Normal Distribution

The normal distribution is a hypothetical distribution of individual scores on any test. The normal distribution is a symmetrically shaped bell curve with the largest number of scores in the middle of the distribution and the smallest number of scores in the tails of the distribution (Figure 1). Because the normal curve is symmetrical, the measures of central tendency (i.e., the mean, median, and mode) will be the same. The mode is the most frequent score in the distribution, the mean is the average score in the distribution, and the median is the middlemost score or the midpoint of the distribution (Cohen & Swerdlik, 1999).

Figure 1 Normal Curve

The normal distribution can be “divided into areas defined in standard deviation units” (Cohen & Swerdlik, 1999, p. 107). Figure 1 shows a hypothetical distribution of mathematics achievement test scores with a mean of 100 and a standard deviation of 15. In this example, 34.13% of all scores will fall between the mean and 1 standard deviation above or below the mean. Stated another way, approximately 34% of all mathematics achievement test scores will lie between the scores of 85 and 100, or 100 and 115. Continuing with this example, 13.59% and 2.14% of all scores will lie between 1 and 2, and 2 and 3 standard deviations above or below the mean, respectively. In other words, approximately 14% of all scores will fall between 70 and 85, or 115 and 130, and approximately 2% of all scores will lie between 55 and 70, or 130 and 145. These relationships between the percentages of test scores and standard deviation units are important when discussing test score interpretations.