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# Comparing with z-scores

Use standardized scores—also called z-scores— to compare data points from different distributions.

## Want to join the conversation?

• Is it very scientific method? Because there is no way of knowing percentile.
Without percentile there is no way of knowing how he faired relatively in both exam.
• If you know the z-score and it's a normal distribution, you can figure out the percentile using a z-table.
• Why did he do better on the LSAT if he's 2.1 SD above the mean? Doesn't this mean he's 2.1 SD far from the mean? Comparable with 1.9, shouldn't he do better on the MCAT since it's closer to the mean?
• The mean is the average result. Would you rather score average on a test, or above average?
• why can he estimate which test score is better based on only the z-score?
what exactly does the z-score show?
• z-score is like percentile. It can be used to compare different data sets with different means and standard deviations. It is a universal comparer for normal distribution in statistics. Z score shows how far away a single data point is from the mean relatively. Lower z-score means closer to the meanwhile higher means more far away. Positive means to the right of the mean or greater while negative means lower or smaller than the mean.
• What if I am looking for the standard deviation and I have the Z Score and the Mean?
(1 vote)
• You would just plug in the Z-score and mean into our formula, and then solve for the standard deviation using algebra. You also will need the number that the Z-score comes from (in our case, that is Juwan's test scores). Let's use the LSAT example from the video.

(172 - 151) / X = 2.1
21 / X = 2.1
21 = 2.1 * X
10 = X

We get that the standard deviation is 10. Hope this helps! :)
• Hi! I have a question. Why Khan writes 11.9 / 6.4 is less than 2?
(1 vote)
• Why do we assume it is a normal distribution/relative close to a normal distribution? Would comparing z-scores not apply if it was not a normal distribution?
• Comparing z-scores assumes a normal distribution or a distribution that is approximately normal because the concept of z-scores is based on the properties of the standard normal distribution. While z-scores can still provide some relative comparison in non-normal distributions, their interpretation might be less straightforward due to potential skewness or other deviations from normality.
(1 vote)
• Why do we divide the difference between the Juwan's score on each test and the mean, by the standard deviation of that test?

To be more clear, how come we don't do something like this:

Let's say the LSAT's score range is 5x bigger than the MCAT's score range. This means that the LSAT mean should be 5x bigger than the MCAT mean. If Juwan's scores on the two tests are comparable, his score on the LSAT should be 5x his score on the MCAT.

Then, we can compare these two values:
1. ((Juwan's LSAT)-(LSAT mean)) ÷ 5
2. ((Juwan's MCAT)-(MCAT mean))

And whichever one is greater is the test he did better on.

I'm essentially wondering, what's the benefit of dividing the mean-to-data-point difference by the standard deviation? What's the benefit of comparing z-scores rather than just the difference (corrected for scale)? Also, if there's anything I might be misunderstanding, please let me know.

Thank you!
(1 vote)
• So the z-score has application involving the normal distribution and hypothesis testing.
• why dont the standard deviations be subtracted by the mean to be the final answer
(1 vote)
• because the standard deviation is how far the number is from the group the average and if it subtracted by the mean the sd wont be able to be interpreted