## 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 $$\bar{x}_P - \bar{x}_C$$.