Binomial Standard Deviation Calculator

This calculator computes the standard deviation for a binomial distribution. You enter the number of trials (n) and the probability of success on a single trial (p), and it returns the standard deviation, variance, mean, and coefficient of variation. The standard deviation is the square root of the variance, and for a binomial distribution it is √(n·p·(1−p)).

Formula

The standard deviation of a binomial distribution is given by:

σ = √(n · p · (1 − p))

Where:

• n = number of trials (a positive integer)
• p = probability of success on each trial (a number between 0 and 1)
• σ = standard deviation (same units as the count of successes)

The variance is σ² = n · p · (1 − p), and the mean (expected value) is μ = n · p. The coefficient of variation is σ / μ, which expresses the spread relative to the mean.

Worked Example

Using the default values from the calculator: n = 100 trials, p = 0.5.

1. Compute the variance: n · p · (1 − p) = 100 × 0.5 × 0.5 = 25.
2. Take the square root: √25 = 5.
3. So the standard deviation is 5.
4. The mean is n · p = 100 × 0.5 = 50.
5. The coefficient of variation is 5 / 50 = 0.1 (or 10%).

The calculator shows the same result: standard deviation = 5.0000, variance = 25.0000, mean = 50, coefficient of variation = 0.1000.

When to Use the Binomial Standard Deviation

Use this formula when you have a fixed number of independent trials, each with the same probability of success, and you want to know how much the number of successes is expected to vary from the mean. Common examples include:

• Quality control: counting defective items in a batch of fixed size.
• Surveys: number of respondents who answer “yesâ€� in a sample of fixed size.
• Games: number of wins in a fixed number of independent rounds.
• Biology: number of offspring with a certain trait in a fixed litter.

The binomial standard deviation is also used to construct confidence intervals for a proportion and to perform hypothesis tests about proportions.

How to Read the Result

The standard deviation tells you the typical distance of the number of successes from the mean. For n = 100 and p = 0.5, the mean is 50 and the standard deviation is 5. That means in many repetitions of 100 trials, the number of successes will often fall between 45 and 55 (one standard deviation from the mean), and usually between 40 and 60 (two standard deviations).

A larger standard deviation means more spread. The standard deviation increases with n, but the relative spread (coefficient of variation) decreases as n grows. For example, with n = 100 and p = 0.5, the CV is 10%; with n = 400 and p = 0.5, the CV drops to 5%.

Common Mistakes and Misconceptions

• Using the wrong p: p must be the probability of success, not failure. If you enter 0.7 instead of 0.3, you get the same standard deviation because 1−p swaps, but the mean will be wrong.
• Forgetting to subtract p from 1: The formula uses (1−p), not p alone. Leaving out that factor gives the standard deviation of a Poisson-like approximation, which is incorrect for a binomial.
• Rounding too early: If you round n·p·(1−p) before taking the square root, you may introduce small errors. Keep full precision until the final step.
• Confusing population and sample standard deviation: The binomial standard deviation is a population parameter because it describes the distribution of all possible outcomes. It is not a sample standard deviation computed from data.
• Assuming normality for small n: The binomial distribution is approximately normal when n·p and n·(1−p) are both at least 5. For smaller n, the distribution is skewed, and the standard deviation alone does not fully describe the spread.

Limits of the Formula

The formula assumes independent trials with a constant probability of success. If trials are not independent (e.g., sampling without replacement from a small population), the binomial model does not apply; you would use the hypergeometric distribution instead. Also, p must be between 0 and 1, and n must be a positive integer. The standard deviation is always nonnegative; it is zero only when p = 0 or p = 1, meaning there is no variability.

For large n and p not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution with the same mean and standard deviation. This is useful for quick probability calculations, but the exact binomial standard deviation is always the one given here.

Frequently Asked Questions

What is the difference between binomial standard deviation and sample standard deviation?

The binomial standard deviation is a theoretical value that describes the spread of the number of successes in a binomial experiment. It is calculated from the parameters n and p. A sample standard deviation is computed from actual observed data and uses a different formula (dividing by n−1 for a sample). They serve different purposes: one predicts variability, the other measures observed variability.

Can I use this calculator for probability of failure instead of success?

Yes. If you replace p with the probability of failure (1−p), the standard deviation remains the same because the formula uses p(1−p), which is symmetric. The mean will change, but the spread is identical.

What if p is 0 or 1?

If p = 0, the number of successes is always 0, so the standard deviation is 0. If p = 1, the number of successes is always n, so the standard deviation is also 0. The calculator will show 0 for these cases.

How do I find the standard deviation for a proportion instead of a count?

To get the standard deviation of the sample proportion (number of successes divided by n), divide the count standard deviation by n: σ_proportion = √(p(1−p)/n). The calculator gives the count standard deviation; you can divide by n yourself if needed.

Why is the standard deviation larger when p is 0.5?

The product p(1−p) reaches its maximum at p = 0.5. At that point, the distribution is most spread out because there is maximum uncertainty about whether a trial will be a success or failure. As p moves toward 0 or 1, the outcome becomes more predictable, so the standard deviation decreases.

Does the binomial standard deviation depend on the order of successes?

No. The binomial distribution only counts the total number of successes, not their order. The standard deviation is the same regardless of whether successes occur early or late in the sequence.

Related

• Bonferroni Adjustment Calculator: Formula and Examples
• Binomial Distribution Calculator: Probability and Formula
• Binomial Formula Calculator: Expand (a + b)^n

in n | Number of trials (n) | - | 100 | 1
in p | Probability of success (p) | - | 0.5 | 0
out sd | Standard deviation | - | sqrt(n * p * (1 - p)) | 4
out var | Variance | - | n * p * (1 - p) | 4
out mean | Mean (expected value) | - | n * p | 2
out cv | Coefficient of variation | - | (sqrt(n * p * (1 - p))) / (n * p) | 4