Binomial setting and what “standard deviation” measures
The binomial standard deviation formula applies when a random variable \(X\) counts the number of successes in \(n\) independent trials, each with success probability \(p\). This is written \(X \sim \mathrm{Bin}(n,p)\). The standard deviation quantifies the typical spread of \(X\) around its mean.
| Quantity | Formula for \(X \sim \mathrm{Bin}(n,p)\) | Interpretation |
|---|---|---|
| Mean | \(\mu = \mathbb{E}[X] = np\) | Expected number of successes |
| Variance | \(\sigma^2 = \mathrm{Var}(X) = np(1-p)\) | Spread measured in squared units |
| Standard deviation | \(\sigma = \sqrt{np(1-p)}\) | Spread measured in the same units as \(X\) |
Step-by-step derivation of the binomial standard deviation formula
A binomial count can be expressed as a sum of Bernoulli indicator variables. Let \(X_i\) be the indicator of success on trial \(i\), so \(X_i=1\) for success and \(X_i=0\) for failure. Then
\[ X = X_1 + X_2 + \cdots + X_n. \]
Step 1: Mean and variance of a single Bernoulli trial.
For \(X_i \sim \mathrm{Bernoulli}(p)\),
\[ \mathbb{E}[X_i]=p, \qquad \mathrm{Var}(X_i)=p(1-p). \]
Step 2: Use independence to add variances.
Because the trials are independent, the covariance terms 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: Take the square root to obtain the standard deviation.
\[ \sigma=\sqrt{\mathrm{Var}(X)}=\sqrt{np(1-p)}. \]
Notation note: \(1-p\) is often written \(q\), so the same result appears as \(\sigma=\sqrt{npq}\).
Worked example
Suppose \(n=10\) independent trials with \(p=0.50\) (success is equally likely as failure). Then the binomial mean and the binomial standard deviation formula give:
\[ \mu=np=10\cdot 0.50=5, \qquad \sigma=\sqrt{np(1-p)}=\sqrt{10\cdot 0.50\cdot 0.50}=\sqrt{2.5}\approx 1.581. \]
Interpretation: counts near \(5\) are most typical, and values roughly within about \(1.6\) of \(5\) are comparatively common (subject to discreteness and the distribution’s shape).
Visualization: binomial probabilities with mean and ±1 standard deviation
Common checks and practical notes
The formula \(\sigma=\sqrt{np(1-p)}\) requires independent trials with constant success probability \(p\). If the binomial model is appropriate, \(\sigma\) increases with \(n\) but is largest near \(p=0.50\) and decreases as \(p\) approaches \(0\) or \(1\).