The z and t Distributions

The key benefit of the sampling distribution being normally distributed is that the standard normal curve can be used. Everything we did in Chapter 6 with respect to using raw scores to find z scores, z scores to find areas, and areas to find z scores can be done with the sampling distribution. The applicability of z , though, is contingent on N being large. “Large” is a vague adjective in statistics because there is no formal rule specifying the dividing line between small and large samples. Generally speaking, large samples are those containing at least 100 cases. When N ≥ 100, the sampling distribution is assumed to be normally distributed, and the standard normal curve can be used. This requirement reduces the overall usefulness of the z distribution; it turns out that the standard normal curve makes somewhat rigid demands, and many real-world data sets fall short of these expectations.

Although it is generally not advisable to work with samples smaller than 100, there are times when it is unavoidable. In these situations, the z distribution cannot be employed and researchers turn to the t distribution instead. The t distribution—like the z curve—is symmetric, unimodal, and has a constant area of 1.00. The key difference between the two is that t is a family of several different curves rather than one fixed, single curve like z. The t distribution changes shape depending on the size of the sample. When the sample is small, the curve is wide and flattish; as the sample size increases, the t curve becomes more and more normal until it looks identical to the z curve.

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