set.seed(123)
d <- data.frame(y = c(rnorm(100), rnorm(100)+.5, rnorm(100)-1),
                x = rep(c("A", "B", "C"), each = 100))

ggplot(d, aes(x, y)) + geom_boxplot()

library(dplyr)

d %>% group_by(x) %>%
  summarize(middle = median(y),
            mean = mean(y),
            sd = sd(y),
            lower = quantile(y, probs = .25),
            upper = quantile(y, probs = .75),
            ymin = max(mean - 2*sd, min(y)),
            ymax = min(mean + 2*sd, max(y)),
            outliers = list(y[y<ymin | y > ymax])) %>%
  ggplot(aes(x, ymin = ymin, lower = lower,
             middle = middle, upper = upper, ymax = ymax,
             outliers = outliers)) +
  geom_boxplot(stat = "identity")