Mathematics
The Binomial Distribution and Its Approximation to Normal
Quick fact
When n is large enough, the binomial distribution’s shape is almost identical to a normal distribution with the same mean and variance.
Why this is interesting
Ever wondered why flipping a coin 100 times gives a bell-shaped curve of outcomes? It’s a binomial distribution quietly turning into a normal one.
Read the full explanation
Understanding The Binomial Distribution and Its Approximation to Normal
Imagine you flip a fair coin 10 times. The number of heads you get can be 0, 1, 2, ..., or 10. Some outcomes are more likely than others. For example, getting 5 heads is much more likely than getting 0 or 10. If you plot the probability of each outcome, you’ll see a shape that peaks in the middle and drops off at the ends. This is called a binomial distribution. It’s built from simple, independent trials—each flip is a trial with two outcomes (heads or tails), and the probability of heads is the same every time. The rule for calculating the exact probability of getting exactly k heads in n flips is: P(k successes) = nCk p^k (1-p)^(n-k). Here, nCk is the number of ways to choose which flips are heads. Now, if you increase n to 100, the distribution becomes smoother and starts to look like the famous bell-shaped curve. That bell is the normal distribution. This connection is not a coincidence: it’s a result of the Central Limit Theorem.
A deeper explanation
The binomial distribution has a mean (expected value) of μ = np and a variance of σ² = np(1-p). The mean is simply the average number of successes you’d expect over many repetitions. The variance tells you how spread out the outcomes are. The normal approximation works because, as n grows, the binomial distribution becomes symmetric and approaches a continuous curve. The rule of thumb is that the approximation is good when both np and n(1-p) are at least 5 (sometimes 10). To use the normal approximation, you first standardize the variable: z = (k - μ) / σ. Then you use standard normal tables to find probabilities. Since the binomial is discrete and the normal is continuous, we apply a continuity correction: we adjust the value k by 0.5. For example, P(X ≤ k) is approximated by P(Z ≤ (k + 0.5 - μ)/σ). This correction accounts for the gap between integer outcomes. This concept matters because it lets us compute probabilities for large n without summing many terms. It’s used in quality control, survey analysis, and many statistical tests.