3).

These different factors can occur in any and all combinations, so there are a lot of potential scenarios.

In the following sections, we review situations you may frequently encounter when analyzing

biological data, and advise you as to how to select the most appropriate testing approach given the

situation.

Comparing the mean of a group of numbers to a hypothesized value

Sometimes you have a measurement from the literature (called a historical control) that provides a

hypothesized value of your measurement, and you want to statistically compare the average of a group

to this mean. This situation is common when you are comparing a value that was calculated based on

statistical norms derived based on the population (such as the IQ test, where 100 has been scaled to be

the population mean).

Typically, comparing a group mean to a historical control warrants using the one-group Student t test

that we describe in the later section “Surveying Student t tests.” For data that are not normally

distributed, the Wilcoxon Signed-Ranks (WSR) test can be used instead, although it is not used often so

we do not cover it in this chapter. (If you need a review on what normally distributed means, see

Chapter 3.)

Comparing the mean of two groups of numbers

Comparing the mean of two groups of numbers is probably the most common situation encountered in

biostatistics. You may be comparing mean levels of a protein that is a hypothesized disease biomarker

between a group of patients known to have the disease and a group of healthy controls. Or, you may be

comparing a measurement of drug efficacy between two groups of patients with the same condition

who are taking two different drugs. Or, you may be comparing measurements of breast cancer treatment

efficacy in women on one health insurance plan compared to those on another health insurance plan.

Such comparisons are generally handled by the famous unpaired or “independent sample”

Student t test (usually just called the t test) that we describe later in the section “Surveying

Student t tests.” Importantly, the t test is based on two assumptions about the distribution of the

measurement value being tested in the two groups:

The values must be normally distributed (called the normality assumption). For data that are not

normally distributed, instead of the t-test, you can use the nonparametric Wilcoxon Sum-of-Ranks

test (also called the Mann-Whitney U test and the Mann-Whitney test). We demonstrate the

Wilcoxon Sum-of-Ranks test later in this chapter in the section “Running nonparametric tests.”

The standard deviation (SD) of the values must be close for both groups (called the equal

variance assumption). As a reminder, the SD is the square root of the variance. To remember why

accounting for variation is important in sampling, review Chapter 3. Also, Chapter 9 provides

more information about the importance of SD. If the two groups you are comparing have very

different SDs, you should not use a Student t test, because it may not give reliable results,

especially if you are also comparing groups of different sizes. A rule of thumb is that one group’s

SD divided by another group’s SD should not be more than 1.5 to quality for a Student t test. If you