Before we show you how you can apply the z score in real life, we believe that it is important that you understand what a z score is.
When you have a variable that is normally distributed, the mean will be the center of the distribution and the standard deviation will show the degree of variability that exists. So, when you already know that the value of a variable is normally distributed, it is easy to calculate the z score. After all, you just need to use the following formula:
Z Score = ( Value – Mean) / Standard Deviation
After determining the z score, you may be interested in finding out the probability of a specific value to occur. So, you will need to use a z score table for that. Notice that not all z score tables are the same.
A Real-Life Example – Newborns
Now that you already know what a z score is and how you can calculate it, it is time to take a look at a real-life example with the weight of newborn babies.
Let’s say that the mean weight of newborns is 7.5 pounds and that the standard deviation is 1.25 pounds.
Let’s assume that you want to discover the probability for a newborn to weight less than 6 pounds.
Here’s how to proceed:
The first thing that you need to do is to calculate the z score. So, you will need to use the formula we mentioned above:
Z Score = (6 – 7.5) / 1.25
Z Score = -1.20
Now that we already have the z score value, we want to know the probability of newborns to weight less than 6 pounds. So, you need to grab your z score table and look at the rows first. On the left side, you will need to look for the right of the decimal point of -1.2. Then, you’ll need to look at the digit that is two spots to the right of the decimal point, which is 0. And this tells you the column that you need to look at.
The probability will be the one that is the intersection of the row and column that you looked at. In this specific example, the probability is 0.1151. So, we can then say that the probability of a baby weighing less than 6 pounds is 0.1151.
However, you can try to answer different questions using the same data. Let’s say that you now what to know the probability of a newborn to weigh more than 10 pounds.
In this case, the z score would be:
Z Score = = (10 – 7.5) / (1.25)
Z Score = 2.00
Now that you have the z score, it is time to determine the probability to answer your question.
So, you need to grab the z score tables again and look at the rows first and then at the columns. In case you didn’t notice, in these z score tables, you have z scores that can either be positive or negative. So, make sure that you are using the right value or you will get a strange probability.
In this second case, you will get a probability of 0.9772. This value shows you that the area underneath the curve to the left of +2 standard deviations is where your z score is located. So, in order to get the right side of that line, you will need to subtract that value from 1:
1 – 0.9772 = 0.0228
So, we can then conclude that the probability for a newborn to weigh more than 10 pounds is equal to 0.0228.