Pop_sd <- sd(height)*sqrt((length(height)-1)/(length(height))) It should be noted that the sd function in R uses the sample standard deviation and not the population standard deviation, though with 25,000 samples the different is rather small. The first step to find the z-score is to find the population mean and standard deviation. The z-scores are also listed on this normal distribution to show how the actual measurements of height correspond to the z-scores, since the z-scores are simple arithmetic transformations of the actual measurements. The data set doesn’t deviate much from the theoretical distribution. The red line represents the theoretical normal distribution, while the blue area chart reflects a kernel density estimation of the data set obtained from UCLA. Unfortunately, the site doesn’t give much detail about the data, but it is an excellent example of normal distribution as you can see in the graph below. They have height measurements from children from Hong Kong. To demonstrate how this is calculated and used, I found a height and weight data set on UCLA’s site.

So a z-score of 2.0 means the measurement is 2 standard deviations away from the mean. Z-scores are a stand-in for the actual measurement, and they represent the distance of a value from the mean measured in standard deviations. In order to get the distributions standardized, the measurements can be changed into z-scores.

Obviously, the means and standard deviations of these measurements should all be completely different.

Normal distributions are convenient because they can be scaled to any mean or standard deviation meaning you can use the exact same distribution for weight, height, blood pressure, white-noise errors, etc. I’ve included the full R code and the data set can be found on UCLA’s Stats Wiki

Standard normal table with Z score