The Difference between Two Means

This section covers how to test for differences between means from two separate groups of subjects and gives an example of opinions on animal research. The detailed testing procedure is carried out using the standard steps in hypothesis testing.

Computations for Unequal Sample Sizes (optional)

The calculations are somewhat more complicated when the sample sizes are not equal. One consideration is that MSE, the estimate of variance, counts the group with the larger sample size more than the group with the smaller sample size. Computationally, this is done by computing the sum of squares error (SSE) as follows:

S S E=\sum\left(X-M_{1}\right)^{2}+\sum\left(X-M_{2}\right)^{2}

where M_{1} is the mean for group 1 and M_{2} is the mean for group 2 . Consider the following small example:

Table 4. Unequal n.

Group 1 Group 2
3 2
4 4

M_{1}=4 \text { and } M_{2}=3


Then, MSE is computed by: MSE =\mathrm{SSE} / \mathrm{df}

where the degrees of freedom (df) is computed as before: \mathrm{df}=\left(\mathrm{n}_{1}-1\right)+\left(\mathrm{n}_{2}-1\right)=(3-1)+(2-1)=3 MSE =S S E / d f=4 / 3=1.333

The formula

The formula

s_{M_{1}-M_{2}}=\sqrt{\frac{2 M S E}{n}}

is replaced by

s_{M_{1}-M_{2}}=\sqrt{\frac{2 M S E}{n_{h}}}

where n_{h} is the harmonic mean of the sample sizes and is computed as follows:

\mathrm{n}_{\mathrm{h}}=\dfrac{2}{1 / n_{1}+1 / n_{2}}=\dfrac{2}{1 / 3+1 / 2}=2.4




t=(4-3) / 1.054=0.949

and the two-tailed \mathrm{p}=0.413.

R code

Data file
t.test(data$WRONG ~ data$GENDER,var.equal=TRUE)

Two Sample t-test

data: data$WRONG by data$GENDER
t = 2.5335, df = 32, p-value = 0.01639
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
0.2882231 2.6529534
sample estimates:
mean in group 1 mean in group 2
5.352941 3.882353