Binomial model and the meaning of standard deviation
The phrase standard deviation formula for binomial distribution refers to the spread of a binomial random variable \(X\), where \(X\) counts the number of successes in \(n\) independent trials, each trial having success probability \(p\). This is written \(X \sim \mathrm{Bin}(n,p)\).
\[ \mu=\mathbb{E}[X]=np, \qquad \sigma=\sqrt{\mathrm{Var}(X)}. \]
Main result (formula)
\[ \mathrm{Var}(X)=np(1-p) \qquad\Longrightarrow\qquad \sigma=\sqrt{np(1-p)}. \]
The expression \(1-p\) is often denoted by \(q\), so the same standard deviation formula is also written \(\sigma=\sqrt{npq}\).
Step-by-step derivation from Bernoulli trials
A binomial count is a sum of independent indicator variables. Define \(X_i\) as the outcome of trial \(i\): \(X_i=1\) for success and \(X_i=0\) for failure. Then
\[ X=\sum_{i=1}^{n} X_i. \]
Step 1: Variance of one Bernoulli trial.
Each \(X_i\) has \(P(X_i=1)=p\) and \(P(X_i=0)=1-p\). Compute \(\mathbb{E}[X_i]\) and \(\mathbb{E}[X_i^2]\):
\[ \mathbb{E}[X_i]=1\cdot p+0\cdot(1-p)=p, \qquad \mathbb{E}[X_i^2]=1^2\cdot p+0^2\cdot(1-p)=p. \]
Therefore
\[ \mathrm{Var}(X_i)=\mathbb{E}[X_i^2]-\big(\mathbb{E}[X_i]\big)^2 =p-p^2 =p(1-p). \]
Step 2: Add variances using independence.
Independence implies covariances are zero, so the variance of a sum is the sum of variances:
\[ \mathrm{Var}(X)=\mathrm{Var}\!\left(\sum_{i=1}^{n}X_i\right) =\sum_{i=1}^{n}\mathrm{Var}(X_i) =\sum_{i=1}^{n}p(1-p) =np(1-p). \]
Step 3: Convert variance to standard deviation.
\[ \sigma=\sqrt{\mathrm{Var}(X)}=\sqrt{np(1-p)}. \]
Worked example
Suppose a quality-control process produces a defective item with probability \(p=0.20\) per item, independently across items, and a sample of \(n=25\) items is inspected. Then \(X\sim\mathrm{Bin}(25,0.20)\).
| Quantity | Computation | Value |
|---|---|---|
| Mean | \(\mu=np=25\cdot 0.20\) | \(\mu=5\) |
| Variance | \(\sigma^2=np(1-p)=25\cdot 0.20\cdot 0.80\) | \(\sigma^2=4\) |
| Standard deviation | \(\sigma=\sqrt{np(1-p)}=\sqrt{4}\) | \(\sigma=2\) |
Interpretation: the expected number of defectives is \(5\), and the typical fluctuation around that center is about \(2\) defectives (subject to the discreteness of counts).
Visualization: how \(\sigma=\sqrt{np(1-p)}\) changes with \(p\) (fixed \(n\))
Key takeaways
The standard deviation formula for binomial distribution is \(\sigma=\sqrt{np(1-p)}\). It follows from writing the binomial count as a sum of independent Bernoulli trials and using variance additivity. Variability is largest near \(p=0.50\) and smaller when successes are very rare or very common.