Understanding the Interpretation of Z-Scores: Insights and Examples

[bhargavsws]

TL;DR Summary

How do z-scores work, and what information do they provide about a data point in a distribution? Additionally, how can I interpret a z-score in terms of its relationship to the mean and standard deviation?

I've been exploring the concept of z-scores and would like a deeper understanding of their practical application. I used a z-score calculator (https://zscorecalculator.org) for a dataset with a mean of 75 and a standard deviation of 8. One of my data points has a z-score of -2.5. Can you walk me through the interpretation of this specific z-score? How does it relate to the mean and standard deviation, and what insights does it offer about the position of this data point in the distribution? Any detailed explanation or example would be incredibly helpful in solidifying my understanding of z-scores.

 

[FactChecker]

The z-score is the number of standard deviations of the sample point from the mean. So your example point is 2.5 standard deviations (std dev=8) below the mean (mean=75).
Based on the text in the link you gave, I would be skeptical of the calculations and would check them. They are dividing the sample standard deviation by $\sqrt n$. I think the text is wrong but the calculation might still be correct, I didn't check it.
I would be more confident of the calculations in this for the sample standard deviation and then this for the z-score. At least the text in those websites is correct. If you get the same results there as you got from your website, then yours is also doing a correct calculation even though the text is wrong.

CORRECTION: I see what your website was doing. The parts that I thought had wrong text were evaluating the sample mean, not a single sample point. Their text might be correct.

 

[FactChecker]

For your particular sample with a -2.5 z-score, you can use the website I gave to see what the probabilities of that (or more extreme) are, assuming that you are dealing with a normal distribution. For -2.5, it gives the results below. Keep in mind that if you have a lot of sample data, you are likely to see some extreme cases.

 

[bhargavsws]

The z-score is the number of standard deviations of the sample point from the mean. So your example point is 2.5 standard deviations (std dev=8) below the mean (mean=75).
Based on the text in the link you gave, I would be skeptical of the calculations and would check them. They are dividing the sample standard deviation by $\sqrt n$. I think the text is wrong but the calculation might still be correct, I didn't check it.

Hi,

The question is not regarding finding z-score probabilities. I want to know the interpretation of the specific z-score and how it is defined on a graph. Probabilities and interpretation both are different things.

 

[Dale]

I am not sure what you are looking for. As @FactChecker mentioned the interpretation of a z score of -2.5 is that the data point is smaller than the mean by an amount equal to 2.5 times the standard deviation. For a normally distributed variable 98.8% of the values will be closer to the mean than that. There really isn’t any deeper interpretation.

 

[FactChecker]

Hi,

The question is not regarding finding z-score probabilities. I want to know the interpretation of the specific z-score and how it is defined on a graph. Probabilities and interpretation both are different things.

The statistical meaning is fairly simple. The importance of a particular z-score depends on the subject matter. Only a subject matter expert can really interpret the result and it might not depend so much on the z-score as on something particular for that subject.

 

[Agent Smith]

Can someone give us an instance of a skewed distribution?
And for a skewed distribution, is the standard deviation less useful with respect to a z score?

 

[gleem]

Since z is related to the distribution of means, it would seem to me that since the distribution of means, even for a skewed distribution, tends to be normal due to the Central Limit Theorem, the z score is largely unaffected.

EDIT: When the z parameter (n=1) is the comparison of the a value in a distribution to the mean of that distribution it is referred to as t, the parameter of the Student's t-test. This test is only valid for a normal distribution.

 

[gleem]

I've been exploring the concept of z-scores and would like a deeper understanding of their practical application.

As for practical applications, you can use z to test for the significance of the difference between two means similar to the way it was used above. example: You have a product that has a manufactured value of a critical parameter of M with a standard deviation of σ_M. You have another sample of a product stored for some time with a mean N of that parameter. You want to know if they are significantly different.

Another application is the comparison of two proportions like the proportion of a group of persons reacting in a specified way to two different stimuli.

 

[FactChecker]

Can someone give us an instance of a skewed distribution?

It's easy to make up examples of a skewed distribution. Just take a symmetric normal distribution and put an non-symmetric condition on it. For instance, a normal distribution with a mean of 1 and say that any negative value dies not count. Also, the standard , well-known distributions which can not be negative, like the gamma distributions are skewed.

And for a skewed distribution, is the standard deviation less useful with respect to a z score?

Good question. It can be used as a quide, but with caution. It is not as precise as it would be with a normal distribution.

 

[Dale]

Can someone give us an instance of a skewed distribution?

A good example is the unit exponential distribution. It is strictly positive with a mean of 1 and a standard deviation of 1.

And for a skewed distribution, is the standard deviation less useful with respect to a z score?

So if you use the z score it suggests that 32 % of the data will be negative, when 0 % is negative.

 

[statdad]

Can someone give us an instance of a skewed distribution?
And for a skewed distribution, is the standard deviation less useful with respect to a z score?

Not very useful, and the greater the skewness the less useful the standard deviation and the mean are. Once you get away from the textbook ideal of data being normally distributed you should also get away from using the mean and standard deviation, as both are impacted by the skewness.