Question

Using an example, describe how a z-score can be used in making comparisons between two or more distributions (Hint: make sure you include the means and standard deviations) in your example.

Answer 1

A z-score is a measure of position that indicates the number of standard deviations a data value lies from the mean. It is the horizontal scale of a standard normal distribution. The z-score is positive if the value lies above the mean, and negative if it lies below the mean.

The standard score (more commonly referred to as a z-score) is a very useful statistic because it (a) allows us to calculate the probability of a score occurring within our normal distribution and (b) enables us to compare two scores that are from different normal distributions. The standard score does this by converting (in other words, standardizing) scores in a normal distribution to z-scores in what becomes a standard normal distribution. To explain what this means in simple terms, let's use an example (if needed, see our statistical guide, Normal Distribution Calculations, for background information on normal distribution calculations).

Example :-

1.) A group of 100 people took some IQ test. My score was 5. So is that good or bad? At this point, there's no way of telling because we don't know what people typically score on this test. However, if my score of 5 corresponds to a z-score of 0.91, you'll know it was pretty good: it's roughly a standard deviation higher than the average (which is always zero for z-scores).
What we see here is that standardizing scores facilitates the interpretation of a single test score.

2.) In a university, a student claims that the average failing rate is eual of both departments of Mathematics and Statistics.

He takes a sample of 30students from both departments and find out that the average failing rate in department of maths is 0.15 with standard deviation being 0.07 whereas in department of stats is 0.22 with standard dev being 0.05,

now using z score he can easily check his claim.

Like he will first find the z score by using the following formula where according to his claim mean1-mean2 will be 0 and hence he would be able to find the Z score.

And then, finally he'll compare this with pvalue or critical value and thwn verify its claim that whether it was correct or not.

I hope i have answered your question very well, do like it and cooment, if you have any query.

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