Mean and Standard Deviation of a Binomial

The mean of a binomial distribution sits at n times p because that is just the long-run average. If you take n trials and each one succeeds with probability p, you expect a fraction p of them to come up successes, so np is the natural balancing point of the histogram. Shoot 100 free throws at a 78 percent success rate, and you expect 78 makes on average, not because every attempt gives exactly that, but because the outcomes cluster around that value over many repetitions.

The standard deviation formula, √(n times p times (1-p)), comes from the same place every standard deviation comes from: the spread of the underlying trials, added up across n independent trials. Each single trial (a Bernoulli trial) has variance p times (1-p), and because the trials are independent, variances add directly when you sum them, giving n times p times (1-p) as the binomial variance. Take the square root to get back to the original units of count.

These two numbers connect binomial thinking to the normal distribution coming next. Once n is large enough and p is not too close to 0 or 1, the lumpy binomial histogram starts looking smooth and bell-shaped, and you can use the normal distribution with mean np and standard deviation √(np(1-p)) to approximate probabilities instead of computing exact binomial ones. Getting comfortable with mean and standard deviation here is what makes that approximation make sense later.