With a normal distribution of a variable, the mean is the center of the distribution, while the standard deviation shows the level of variability that exists. When you know the value of a variable that is normally distributed, you can calculate the z-score by using the formula:
When you are interested in determining the probability of a specific value occurring, you can use a z-table to look up areas for a calculated z-score to determine the area under the curve of any normal distribution. This can tell you the chance of a value occurring. It should be noted that not all z-score tables are the same.
Take a look at the weight of newborn babies. Suppose that the mean weight of newborns is 7.5 pounds and the standard deviation is 1.25 pounds. Say you’re interested in determining the probability that a newborn weighs less than 6 pounds. How do you do that?
The first thing you do is calculate the z-score. To figure out the z-score, take the difference between 6 and 7.5 to arrive at -1.5. When you divide -1.5 by the standard deviation of 1.25, you wind up with a z-score of -1.2.
When you’re looking at a z-table, the first step is to determine which row you want to look at. That’s going to be represented by the spot to the left of the decimal point and the first spot to the right of the decimal point, which is -1.2. The second step is to look at the digit that is two spots to the right of the decimal point, which is a 0. This determines the column you look at.
The point at which both the row and the column intersect one another indicates the probability. The intersection in this scenario is 0.1151. That tells you that the probability associated with the baby weighing less than 6 pounds is 0.1151.
What if you’re interested in looking at the probability that a newborn might weight more than 10 pounds? You could calculate the z-score much the same way you did with the previous calculation:
Probability that a newborn weighs > 10 lbs z-score = (10 - 7.5)/(1.25) = 2.00
The same process is involved in looking at the z-table. Look at 2.0 and which row that entails. Since there are no numbers other than 0 to the right of the decimal point, you look at the very first column, which is the 0.00 column. Note, this table looks different from the one above. The z scores in the left column are negative instead of positive. For a full PDF on z-score tables, please see the attachment below this tutorial.
You wind up with a probability of 0.9772. Remember, you are trying to determine the probability that a newborn weighs more than 10 pounds. The value given to you here is telling you the area underneath the curve to the left of +2 standard deviations, as you see on the graph right here:
You will need to subtract that value from 1 to get the area that’s on the right-hand side of that line. The probability that a newborn would be greater than 10 pounds would be 1 minus 0.9772. The probability is equal to 0.0228.
Take a look at something a little bit different: the average lifespan of car tires. Suppose that the population mean is 60,000 miles, and the standard deviation has been determined to be 8,000 miles. Imagine you’re interested in determining the probability that a tire lasts less than 45,500 miles.
When you look that up on the z-table, remember that first you start with the -1.8. Since the second decimal point is a 0.01, you look at the second column.
The probability is equal to 0.0351.
What if you were interested in determining the probability that a tire would have a lifespan greater than 80,300 miles?
Using the z-table, find the row that represents 2.5 and the column that represents 0.04:
The probability provided to us is 0.9945. That’s not the answer you’re looking for here. That would be the area underneath the normal distribution curve up to 2.54 standard deviations. You’re interested in the area to the right of that, and so you would need to subtract that value from 1. When you do, you get a probability 0.0055. In other words, it is very unlikely in this particular case that a tire would have a lifespan greater than 80,300 miles.
Source: This work is adapted from Sophia author Dan Laub.