Lets say, for example, that your data set has a mean of 50 and a standard deviation of 10, and you want to calculate the Z. Where X is the data point youre interested in, is the mean of your data set, and is the standard deviation of your data set. These are the left-tailed p values for the z-score. To calculate the Z score for a single data point, you use the formula: Z (X - ) /. A z table is a table that allows you to find the probability of a value being to the left of a z-score in a normal distribution.Įach entry in the z table represents the area under the normal distribution bell curve to the left of z. A z of greater than 3 or less than -3 generally indicates that the raw score is an outlier. A z value of 0 means that the raw score is equal to the mean.Ī very large z-score also tells us that the raw score is unusual, while a smaller z-score indicates that it might fall closer to the middle of the distribution. For example, 68.27 of values would fall between -1 and 1 standard deviations of a Z distribution. This is very similar to the one-sample t-test formula.Īs noted above, the z-score is equal to the distance of a value from the mean in standard deviations, but what does that actually tell us? There are a few things we can take away from the z-score after we calculate it.įirst, a positive z-value means that the raw score is greater than the mean, while a negative z-value means that the raw score falls below the mean. The z-score for the sample is equal to the sample mean x̄ minus the population mean μ, divided by the standard error of the mean, which is equal to the population standard deviation σ divided by the square root of the number of observations n in the sample.
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