Question

If you wanted to calculate a z score for a hwy of 27, how would it be affected by the standard deviation for hwy?

A) If the standard deviations large, the z score should also be very large and positive
B) If the standard deviations is large, the absolute z score should also be large but we won't be able to tell if it is positive or negative
C) If the standard deviation is large, the z score should be small and positive
D) Standard deviation and z score are unrelated because they measure different things about the distribution

Answer 1

To understand how the standard deviation affects the calculation of a z score, let's first consider the formula for calculating a z score:

The z score is calculated as (x - μ) / σ

In this formula:
- x denotes the value for which we are calculating a z score.
- μ refers to the mean (average) value of the population or sample.
- σ represents the standard deviation, which measures the amount of variation or dispersion in the set of values.

Here, if the standard deviation (σ) is large, this leads to the denominator of the formula becoming large.

Now, in regular arithmetic division, say with the formula A/B, if the denominator B becomes large, the result of A/B would decrease.

Applying this concept to the calculation of a z score, we can conclude that a larger standard deviation (σ) would result in a smaller z score.

Therefore, the correct answer to the question "if the standard deviation for hwy is large, how would it affect the z score for a hwy of 27" is:

C) If the standard deviation is large, the z score should be small and positive.

This aligns with the fundamental understanding and calculations in statistics.