## Statistics

### T-test

A T-test is performed to compare the averages of two groups of data. How do you know that the difference between the averages you observed is "real"? It could be that you just happened to see a difference because your sample size is small and by chance you ended up with larger numbers in one group and smaller numbers in the other. The T-test allows us to say with 95% certainty whether they are truly different. Using Microsoft Excel, the following function returns the probability associated with a Student's t-test:

#### =TTEST(Array1,Array2,Tails,Type)

Where
• Array1 is the range of data for the first group
• Array2 is the range of data for the second group
• Tails refers to a one-tailed test or a two-tailed test. If we have background information that would allow us to predict that one group should have a higher value, we use 1. If we really don't know which one should end up being higher, we use 2.
• Type can be one of three things
1. paired T-test: Used when the same animals are used in groups 1 and 2, and the measurement is repeated. Example: you measure respiration rate in fish at rest, then the same fish during exercise.
2. unpaired, equal variance: different animals are in each group, and the variation within each group is about the same. To examine the variance of each group, use =VAR(array1) and =VAR(array2) and eyeball the results. Technically, an F-test should be performed, but let's not go there right now.
3. unpaired, unequal variance: different animals are in each group, and the variation within each group is different.
The number that you get from the function is called the P-value, which ranges from 0 to 1 (if you get a number outside these limits, then you have not set up the function properly.) P represents the probablility that the two means are the same. If P is high, then there is probably no difference between the two groups; if P is low, then there may be a difference. By convention, we us 5% as a cut-off, so if P is less than 0.05, we say that the difference between the means is statistically significant because the chance that the averages are the same is less than 5%. We can live with that level of uncertainty.

#### Example

Suppose we do an experiment wherein we measure respiration rate in 17 fish at rest, then the same fish during exercise. We get the following data:
 A B 1 Respiration (/min) 2 Rest Exercise 3 78 84 4 120 92 5 88 116 6 39 100 7 80 84 8 56 104 9 52 104 10 72 120 11 96 96 12 88 162 13 90 108 14 96 156 15 80 104 16 102 138 17 100 140 18 86 122 19 84 110
Our experiment and data are set up as follows:
• Array1 is the range of data for the rest group (A3:A19)
• Array2 is the range of data for the exercise group (B3:B19)
• We know that exercise causes increased respiration in many species, so we choose a one-tailed test (1)
• The same animals are used both groups. A3 is for fish1 at rest, B3 is for fish1 during exercise, and so on. Therefore we use a paired T-test (1)
Our formula is completed as follows:

=TTEST(A3:A19,B3:B19,1,1)

The result is 6.82779E-05, or 6.82779 x 10-5, which translates as 0.0000683. This P-value means that we would expect to get these results by random chance only about 0.007% of the time. The difference between the means is highly significant. We can say that the two means are significantly different. This is a term of art--we must never say this unless a statistical test has actually shown it to be so.

[ 8/23/01 jrc]