Mean and standard deviation of the binomial
Two formulas worth knowing by heart. The common mistake: forgetting the square root for the standard deviation.
Worked example
X is binomial with n = 25 and p = 0.40. Work out the expected value and the standard deviation.
Worked solution
- 1.E(X) = n · p = 25 · 0.40 = 10.00
- 2.σ = √(n · p · (1 − p)) = √6.0000 = 2.45
Typical mistake: Giving the variance n·p·(1−p) as the standard deviation (forgetting the root).
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task
X is binomial with n = 80 and p = 0.50. Work out the expected value and the standard deviation.
E(X) (2 decimal places)
σ (2 decimal places)
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
- P(X ≤ k) = P(X = 0) + … + P(X = k)
- E(X) = n · p, σ = √(n · p · (1 − p))
Typical mistake: Giving the variance n·p·(1−p) as the standard deviation (forgetting the root).
Frequently asked questions
What is the typical mistake in "Mean and standard deviation of the binomial"?
Giving the variance n·p·(1−p) as the standard deviation (forgetting the root).
How do I work out "at least k"?
Use the complement: P(X ≥ k) = 1 − P(X ≤ k − 1).