Student's T-test: comparing two means A set of measurements can be analysed in a number of ways. Mean, mode and median all give some idea of the “middle” value. The range of a sample (highest measurement – lowest measurement) does not describe how the measurements are distributed – whether they are clustered together or evenly spread. However, a line plot of the same data shows the distribution. Figure 1: Example set of data 2 19 7 6 13 14 10 12 5 8 4 9 8 10 8 5 3 6 Figure 2: Line plot 4 3 2 1 0 0 5 10 15 20 A normally distributed population is shown in figure 3. The measurements are spread symmetrically about the mean. The spread of the data can be measured by calculating standard deviation. Figure 3 Standard deviation σ = ∑ (x n-1 )² is the sample mean, n is the number of observations in the sample For a normally distributed population, 66.7% of the readings will lie within 1 standard deviation of the mean. The greater the variability of the data, the larger the standard deviation will be. The Student’s t-test is used to find out if there is a significant difference between the means of two sets of data. It looks at the degree of overlap between the two samples. This test is used: 1. for data that is normally distributed around the mean (if not use MannWhitney U test) 2. to test for differences in sample means 3. when you have unmatched samples 4. if the measurements are at interval level – length, mass etc. 5. when there are between 20 and 30 measurements per sample It is calculated from the formula: t= 1- 2 (σ1)² + (σ2)² n n Where, 1 = mean of sample 1 2 = mean of sample 2 σ1 = standard deviation of sample 1 σ2 = standard deviation of sample 2 n = number in sample Degrees of freedom = (n1 + n2) – 2 To determine whether the calculated value of t demonstrates a statistically significant difference between the 2 samples, it is compared to published tables. If it is greater than the critical value, at 5% probability and at the appropriate degrees of freedom, then the null hypothesis can be rejected. (critical value table from: Fowler, J. and Cohen L., 1995. Practical Statistics for Field Biology, pp227.) This test can be used to analyse results from beach profile data at two different sites, comparing Environmental Quality scores in two different parts of a settlement.