Statistics question: Difference between Z and z

Hello,

I have a straight forward question, that I hope you can help with.

What is the difference between using the following formulas and when do you use one over the other:

Z = (X - μ) / σ, and

z = (X-bar - μ) / (σ / √n)

In the CFA material, question 3B for reading 10, they use the formula z = (X-bar - μ) / (σ / √n) to calculate the probability of a -2.0 percent or lower average return, given a population mean and standard deviation.

Could someone explain why not use the other formula for this?

I believe that Z is simply to normalize in order to obtain a standard normal distribution. Once you normalize, you will have a N–(0,1) and be able to use probabilities based on a standard normal distribution.

For the z, I think this is the formula for the test statitics. So basically, they are used for two different purposes. One is to obtain a normalized variable in order to be able to use a standard normal distribution. The other is used for hypothesis testing.

Not entirely correct. This is associated with another post from yesterday: http://www.analystforum.com/forums/cfa-forums/cfa-level-i-forum/91317258

Use Z = (X - μ) / σ when you know the population mean and variance.

In practice, you never know the population mean and variance. Hence, we resort to sampling. When you sample data, there are estimation errors. The standard error of the standard deviation (think of this as standard deviation of standard deviation) is σ / √n if you know the population variance; if you don’t know the population variance, you resort to using the sample variance, s / √n.

Use z = (X-bar - μ) / (σ / √n) when population variance is known. Again, the idea is, you’re trying to obtain a point estimate of the population - you want to be as accurate and precise as possible. You can’t get any better than the estimate itself (i.e. the population mean or variance).

Use z = (X-bar - μ) / (s / √n) when sample variance is known.

The CFAI curriculum does a poor job of explaining this, but just remember to use the population mean/variance if they’re already known. They could have an example that supplies all four values - population mean, population variance, sample mean, and sample variance. Here, you already know your point estimates, so why would you use the sampled data? Think of this in terms of S&P 500 returns - if you already know the mean and standard deviation of the entire index, why would you use sampling! So when you know all four values, use Z = (X - μ) / σ.

Hope this helps.

Also, don’t confuse this topic with hypothesis testing. The above explanation mostly focuses on confidence intervals and looking up probabilities using the z-table.

Thanks for the rectification Aether

Thanks, Aether, that makes it clearer. I agree with you that this is not very well explained in the book but your answer helped.