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What Are And How To Calculate Z Scores - Z Score Statistics Formula Explained
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In this video we discuss what are z scores and how to calculate z scores using a formula, which uses the mean and standard deviation.
Transcript/notes
Z scores
A z score or standard score for a value is the number of standard deviations that value falls above or below the mean.
Z scores give us a way to compare 2 different variables from different data sets to one another based on where they are located in relation to their individual means.
The formula for the z score is, z = a value minus the mean, divided by the standard deviation. A z score with a positive value is above the mean, a negative value is below the mean, and a z score of zero means it is equal to the mean. On the screen is the formula for both samples and populations.
For an example, let’s say that Karen here is competing for a promotion at work with 100 other people, and recently took a practice exam.
The practice and real exam are based on 3 categories, leadership, quickly processing information and data analysis.
On the practice exam she scored a 22 in leadership, a 12 in info processing, and a 34 in data analysis.
The group results for the practice exam were given to all applicants, for leadership the mean was 18, and standard deviation was 4.8, for info processing, the mean was 9 with a standard deviation of 1.8, and for data analysis, the mean was 32 with a standard deviation of 1.8.
The good thing is is that Karen was above the mean in all 3 categories, but she has 3 weeks to study for the real exam and wants to put most of her study time into her weakest categories.
To find this out, she can calculate her z scores for each of the categories to give her an idea of what she is weakest at, compared to other applicants.
Using the z score formula for leadership, she scored a 22, subtract the mean of 18, and divide that result by 4.8, the standard deviation, which gives us 0.83. For info processing she scored a 12, minus the mean of 9, and divide by a standard deviation of 1.2, and we get 2.5. And for data analysis she got a 34, minus a mean of 32, and divide by a standard deviation of 1.8, and we get 1.11.
Since her z score of 2.5 in info processing is her highest of the 3, her relative position in this category is higher than the other categories, and her z score of 0.83 is lowest in leadership, her relative position in this category is her lowest.
So, based on this data, she should spend most of her study time on leadership, then on data analysis and lastly on info processing techniques.
To make this a little clearer, we can draw a line and mark x bar, the mean, in the middle.
And we can put in 6 other marks, being x bar plus and minus 1, 2, and 3s, where s represents standard deviation. Now we can estimate and draw a line at x bar plus 0.83, which is her z score in leadership, draw another line at x bar plus 1.11, her z score in data analysis, and another line at 2.5, her z score in info processing.
If you think of the right end of the line as being the top ranked applicants hopefully this gives you a visual perspective of her z scores.
Timestamps
0:00 Z Scores Explained/Overview
0:16 Z Score Formula Explained
0:33 Z Score Example Problem
2:01 Z Score Interpretation
Transcript/notes
Z scores
A z score or standard score for a value is the number of standard deviations that value falls above or below the mean.
Z scores give us a way to compare 2 different variables from different data sets to one another based on where they are located in relation to their individual means.
The formula for the z score is, z = a value minus the mean, divided by the standard deviation. A z score with a positive value is above the mean, a negative value is below the mean, and a z score of zero means it is equal to the mean. On the screen is the formula for both samples and populations.
For an example, let’s say that Karen here is competing for a promotion at work with 100 other people, and recently took a practice exam.
The practice and real exam are based on 3 categories, leadership, quickly processing information and data analysis.
On the practice exam she scored a 22 in leadership, a 12 in info processing, and a 34 in data analysis.
The group results for the practice exam were given to all applicants, for leadership the mean was 18, and standard deviation was 4.8, for info processing, the mean was 9 with a standard deviation of 1.8, and for data analysis, the mean was 32 with a standard deviation of 1.8.
The good thing is is that Karen was above the mean in all 3 categories, but she has 3 weeks to study for the real exam and wants to put most of her study time into her weakest categories.
To find this out, she can calculate her z scores for each of the categories to give her an idea of what she is weakest at, compared to other applicants.
Using the z score formula for leadership, she scored a 22, subtract the mean of 18, and divide that result by 4.8, the standard deviation, which gives us 0.83. For info processing she scored a 12, minus the mean of 9, and divide by a standard deviation of 1.2, and we get 2.5. And for data analysis she got a 34, minus a mean of 32, and divide by a standard deviation of 1.8, and we get 1.11.
Since her z score of 2.5 in info processing is her highest of the 3, her relative position in this category is higher than the other categories, and her z score of 0.83 is lowest in leadership, her relative position in this category is her lowest.
So, based on this data, she should spend most of her study time on leadership, then on data analysis and lastly on info processing techniques.
To make this a little clearer, we can draw a line and mark x bar, the mean, in the middle.
And we can put in 6 other marks, being x bar plus and minus 1, 2, and 3s, where s represents standard deviation. Now we can estimate and draw a line at x bar plus 0.83, which is her z score in leadership, draw another line at x bar plus 1.11, her z score in data analysis, and another line at 2.5, her z score in info processing.
If you think of the right end of the line as being the top ranked applicants hopefully this gives you a visual perspective of her z scores.
Timestamps
0:00 Z Scores Explained/Overview
0:16 Z Score Formula Explained
0:33 Z Score Example Problem
2:01 Z Score Interpretation
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