In this video I explain how to use a z-table or a standard normal table in order to calculate the percentage of scores below, above, or between z-scores for a normal distribution. Finding the areas under the curve associated with different z-scores allows for the calculations of probability in a normal distribution is a fundamental underlying concept in hypothesis testing. Z-table from www.z-table.com Normal curves created using www.desmos.com
Video Transcript
Hi I’m Michael Corayer and this is Psych Exam Review. In the previous video we saw how we can use just one version of a normal curve, the standard normal curve, in order to estimate probabilities and we saw how we can convert between our particular measurements, or our raw scores, and standard deviation units or Z scores, and this is what allows us to estimate probabilities more easily. And we mostly did this using the Empirical Rule or the 68, 95, 99.7 Rule, which tells us the percentage of scores that fall within one, two, and three standard deviations of the mean of the distribution.
But of course not all z-scores are going to fall neatly at one two or three standard deviations. We could have a z-score of 1.72 or -2.16 and so in order to know the probabilities associated with these we won’t be able to get them just by memorizing the empirical rule. So instead what we’re going to have to do is look to a table that gives us the precise probabilities that are associated with particular z-scores and this is what’s known as a z table or a standard normal table.
The first step for using a z table is to make sure that you know what the Z table is presenting. They all contain essentially the same information but there are variations in how it can be presented. So most tables will give the percentage of scores falling below a particular z-score although some tables will give the percentage falling above the z-score or even between the mean and the particular particular z-score. So it’s important to check which type of table you’re looking at. And for these examples I’ll be using a z table that tells us the area under the curve below each z-score, which means it’s telling us the percentage of scores that have been accounted for up to that point in the distribution.
So here’s an example of a z table from z-table.com and if you have a statistics textbook you’ll probably find a similar table somewhere in the appendices. And you’ll see the values should be the same, because any normal curves have the same probabilities associated with the same Z scores. Now this table here is in a condensed format that uses rows and columns for looking up each z-score to two decimal places, but you may have a table that lists each z-score to two digits as its own row, but obviously this would take up more space.
To use a table like this, first we go to the row for the first two digits of our z-score and then we find the column for the last digit of the z-score. So to find the probability associated with a z-score of 1.23 I would first find the row starting with 1.2 and then I would find the column for .03 and the box where this row and column meet contains the percentage of scores that fall at or below that z-score. So I can see if I have a z-score of positive 1.23 then .8907 or 89.07% of scores would be at that point or below, meaning it’s telling us the percentage of scores that have been accounted for up to that point in the distribution.
If we wanted to know the percentage of scores falling at or above that z-score we could just subtract this value of .8907 in the table from 1.00, since the total area under the curve always equals 1. So if we’ve accounted for .8907 or 89.07% of the scores falling below this particular z value, that means that .0913 or 10.93% of scores would have to fall at or above a z-score of 1.23.
You might notice that this table only shows positive Z values, so if we want to look up a negative z-score we can look to a different table. And this is going to show us the percentage of scores falling below a particular negative z-score. So if I wanted to know the percentage of scores falling below a z-score of -2.72 I would look this up in the table in exactly the same way and I’d find that this value is 0.0033 meaning only 33% of scores would fall below this point. And again, we could find the value above this point by simply subtracting .0033 from 1.00 and that would tell us that 99.67% of scores would fall above a z-score of -2.72.
Now it’s not entirely necessary to have both a positive and negative Z table; it makes it easier to look things up quickly but it’s actually redundant. Each of these is actually showing us the same exact information. So if you only had a positive Z table you could still find the percentages associated with negative z-scores because a normal distribution is perfectly symmetrical. So if I looked up the z-score of 0.4 and I see .6554 this tells me that 65.5 4% of scores fall below a z-score of positive 0.4. If I wanted to know the percentage falling below a z-score of negative 0.4 I could subtract .5 from my probability that would remove the 50% of scores below the mean telling me that 0.1554 or 15.54% of scores are between the population mean and a z-score of positive 0.4. Since the curve is symmetrical this means that 15.54% of scores will also be between the mean and a z-score of negative 0.4.
So if I want to find the percentage of scores below a z-score of 0.4 I could subtract 0.1554 from 0.5, which is the percentage at or below the mean, 50%, and I get a value of 0.3446 and if we double check our negative Z table we can see that this is the correct value for the percentage of scores falling below a z-score of 0.4. Now you probably won’t ever have to do this since you can easily find both positive and negative Z tables or you can even use online calculators that will simply tell you the percentage of scores associated with any z-score without having to look across rows and columns, but doing this helps you to think about what Z tables are showing conceptually and it can serve as a reminder that the curve is perfectly symmetrical. And hopefully this gives you a more firm grasp on the idea of z-scores.
In addition to being able to find the percentage of scores that fall below or or above a particular z-score, we can also use a z table in order to find the percentage of scores that falls between two z-scores. So in order to do this all we have to do is find the percentage of scores that fall below the higher z value and then subtract the percentage of scores that fall below the lower z value and then this will give us the percentage of scores that falls between those two points.
For example, let’s say that we wanted to know how many scores fall between the z-scores of -1.32 and +2.14. So first we’d find the percentage of scores that fall below +2.14 and if we look in the table this would be 0.9838. Next we’d find the scores falling below a z-score of -1.32 and this is 0.0934 and if we subtract 0.0934 from 0.9838 we would get 0.8904 and this would mean that 89.04% of scores in a normal distribution will fall between a z-score of -1.32 and +2.14. And as a reminder this means that if we randomly select a score from the population it would have an 89.04% chance of falling somewhere between these two z-scores.
Of course, if we want to do this for raw scores, assuming we believe we have a normally distributed population, then we have to convert them to z-scores. So let’s say that we’re looking at a distribution of IQ scores with a population mean of 100 and a standard deviation of 15. I might ask, what’s the probability that a randomly selected person from the population will have an IQ between 90 and 110? To figure this out first we need to convert these raw scores of 90 and 110 into standard deviation units or Z scores.
So the formula for a z-score is the raw score minus the population mean divided by the standard deviation. So for a raw score of 90 this would be 90 – 100 or – 10, divided by 15 equals -0.667 and for 110 it would be 110 minus 100 or 10, divided by 15, which would equal +0.667. Now we can look up these z-scores in the table and and see how many scores fall below each. In this case we’ll have to round our value of 0.667 to 0.67, since this table only goes to two digits after the decimal place. And if I look in the tables I’ll find that for a z-score of +0.67 we get a value of 0.7486 and for a z-score of -0.67 we’ll have a value of 0.2514.
So if I subtract 0.2514 from 0.7486 I’ll get a result of 0.4972 meaning that approximately 49.72% of scores would be expected to fall between the raw IQ scores of 90 and 110. Or we could say that there’s about a 49.72% chance that a randomly selected person from the population would fall into this range of 90 to 110 for their IQ.
Okay, so I hope this helped you to understand how to use a z table and what it’s used for. Let me know in the comments if this was helpful, ask other questions that you still have, be sure to like and subscribe, and check out the hundreds of other psychology and statistics tutorials that I have on the channel. Thanks for watching!
