Describe what a z-score is in terms of the sample mean and sample standard deviation.?

It is a normalised unit for the x-axis such that any measure being sampled can be compared to a common reference. Since x, mean and sd are in the same units, then

(x - mean)/sd is dimensionless, ie a pure number and is in fact the number of sd's that x is away from the mean.

In simple terms, the z-score represents how many standard deviations you are away from the mean. That's true of a population or a sample.

For example, let's say the average IQ is 100 and the standard deviation is 15.

If you have an IQ of 130, then you are *2* standard deviations above the mean. That's a z-score of 2.

On the other hand, if you have an IQ of 85, then you are *1* standard deviation below the mean. That's a z-score of -1.

The formula is:

z = (x -  μ) / σ

x : raw score

μ : mean

σ : standard deviation

Using the first example above.

z = (130 - 100) / 15

z = 30 / 15

z = 2

Using the second example above:

z = (85 - 100) / 15

z = -15 / 15

z = -1

The Z-score of a value is the difference between that value and the mean, divided by the standard deviation.

If the mean is 5 and the standard deviation is 1.5, then the Z-score of 3 is -1.33….