We can see that the mean is 31 and the standard deviation is 1.5. This means that the time in the 200m race is two standard deviations less than the mean time.
This results in the mean of 0 and a standard deviation of 1. The z-score can be used to normalise a set of values in a normal distribution by calculating the z-score of every value in the data set. The Probability Density Function of the Z-Distribution Calculate the probability of a particular score occurring.Compare scores that have different means and standard deviations.It is a value that describes how many standard deviations a result is from its mean.Ĭalculating the z-score is useful because it allows us to: The z-score is calculated using the following formula:Ī z-score does not have any units. This is because an outlier can be defined as a value that is more than 3 standard deviations above or below the mean.Ĭalculating the z-score is used to compare scores from different sets of data that have different means and standard deviations. Z-scores greater than +3 or less than -3 are considered outliers. Z-scores close to zero indicate that the result is close to the mean, whereas larger or very negative z-scores indicate that the result is further from the mean. Negative z-scores indicate raw scores that are below the mean and positive z-scores indicate raw scores which are above the mean. In other words, the raw score is zero standard deviations away from the mean and is therefore equal to the mean. This can be seen in the diagram below in which the position of the z-scores are labelled on the horizontal axis.Ī z-score of zero means that the raw score is identical to the mean. For example, a z-score of 2 means two standard deviations above the mean and a z-score of -1 means one standard deviation below the mean. Negative z-scores indicate a position below the mean.
The z-score describes how many standard deviations a raw score is above the mean.
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