# How to Calculate Z-Scores in Statistics

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A Z score is a statistical calculation that you can perform on a single data point that belongs to larger data set. The Z score tells you how far -- in standard deviations -- that data point is from the mean, or average, of the data set.

The score can be helpful in predicting the probability that any new data will be at, above, or below the point for which you calculated the score. Z scores have applications in business, the sciences and in just about any other discipline that involves data analysis.

## The Data Set

• Imagine that Mr. Smith's gym class has 20 students whose running abilities are a representative random sample of the entire school. Today, in Mr. Smith's class, the students ran a 200-meter dash. The students' finishing times form a normal distribution -- or bell curve -- with an average score of 75 seconds a standard deviation of 10 seconds. The data set below shows their finishing times:

{80, 76, 83, 77, 81, 97, 64, 83, 69, 63, 60, 53, 76, 98, 67, 75, 89, 74, 59, 91}

## The Question

• Now, imagine that Mr. Smith wants to know the probability that a random student selected from the rest of the school population could finish the dash in 90 seconds or less. Calculating the Z score on that new data point -- 90 seconds -- is the first step in figuring out the probability.

## Calculating the Z Score

• The formula for calculating a Z score involves the data point in question, the mean of the data set, and the standard deviation of the data set. In the equation for the Z score shown below, x stands for the data point, m stands for the mean of the data set, and sd stands for the standard deviation of the data set.

Z = (x - m) / sd

The mean of the students' running times is 75 seconds, the standard deviation is 10 seconds, and the data point in question is 90 seconds. The Z score calculation for this data point looks like this:

Z = (90 seconds - 75 seconds) / 10 seconds = 1.50

## Finding the Probability

• The Z score for a finishing time of 90 seconds based on Mr. Smith's data set is 1.50. This means that a time of 90 seconds is 1.50 standard deviations above the mean -- a negative Z score indicates that the new data point falls below the mean.

To figure out the probability of a new data point falling at or below 90 seconds, you must consult a Z distribution table, such as the chart from the Columbia University Business School.

Locating the score of 1.5 in the first column and matching it to the zero in the first row, because zero is the next digit after the 5 in the Z score, reveals a probability of 0.9332.

You can interpret this answer to mean that a student chosen at random from the rest of the school has a 93.32 percent chance of finishing the 200-meter dash in 90 seconds or less. Mr. Smith's question has been answered!

## One Step Further

• What if Mr. Smith wanted to know the probability that a randomly selected student would take longer than 90 seconds to finish the dash. To figure this out, simply subtract the probability for the 90 second Z score from 1, as shown below.

1 - 0.9332 = 0.0668

There is a 6.68 percent chance that it would take a randomly selected student longer than 90 seconds to finish the dash.

## References

• Photo Credit DAJ/amana images/Getty Images
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