A z score measures the relative location of a score within a distribution. The distribution of z scores follows the shape of the theoretical normal distribution. The z score constitutes the simplest statistic that can be used for the purposes of testing hypotheses about a mean.

The resources in this section focus exclusively on the z statistic and the statistical procedures for which it can be used.

  • z Scores: An Introduction
  • Software Tutorial: Computing z Scores (SPSS)
    Software Tutorial: Computing z Scores (SPSS)

  • z Procedures: Confidence Intervals for the Population Mean
    z Procedures: Confidence Intervals for the Population Mean

    A confidence interval is a range of values that a parameter may take in the population. This range is estimated using information collected from a sample, such as the mean, the degree to which values vary across individuals, or the sample size. For instance, a researcher may be interested in estimating the achievement motivation of first year college students. The researcher must select a random sample of students, administer a motivation scale, and then compute an average score for the entire sample. Results from the sample can then be used to make an inference about the motivation of the entire population of first year college students. The narrated presentation bellow provides an introduction to the topic of confidence intervals and demonstrates how to estimate the population mean of a normally distributed variable after computing the mean for a specific sample. The software tutorial shows how to calculate confidence intervals using SPSS.

    Diana Mindrila, Ph. D.
    Phoebe Balentyne, M.Ed.

  • z Procedures: Testing a Hypothesis about the Population Mean
    z Procedures: Testing a Hypothesis about the Population Mean

    A hypothesis is a statement about a parameter such as the population proportion or the population mean. To determine whether this statement is true, researchers use tests of significance to compare observed values of a statistic to the given parameters. Results from such tests show whether the difference between the sample statistic and the given parameter are statistically significant. The results of a significance test are expressed in terms of a probability that indicates the extent to which data from the sample and the hypothesis agree. The narrated presentation provides an introduction to this topic. It demonstrates how to formulate hypotheses, and how to conduct a test of significance for a population mean using the properties of the normal distribution.

    Diana Mindrila, Ph. D.
    Phoebe Balentyne, M.Ed.

  • z Procedures: Practical Issues
    z Procedures: Practical Issues

    To be able to use the z procedures, certain assumption must be met. First, data should have a normal distribution. Second, the sample must have an adequate size, and individuals must be randomly selected. However, these conditions are often difficult to meet in practice. The following narrated presentation describes the necessary conditions for making inferences based on the z procedures, demonstrates how to determine the sample size needed for a certain level of error, and discusses the notions of Type I and Type II error, and the power of a significance test.

    Diana Mindrila, Ph. D.
    Phoebe Balentyne, M.Ed.

    Inference in Practice (Notes)