Z Score

Z scores are also called standard units or standard scores. A z score standardizes values of a random variable from a normal (or presumed normal) distribution for comparison with known probabilities in a standard normal probability distribution table or z table. A z score is unitless and is simply a measure of the number of standard deviations a value is from the mean. The z score is especially useful to indicate where a particular data point is relative to the rest of the data. A positive z score indicates that the point is above the mean, while a negative z score indicates that the point is below the mean. The z score removes varying units (e.g., pounds or kilograms) and allows for easy determination of whether a particular result is unusual.

Most significance testing, hypothesis testing, and confidence intervals are based on an assumption that the data are drawn from an underlying normal distribution. However, the probabilities are dependent on the mean, μ, and standard deviation, σ, of the distribution. Since it is physically impossible to calculate the probabilities associated with infinitely many pairs of μ and σ, the z score allows a researcher to compare the sample data with the standard normal distribution. The standard normal distribution is a normal distribution with μ = 0 and σ = σ2 = 1, also denoted N(0,1).

The z score finds the point z on the standard normal curve that corresponds to any point x on a nonstandard normal curve. To convert an x value from [Page 1110]the original scale to a z score, center the distribution by subtracting the mean and then rescale by dividing by the standard deviation. The formula for this conversion is

In a population, the formula becomes z = (x − μ)/σ. In a sample, the formula becomes . This standardization ensures that the resulting distribution has a mean of 0 and standard deviation of 1. See Figure 1 for an illustration of how the x scales and z scores compare for a normal distribution with mean μ = 5 and standard deviation σ = 2.

If the assumption of normality for the data is not grossly violated, then the z score will allow the researcher to compare the data with known probabilities and draw conclusions. If x did not have at least an approximately normal sampling distribution, then further use of the z score may result in erroneous conclusions. The central limit theorem does not apply since the researcher is interested in individual x values, not the mean of the x values.

Using the relationship between the z and x scales, a researcher can use a standard normal table or z table to find the area under any part of any nonstandard normal curve. To calculate the area or probability of an x value occurring between two numbers a and b in the x scale, use

In Figure 1 (μ = 5, σ = 2), to find the probability of an x value between 3 and 10, use

Figure 1 Comparison of x Scales and z Scores

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