So here we say that the t-test is used to test the mean or average between populations, one of which could be the standard. Now, in order to test the similarities and differences between these populations, we utilize the t-score. To find our t-score, we employ the t-score formula. We use the t-score formula when we don't know what the population standard deviation is. Remember, your standard deviation is just s. As the total number of measurements increases, our standard deviation transitions into our population standard deviation, which is σ. Here, this t-score formula is used when the sample size is less than 30. Size greater than 30 would often utilize software like Microsoft Excel in order to do those large calculations.
Here, our t-score formula is: t = ( x ¯ - μ 0 ) s n , which represents our population average divided by our standard deviation s divided by the square root of the number of samples or measurements, which is n. How does the t-score tell us about similarities and differences between our populations? We're going to say here that the larger the t-score, then the more different the populations are; the smaller the t-score, then the more similar they are to one another.
When we're comparing different populations to one another, we could use three variations of a t-calculated value. Here we can look at our t-calculated for equal variance, for unequal variance, and for paired data. Remember, your variance is just your standard deviation squared. We're going to say here when our standard deviation is equal for both populations, we use these two sets of data. Here, t-calculated will equal the absolute value of the average of population 1 minus the average of population 2 divided by s_{pooled} times the square root of the measurements divided by the measurements added together. s_{pooled} would have its own formula here where we're dealing with the standard deviation squared of population 1 times the number of measurements of Population 1 minus 1 plus the standard deviation of Population 2 squared times its number of measurements minus 1. Here on the bottom, we'd have the measurement of Population 1 plus the measurement of Population 2 minus 2. That bottom portion also represents the degrees of freedom involved.
When the variances are not equal between my two populations, then I use these two sets of data. Again, we use 1 to help us figure out what our t-calculated would be. Notice the differences in the formula whether the variances are equal or not. You're not going to be expected to memorize these. Normally, you're given a formula sheet in which you can use them. But always refer to your professor just in case. These calculations, these formulas can get very complicated, so it's always best just to give them to you.
Paired data is used when we have two populations done by completely different methods. Let's say you're trying to analyze the reactivity or fluorescence of some type of material. You have a bunch of different methods that can be used that are very different from one another. In that case, we'd rely on paired data to figure out t-calculated. For the first two, we may be looking at two different populations, but we're using the same type of software and the same types of methods in order to figure out our t-calculated. The whole point of figuring out t-calculated at this point would be then to compare it to our t-table. You'd find your T calculated and you can compare it to your T-table. If your T-calculated happens to be greater than your T-table, remember we looked at our t-table when we're dealing with just figuring out confidence intervals. It'd be that same exact table. If t-calculated were found to be greater than t-table, then you would say there is a significant difference in the average or means of the two populations in which you're examining. If your t-calculated were less, then you would say that there is no significant difference between the average or means between the two populations.