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 Interpreting results: One-sample t test

A one-sample t test compares the mean of a single column of numbers against a hypothetical mean that you provide.

The P value answers this question:

If the data were sampled from a Gaussian population with a mean equal to the hypothetical value you entered, what is the chance of randomly selecting N data points and finding a mean as far (or further) from the hypothetical value as observed here?

If the P value is large, the data do not give you any reason to conclude that the population mean differs from the hypothetical value you entered. This is not the same as saying that the true mean equals the hypothetical value. You just don't have evidence of a difference.

If the P value is small (usually defined to mean less than 0.05), then it is unlikely that the discrepancy you observed between sample mean and hypothetical mean is due to a coincidence arising from random sampling. You can reject the idea that the difference is a coincidence, and conclude instead that the population has a mean different than the hypothetical value you entered. The difference is statistically significant. But is the difference scientifically important? The confidence interval helps you decide.

Prism also reports the 95% confidence interval for the difference between the actual and hypothetical mean. You can be 95% sure that this range includes the true difference.

Assumptions

The one sample t test assumes that you have sampled your data from a population that follows a Gaussian distribution. While this assumption is not too important with large samples, it is important with small sample sizes, especially when N is less than 10. If your data do not come from a Gaussian distribution, you have three options. Your best option is to transform the values to make the distribution more Gaussian, perhaps by transforming all values to their reciprocals or logarithms. Another choice is to use the Wilcoxon signed rank nonparametric test instead of the t test. A final option is to use the t test anyway, knowing that the t test is fairly robust to departures from a Gaussian distribution with large samples.

The one sample t test also assumes that the “errors” are independent. The term “error” refers to the difference between each value and the group mean. The results of a t test only make sense when the scatter is random – that whatever factor caused a value to be too high or too low affects only that one value. Prism cannot test this assumption.

How the one-sample t test works

Prism calculates the t ratio by dividing the difference between the actual and hypothetical means by the standard error of the mean.

A P value is computed from the t ratio and the numbers of degrees of freedom (which equals sample size minus 1).