population mean. Similarly, relationships between variables can be tested by measuring their relationship in a sample, and studying how likely it would be to find this relationship if there was no relationship in the population.

### References:

- Agresti, & Finlay, B. (1997) Statistical Methods for the Social Sciences. Prentice-Hall, Upper Saddle River, NJ.
- Healey, F. (2005) Statistics: A Tool for Social Research.
- Thomson Wadsworth, Belmont, Ritchey, F. (2000) The Statistical Imagination: Elementary Statistics for the Social Sciences. McGraw-Hill College, Boston.