Assumptions of the Normal Distribution

Assumptions of the Normal Distribution

Whenever we compute a statistic, there are assumptions that are being made regarding the data that we have for computation (i.e., our sample data). For example, when we compute a mean of a sample of data, we are assuming that the data are not nominal and do not have an outlier or extreme score. If either of these assumptions are violated, then the resulting mean is uninterpretable or misrepresentative of the true data.

Similarly, in order to use the one-sample z-test in the appropriate circumstance to lead to appropriate interpretations, the following assumptions are made, and should be met:

1) The outcome (dependent or criterion) variable is continuous (interval or ratio level of measurement).

2) The population the data are drawn from follow the normal probability distribution. This assumption is bolstered by the central limit theorem, which means that if we do not know the shape of the population data, or even if we think the population may not be normally distributed, as long as we have a sufficiently large sample (at least 30), our sampling distribution will be normally distributed.

3) The sample is a simple random sample from its population. Each individual in the population has an equal probability of being selected in the sample.

4) The population standard deviation is known. This must be the case, otherwise we will be unable to compute the z-score.

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