24.3 Notation: Two independent means

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:	\(\bar{x}_A\)	\(\bar{x}_B\)
Standard deviations:	\(s_A\)	\(s_B\)
Standard errors:	\(\displaystyle\text{s.e.}(\bar{x}_A) = \frac{s_A}{\sqrt{n_A}}\)	\(\displaystyle\text{s.e.}(\bar{x}_B) = \frac{s_B}{\sqrt{n_B}}\)
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 \(\bar{x}_A-\bar{x}_B\).

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 \(\bar{x}_A - \bar{x}_B\). 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 \(\bar{x}_A - \bar{x}_B\) 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 \(\bar{x}_P\) and \(\bar{x}_C\), and the difference between them as \(\bar{x}_P - \bar{x}_C\).