Binomial Distribution vs Normal Distribution

A binomial setting has a fixed number of trials, two outcomes per trial, independent trials, and the same success probability each time. For example, the number of heads in 10 flips of a fair coin is binomial with $n = 10$ trials and success probability $p = 0.5$. The probability of exactly $k$ successes is $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ (n choose k, times p to the power k, times one minus p to the power n minus k). Its mean is $\mu = np$ (n times p).

The normal distribution is a bell-shaped curve centered at its mean $\mu$ (mu) and spread out by its standard deviation $\sigma$ (sigma). Many natural measurements, such as heights or measurement errors, are approximately normal. For example, adult heights cluster near an average, with fewer people far above or below it. The empirical rule says about 68, 95, and 99.7 percent of the data fall within 1, 2, and 3 standard deviations of the mean.