When n is large and p isn't too extreme, a normal curve tracks a binomial distribution closely.
9 min read2 worked examples
When a normal curve tracks a binomial distribution
Consider the binomial random variable $X\sim \mathrm{B}(n,p)$. It can be difficult to calculate probabilities for $X$ when $n$ is large. In certain circumstances you can use a normal distribution to approximate a binomial distribution.
A binomial distribution $\mathrm{B}(30,0.5)$ (bars) with its normal approximation $\mathrm{N}(15, 7.5)$ overlaid (curve).
You need to understand the conditions under which this approximation is valid, and learn the relationship between the values of $n$ and $p$ in $\mathrm{B}(n,p)$ and the values of $\mu$ and $\sigma$ in the normal approximation $\mathrm{N}(\mu,\sigma^2)$.
Key fact
If $n$ is large and $p$ is close to $0.5$, then the binomial distribution $X\sim \mathrm{B}(n,p)$ can be approximated by the normal distribution $\mathrm{N}(\mu,\sigma^2)$ where $\mu=np$ and $\sigma=\sqrt{np(1-p)}$.
Hint
The approximation is only valid when $p$ is reasonably close to $0.5$, since the normal distribution is symmetrical — a binomial distribution with $p$ very close to $0$ or $1$ is heavily skewed and won't be well approximated by a symmetric curve.
Worked example 1
Finding μ and σ for the approximation
A call centre resolves 62% of incoming calls within 5 minutes. 150 calls are sampled, and the number resolved within 5 minutes, $X$, is recorded. a Write down a binomial model for $X$. b Explain why $X$ can be approximated with a normal distribution, $Y\sim \mathrm{N}(\mu,\sigma^2)$. c Find the values of $\mu$ and $\sigma$ in this approximation.
a
$X\sim \mathrm{B}(150, 0.62)$.
b
$X$ can be approximated with a normal distribution since $n$ is large and $p$ is reasonably close to $0.5$.
The binomial distribution is discrete, but the normal distribution is continuous.
Key fact
If you are using a normal approximation to a binomial distribution, you need to apply a continuity correction when calculating probabilities.
The diagrams below show a binomial variable $X\sim \mathrm{B}(16,0.5)$ being approximated by $Y\sim \mathrm{N}(8, 2^2)$:
$\mathrm{P}(X>6) \approx \mathrm{P}(Y>6.5)$
$\mathrm{P}(X=10) \approx \mathrm{P}(9.5<Y<10.5)$
$\mathrm{P}(X\le 4) \approx \mathrm{P}(Y<4.5)$
Worked example 2
Applying the continuity correction
At a large university, 40% of students hold a scholarship. A random sample of 200 students is taken, and $X$, the number holding a scholarship, is recorded. Use a suitable approximation to find $\mathrm{P}(X>90)$.
MethodApproximate, then apply the continuity correction
1
$X\sim \mathrm{B}(200,0.4)$ can be approximated by $Y\sim \mathrm{N}(80, 6.93^2)$, since $n$ is large and $p$ is reasonably close to $0.5$.
$X$ is discrete and $Y$ is continuous, so shift the boundary by $0.5$ towards the tail you're excluding.
$\mathrm{P}(Y>90.5)=0.0648$ (3 s.f.) — very close to the exact binomial value of $0.0655$.
3
$\mathrm{P}(X>90) \approx 0.0648$ (3 s.f.).
Compare with the exact binomial probability $\mathrm{P}(X>90)=0.0655$ — the normal approximation is very close.
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Check yourself
In a large city, 45% of households recycle regularly. A random sample of 120 households is taken, and $Y$, the number that recycle regularly, is recorded. a Explain why $Y$ can be approximated by a normal distribution. b Find the values of $\mu$ and $\sigma$ for this approximation.
a
$n=120$ is large and $p=0.45$ is reasonably close to $0.5$.