A measure used to describe data in statistics includes the z-score of a distribution. What exactly are z-scores and how are they calculated? Read to find out!
In statistics, a concept you need to be familiar with are z-scores. These are also called standard scores, and are used for normal distributions. They are a measure of how far a data point is away from the mean of the entire data set, calculated in amounts of standard deviations.
How to Calculate the Z-Score
The formula to calculate a z-score is:
z = (x - μ) / σ
where μ (mew) = the mean of the data set, and σ (sigma) = the standard deviation.
As a review: To find the mean of the data set, you'd find the sum of all values, then divide by the total number of terms. Thus Standard deviation, is found by finding the average of the differences between each term and the mean, which has a separate formula to follow.
Interpreting the Z-Score
Z-scores range from -3 to 3, because a normal distribution consists of three standard deviations to the left and to the right of the mean, as shown in the standard distribution curve below:
The closer to 0 the z-score is, the closer the data point is to the mean.
So if the z-score is positive, the data point is higher than the value of the mean.
However, if the z-score is negative, the data point is lower than the value of the mean.
The majority of the data points lie within one standard deviation of the mean, as you can see in the shaded area of the graph. In statistics, there is a concept called empirical rule, also known as the 68-95-99.7 rule. Data points that have a z-score between -1 to 1 make up 68% of all data; 95% of the data have z-scores between -2 and 2, and 99.7% of data have z-scores within 3 standard deviations of the mean (-3 to 3).
Usually, the z-score would be used to find the percentage of values that fall below or above a specific value. For instance, in a store where $10 is the mean cost for a shirt, and the costs of all shirts have a standard deviation of $2, a shirt that costs $8 would have a z-score of:
z = (8-10) / 2 = -1
Meaning that this T-shirt costs lower than average, specifically 1 standard deviation lower than average.
Let's say the problem additionally asks what percentage of T-shirts costs less than $8. Using the z-score of 8 in this data set as well as the 68-95-99.7 rule, you can solve for this!
Since 8 is one standard deviation below the mean, the area of the curve that is below z = -1 should be solved for in this problem. To do this, you would find 0.5(68) = 34, since half of the 68% data will be between z-scores -1 and 0. You would subtract this from all the data values below z-score 0. Therefore, you would get:
50-34 = 16% of the shirts cost less than $8
Thank you for reading!
References:
“Empirical Rule.” Investopedia, 29 Aug. 2021, www.investopedia.com/terms/e/empirical-rule.asp.
“What Is the Standard Deviation?” Investopedia, 8 Dec. 2021, www.investopedia.com/terms/s/standarddeviation.asp.
“Z-Score: Definition, Formula and Calculation.” Statistics How To, 20 Nov. 2021, www.statisticshowto.com/probability-and-statistics/z-score.
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