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# Z-statistics vs. T-statistics

## Video transcript

I want to use this video to kind of make sure we intuitively and otherwise and understand the difference between a Z-statistic-- something I have trouble saying-- and a T-statistic. So in a lot of what we're doing in this inferential statistics, we're trying to figure out what is the probability of getting a certain sample mean. So what we've been doing, especially when we have a large sample size-- so let me just draw a sampling distribution here. So let's say we have a sampling distribution of the sample mean right here. It has some assumed mean value and some standard deviation. What we want to do is any result that we get, let's say we get some sample mean out here. We want to figure out the probability of getting a result at least as extreme as this. So you can either figure out the probability of getting a result below this and subtracted that from 1, or just figure out this area right over there. And to do that we've been figuring out how many standard deviations above the mean we actually are. The way we figured that out is we take our sample mean, we subtract from that our mean itself, we subtract from that what we assume the mean should be, or maybe we don't know what this is. And then we divide that by the standard deviation of the sampling distribution. This is how many standard deviations we are above the mean. That is that distance right over there. Now, we usually don't know what this is either. We normally don't know what that is either. And the central limit theorem told us that assuming that we have a sufficient sample size, this thing right here, this thing is going to be the same thing as-- the sample is going to be the same thing as the standard deviation of our population divided by the square root of our sample size. So this thing right over here can be re-written as our sample mean minus the mean of our sampling distribution of the sample mean divided by this thing right here-- divided by our population mean, divided by the square root of our sample size. And this is essentially our best sense of how many standard deviations away from the actual mean we are. And this thing right here, we've learned it before, is a Z-score, or when we're dealing with an actual statistic when it's derived from the sample mean statistic, we call this a Z-statistic. And then we could look it up in a Z-table or in a normal distribution table to say what's the probability of getting a value of this Z or greater. So that would give us that probability. So what's the probability of getting that extreme of a result? Now normally when we've done this in the last few videos, we also do not know what the standard deviation of the population is. So in order to approximate that we say that the Z-score is approximately, or the Z-statistic, is approximately going to be-- so let me just write the numerator over again-- over, we estimate this using our sample standard deviation-- let me do this in a new color-- with using our sample standard deviation. And this is OK if our sample size is greater than 30. Or another way to think about it is this will be normally distributed if our sample size is greater than 30. Even this approximation will be approximately normally distributed. Now, if your sample size is less than 30, especially if it's a good bit less than 30, all of a sudden this expression will not be normally distributed. So let me re-write the expression over here. Sample mean minus the mean of your sampling distribution of the sample mean divided by your sample standard deviation over the square root of your sample size. We just said if this thing is well over 30, or at least 30, then this value right here, this statistic, is going to be normally distributed. If it's not, if this is small, then this is going to have a T-distribution. And then you're going to do the exact same thing you did here, but now you would assume that the bell is no longer a normal distribution, so this example it was normal. All of Z's are normally distributed. Over here in a T-distribution, and this will actually be a normalized T-distribution right here because we subtracted out the mean. So in a normalized T-distribution, you're going to have a mean of 0. And what you're going to do is you want to figure out the probability of getting a T-value at least this extreme. So this is your T-value you would get, and then you essentially figure out the area under the curve right over there. So a very easy rule of thumb is calculate this quantity either way. Calculate this quantity either way. If you will have more than 30 samples, if your sample size is more than 30, your sample standard deviation is going to be a good approximator for your population standard deviation. And so this whole thing is going to be approximately normally distributed, and so you can use a Z-table to figure out the probability of getting a result at least that extreme. If your sample size is small, then this statistic, this quantity, is going to have a T-distribution, and then you're going to have to use a T-table to figure out the probability of getting a T-value at least this extreme. And we're going to see this in an example a couple of videos from now. Anyway, hopefully that helped clarify some things in your head about when to use a Z-statistic or when to use a T-statistic.