This tutorial talks about the standard normal distribution. A standard normal distribution is any normal distribution with a mean of 0 and a standard deviation of 1. Now, you can transform any set of normally distributed data into a standard normal distribution by obtaining a z-score for every value. Now, z-scores, in case you don't remember, the formula for calculating a z-score is the value that you're interested in minus the mean divided by this standard deviation.
Once you have a z-score, you can use a z-table to find area and then percent. A z-table is a chart showing the area under the standard model curve and less than or equal to a particular z-score. So, for example, here's a very rough sketch of our standard normal curve. If this is the value that we're interested in and we obtain a z-score for, the z-table can tell us what the area of this shaded part is. So it's the area under the curve and less than or equal to a particular value.
Now for z-tables, they organize the information by positive and negative numbers and then 0.1, 0.2. And the hundredth go across the top 0.01-- 0.00, 0.01, 0.02, up until 0.09. And then in the interior here, it contains the areas. The z-scores are on the edge.
So here it says a light bulb on average lasts for 500 hours with a standard deviation of 24 hours. Then, they want to know what percent of light bulbs last for less than 540 hours. Now, we would have to assume that our data is normally distributed in order to take it and transform it to go onto our standard normal curve.
Now, I have found a good image of a standard normal curve to start off with. But even if I don't have this image on my paper, I always start by making a quick sketch of something that looks like this-- and it can be pretty rough-- on the side. And then in the middle, I have my mean. And then, I have marks for the standard deviation. And these, again, aren't perfect, but it helps to give me a good image.
Now, in our standard normal curve, we have a mean of 0. And every standard deviation is 1. So 0 plus 1 gets us 1, plus another one is 2, 3, and 4, and so forth.
Now, I can do a quick transformation just by writing my mean and the standard deviations along the bottom. So if my mean is 500, then when I add a standard deviation, I get 524, 548, and 572. And going down, we're going to be subtracting 24 each time. So we have 476, 452, and 428.
Now the score that we're interested in is 540. We're going to need to use a z-score in order to transform it to place it on our standard normal curve. I know it's somewhere around here because it's just before 540, but I want to find out exactly. So I use my z-score. z equals the square I'm interested in is 540 minus the mean divided by the standard deviation.
When we simplify and do the subtraction we get 40 divided by 24. And then when I type that into my calculator, I get 1.66 repeating. The z-table only goes up to the hundredths place. So I'm just going to use 1.67. I'm rounding it. Again, you can either use a z-table, or you can search for the value for a z-score of 1.67. It should have come up when you look it up online.
Now, when I look up this 1.67 on the table, it corresponds to value of 0.9525. And that tells me that the area below this curve up to 540 is 0.9525. Because our mean is 0 and our standard deviation was 1, the area of this whole curve is 1. So when I want to turn that into a percent, it's pretty simple. It's this much out of the total of 1. So I know that that is 95.25%. So 95.25% of our data is less than 540.
If the question wanted to know what percent is more than 540, this area in the white up here, I would subtract that from 100%, because the graph is 100% total. So I subtract 95.25 and I end up with 4.75%. The z-table is always going to be giving me percent less than. So I just subtract from 100 in order to find out what percent is more than. Similarly, you can find out what percent lies between two values by doing a combination of problems.
This has been your tutorial on the standard normal distribution.