“The Student’s t Test is used to compare the mean of two **normally distributed** samples, preferably of equal size and variance. More specifically, the Student’s t Test gives you a probabilistic estimate of the likelihood that your samples were randomly selected from the same population, i.e. that the Null Hypothesis is true. If the Student’s t Test suggests that your samples were probably not taken from the same population, you can conclude, with some certainty (how certain depends on your p value), that your samples were taken from different populations.

Thinking of the Student’s t Test in terms of “signal to noise” ratios, the “signal” is the difference between the mean of two samples (similar to the Z test). All other things being equal, large differences in means are *more* likely to be statistically significant. However, if the distribution of values in the population (or, in this case, the samples) is large, (i.e. there is a lot of noise), the difference in means is less likely to be significant, and more likely to be due to random variation. For the Student’s t Test, the square root of the sum of the sample variances squared, divided by the sample sizes, is used as an estimate of noise.

## Contents

## Mathematical Makeup

### Signal

Absolute value of (sample mean – population mean)

### Noise

sqrt(s1^2/n1 + s2^2/n2) where s = variance and n = number in each sample

### Formual for t

t = [abs(population mean – sample mean)] / [sqrt(s1^2/n1 + s2^2/n2)]

### Getting a p value

Once you have t, you probably want to know if it is “statistically significant.” In order to do this, you have to look it up in a table, or use a computer program. Remember, the t Test assumes normally distributed data. Because the probability of normally distributed occurrences obeys known mathematical formulas, we (or statisticians) can theoretically calculate the likelihood that a given z value occurs simply out of chance. In order to look up the p value that corresponds to your t statistic, you need to know the degree of freedom, which in this case is (n1 + n2 – 2). You also have to decide whether or not you are conducting a one-tailed (ex. x > X) or two tailed (ex. x != X) test. Note that as your degrees of freedom increase (i.e. the sample sizes get larger), the t test approaches the Z test. Whether or not the P Value you find is statistically significant depends on what P Value you find acceptable.

## How to Calculate

### Excel

### MATLAB

### R

## Other Links

## Comparison to the z Test

The major difference between the t Test and the z Test is that with the z Test, one of the groups (the population) is much larger than the other (the sample). So, when thinking about signal:noise ratios, it is easy to calculate noise (which in this case is the Standard Error of the Mean). With the t Test, there is no reference “population,” so you don’t have the luxury of accurately calculating noise. To compensate for that, the t Test approximates noise by using the variances. As the sample population sizes get larger (and the degrees of freedom increase), the t statistic approaches the z statistic, i.e. the tests are essentially the same.”

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