Binomial mean and variance
Summarize the center and spread of a binomial count.
Binomial mean and variance is one of 7 distributions formulas in the probability section of this library, and it is used at ap · university level.
Why binomial mean and variance works
Think of each trial as contributing 1 with probability p and 0 otherwise, so its own average contribution is p. Averages always add, so n trials give np. Each trial also has variance p times 1 minus p, and independent variances add too, which gives the second formula.
What each symbol means
$n$ is trial count and $p$ success probability.
Binomial mean and variance: when it holds
The variable must satisfy the binomial conditions.
When it stops applying
The mean survives dependence but the variance does not. Averages add whether or not the trials are related, so np stays correct, while trials that rise and fall together, such as free throws in one windy game, make the true variance larger than np times 1 minus p.
Binomial mean and variance: a worked example
For $n=100,p=.2$, mean is $20$ and variance $16$.
The mistake to avoid
What people do: Quoting the variance as if it were the standard deviation.
Why it goes wrong: With 100 trials at probability 0.2 the variance is 16, so the standard deviation is 4. Reporting 16 as the spread makes ordinary results look like extreme ones.
Do this instead: Take the square root before comparing anything with the mean, since the mean and the standard deviation are in counts while the variance is in squared counts.
Binomial mean and variance: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ is trial count and $p$ success probability.
- Check the conditions before substituting. The variable must satisfy the binomial conditions.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Distributions
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where binomial mean and variance comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about binomial mean and variance
For a fixed number of trials, when is the variance largest?
At a success probability of one half, where it reaches n divided by 4. A near-certain or near-impossible outcome varies very little from one run to the next.
Does the mean have to be a whole number?
No, and it usually is not. With 10 trials at probability 0.35 the mean is 3.5, which is a long-run average rather than a count you could actually observe.
What is the standard deviation for 100 trials at probability 0.2?
It is 4, since the variance is 16. That tells you counts roughly between 12 and 28 would be unremarkable, being within two standard deviations of the mean of 20.
When can I treat the binomial count as approximately normal?
When both np and n times 1 minus p are at least about 10. With 100 trials at 0.2 those are 20 and 80, so the approximation is comfortable.