24.3 Two independent means: Notation

Since two groups are being compared, distinguishing between the statistics for the two groups (say, Group A and Group B) is important. One way is to use subscripts (Table 24.1).

TABLE 24.1: Notation used to distinguish between the two independent groups

	Group A	Group B
Population means:	${\mu }_A$	${\mu }_B$
Sample means:	${\overline{x}}_A$	${\overline{x}}_B$
Standard deviations:	$s_A$	$s_B$
Standard errors:	$\text{s.e.}\left({\overline{x}}_A\right)=\frac{s_A}{\sqrt{n_A}}$	$\text{s.e.}\left({\overline{x}}_B\right)=\frac{s_B}{\sqrt{n_B}}$
Sample sizes:	$n_A$	$n_B$

Using this notation, the difference between population means, the parameter of interest, is ${\mu }_A-{\mu }_B$. As usual, the population values are unknown, so this parameter is estimated using the statistic ${\overline{x}}_A-{\overline{x}}_B$.

Notice that Table 24.1 does not include a standard deviation or a sample size for the difference between means; they make no sense in this context.

For example, if Group A has 15 individuals, and Group B has 45 individuals, and we wish to study the difference ${\overline{x}}_A-{\overline{x}}_B$. what is the sample size be? Certain not $15-45=-30$.

On the other hand, the standard error of the difference between the means does make sense: it measures how much the value of ${\overline{x}}_A-{\overline{x}}_B$ varies from sample to sample.

For the reaction-time data, we will use the subscripts $P$ for phone-users group, and $C$ for the control group. That means that the two sample means would be denoted as ${\overline{x}}_P$ and ${\overline{x}}_C$, and the difference between them as ${\overline{x}}_P-{\overline{x}}_C$.