A distribution has a standard deviation of o=10. Find the z-score for each of the following location

excluderho

excluderho

Answered question

2022-06-12

A distribution has a standard deviation of o=10. Find the z-score for each of the following locations in the distribution.
a) Above the mean by 5 points b) Above the mean by 2 points c) Below the mean by 20 points d) Below the mean by 15 points
Please don't give me the answer, but explain how I would work this out.

Answer & Explanation

Odin Jacobson

Odin Jacobson

Beginner2022-06-13Added 17 answers

If a given distribution X is normal with some mean μ and standard deviation σ = 10, then the standardized value is given by
Z = X μ σ .
This standardized value is a z-score, and represents the number of standard deviations above/below the mean the original X was observed to be.

For example, suppose that the height X of a particular species of plant is normally distributed with mean μ = 87 centimeters and standard deviation 10 centimeters. Then if I randomly select one such plant from the population and measure it, and find that it is X = 97 centimeters tall, then this corresponds to a z-score of
Z = 97 87 10 = 1.
So in this particular instance, this plant's height is 1 standard deviation above the mean. That makes sense: 87+10=97.

Now, if I don't know what the population mean is, but I know that the standard deviation is still σ = 10, then I can say that if a measurement is 5 units above the mean, that is equivalent to a z-score of...??? (you fill in the blank), because 5 units above the mean is equivalent to ??? standard deviations above the mean.

Similarly, if I say that a measurement is below the mean by 20 points (or units), then that must be how many standard deviations below the mean? Remember that if we are below the mean, then we should express that as a negative z-score.

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