The Normal Curve

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Three ideas about the normal curve help us to understand inferential statistics. First, the normal curve describes the variability of many physical and psychological characteristics. Second, the normal curve may be translated into percentages, allowing us to standardize variables and make direct comparisons of scores on different measures. Third, a distribution of means, rather than a distribution of scores, produces a more normal curve. The last idea is based on the central limit theorem, by which we know that a distribution of means will be normally distributed and less variable as long as the samples from which the means are computed are of a sufficiently large size, usually at least 30.

Standardization, z Scores, and the Normal Curve

The process of standardization converts raw scores into z scores. Raw scores from any normal distribution can be converted to the z distribution. And a normal distribution of z scores is called the standard normal distribution. z scores tell us how far a raw score falls from its mean in terms of standard deviation. We can also reverse the formula to convert z scores to raw scores. Standardization using z scores has two important applications. First, standardized scores—that is, z scores—can be converted to percentile ranks (and percentile ranks can be converted to z scores and then raw scores). Second, we can directly compare z scores from different raw-score distributions. z scores work the other way around as well.

The Central Limit Theorem

The z distribution can be used with a distribution of means in addition to a distribution of scores. Distributions of means have the same mean as the population of individual scores from which they are calculated, but a smaller spread, which means we must adjust for sample size. The standard deviation of a distribution of means is called the standard error. The decreased variability is due to the fact that extreme scores are balanced by less extreme scores when means are calculated. Distributions of means are normally distributed if the underlying population of scores is normal, or if the means are computed from sufficiently large samples, usually at least 30 individual scores. This second situation is described by the central limit theorem, the principle that a distribution of sample means will be normally distributed even if the underlying distribution of scores is not normally distributed, as long as there are enough scores—usually at least 30—comprising each sample. The characteristics of the normal curve allow us to make inferences from small samples using standardized distributions, such as the z distribution.