|Test your knowledge|
In the last blog about Z-score, we talked about the odd of having a sample data point x from a given Normal distribution.
Today, we are not taking only 1 data point, but a set of n data points. We call the mean value of this set , the question is: How likely do we have a sample set with size n whose mean is at least as extreme as ?
Let’s return to the example of IQ-tests. Humans’ IQ is normally distributed with mean and std (which means variance ). This time, says you are not taking the test alone, but with all your classmates. Your class has 20 students, and their average result on IQ-test is 106. The question is: how is your class’s IQ compared to other groups of 20 people?
Remember that in the original version of this example, you are the only one to take the IQ-test, and we compare your result to the full population on Earth. In this modified version, we change the size, from only you (size = 1) to all your class (size = 20).
In fact, we can map the current problem to the same state as the original problem, because the average value of a group of size n (n > 1) also follows Normal distribution, called Normal distribution of the mean. What we need is the mean and std of this distribution, let’s call them and , respectively.
Before computing , let’s revise some properties of Variance:
Ok, let’s continue:
(It worths mentioning that has its own name, called the standard error of the sample.)
Well done! Now we know that:
The distribution of sample mean is a normal distribution with and .
To get the z-score of your class’s IQ, we should use this distribution. Do you remember the formula of Z-score? Apply it here:
or we can say in general:
For our case, and . Thus,
Z-score[class IQ] =
Look it up on the z-table gets us a percentile of 96.3%, which means your class is in top 3.7% of the world (compared to any other groups of 20 people), that’s impressive!
|Test your understanding|
In this blog, we introduced the distribution of sample mean, which is also a normal distribution with the same mean () but different standard deviation (.