one way anova to compare each factor against the mean I have a factor with six levels and then a proportion for each of those levels. I have six months that women became pregnant and then the proportion who smoked during pregnancy, as an example. 
I used an anova and I know that there are significant differences in the group.
I also used a tukey hsd and I looked at the pair-wise differences.
I simply want to know if the difference in each factor is significant against the average for everyone. 
Is that something I can learn with a one way anova? 
 A: First, you should be using a binary regression model (e.g., logistic regression) to get correct inferences. ANOVA is designed for continuous outcomes (not binary), and the standard errors and p-values will only be correct under normally distributed outcomes. You can continue to use ANOVA with binary data as long as you use a robust standard error.
There is a way to code the categorical variable so that the interpretation of the intercept is the mean of the outcomes across the categories (i.e., the grand mean) and the interpretation of each slope is the deviation of each category from the grand mean. This is called deviation or effect coding. In R this can be done using the contr.sum function. For example, you might run the following:
fit <- glm(smoke ~ month, data = data, 
           family = binomial(link = "identity"),
           contrasts = list(month = contr.sum))
jtools::summ(fit, robust = TRUE, digits = 3)

(Ignore the warning you'll get about month being absent.) 
Unfortunately the R output is a little ambiguous and you'll have to do some work do determine which coefficient belongs to which group. You can easily check the probabilities for each group using 
(smoke_by_month <- with(data, tapply(smoke, month, mean)))

and then assess which coefficients correspond to which groups.
One caution is that you won't get a test for one of the months. This is because the result of this test is not independent from the results of the other tests. You can manually calculate the deviations from the grand mean (e.g., with smoke_by_month - mean(smoke_by_month)), but you won't get a hypothesis test for this group. If you want a hypothesis test for this group, relevel the month factor and run the analysis again.
