Statistics Resources

This guide contains all of the ASC's statistics resources. If you do not see a topic, suggest it through the suggestion box on the Statistics home page.

Z-Scores

A z-score tells us the number of standard deviations a value is from the mean of a given distribution.

• negative z-scores indicate the value lies below the mean
• positive z-scores indicate the value lies above the mean

If we know the mean and standard deviation for a distribution, we can find the location of any value on that distribution by converting it to a z-score using the following formula:

Essentially what we're doing here is finding the distance between the score (x) and the mean (µ) and then dividing that distance by the value of one standard deviation to see how many standard deviations it would take to travel that distance.

Example:

IQs are normally distributed with a mean of 110 and a standard deviation of 15. What is the z-score associated with an IQ of 95?

1. Pull out important values and assign appropriate letters from the formula.
• µ = 110
• σ = 15
• x = 95 (because this is the score that we are interested in)
2. Plug the values into their places in the formula.
• z = (95 - 110) / 15
3. Follow order of operations to simplify (solve).
• subtract first: z = -15/15
• then divide: z = -1

Since a z-score tells us the number of standard deviations a value is from the mean, we can interpret this to mean that an IQ of 95 is one standard deviation from the mean. We can be even more specific by taking into account the sign. Since it's negative, that means the value lies below the mean. So, an IQ of 95 lies one standard deviation below the mean.