Binomial Mean and Variance Calculator
With 20 trials and a 0.35 chance of success on each, a binomial count averages 7 successes with a variance of 4.55. The standard deviation shows how far results usually stray from that center, without building a full probability table. Free in your browser.
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Worked example
- N
- 20
- Power (W)
- 0.35
- Mean
- 7
- Variance
- 4.55
- Standard deviation
- 2.1330729008
A binomial distribution describes the number of successes observed across a fixed number of independent trials when every trial has the same chance of success. This calculator accepts the trial count n and success probability p, then returns the distribution's mean, variance, and standard deviation. It is useful for checking homework, planning experiments, summarizing repeated yes-or-no events, and preparing values for later statistical analysis without constructing a full probability table.
Enter trials and probability correctly
Start with the number of trials, written as n. It must be a nonnegative whole number because a binomial experiment cannot contain a fraction of a trial. Then enter p, the probability that one trial is a success.
For example, use 0.35 for a 35 percent chance, not 35. The model assumes that the trial count is fixed before the experiment begins, each trial has only two relevant outcomes, trials are independent, and the probability of success remains constant. Those assumptions matter more than the arithmetic.
Repeated coin flips, inspected products classified as pass or fail, and customers who either respond or do not respond can fit the model when the observations do not influence one another. Drawing cards without replacement usually does not fit because the success probability changes after each draw. The calculator rejects probabilities below 0 or above 1 instead of silently clipping them, so an invalid scale or typing mistake is visible immediately.
Understand the three returned values
The mean is n multiplied by p. It is the long-run average number of successes across many repetitions of the complete experiment, not a guarantee for any single run. The variance is n multiplied by p multiplied by one minus p.
Variance measures spread in squared-success units, which makes it mathematically convenient but sometimes less intuitive to interpret directly. The standard deviation is the square root of the variance, returning the spread to the same unit as the success count. Suppose a process has 20 trials and a success probability of 0.35.
The mean tells you where repeated results tend to center, while the standard deviation gives a practical sense of how far counts commonly move around that center. At p equal to 0 or 1, the result is certain, so variance and standard deviation are both zero. For a fixed n, variance is largest at p equal to 0.5 because uncertainty is greatest when success and failure are equally likely.
These relationships also provide quick checks for interpreting the output.
Use the result without overstating it
These moments summarize the distribution, but they do not replace the full probability mass function when you need the chance of an exact count or a tail event. Use the mean for an expected total, the variance for formulas that combine independent random quantities, and the standard deviation for an interpretable measure of typical dispersion. In quality control, the mean can estimate expected defects only when p represents the defect probability consistently across items.
In campaign planning, it can estimate expected responses only when recipients behave independently enough for the model to be defensible. A large difference between observed data and the returned spread may suggest that p varies between trials, outcomes are correlated, or the binomial assumptions otherwise fail. Keep enough precision for downstream calculations and round only when presenting the final result.
This capability performs direct deterministic arithmetic: it does not sample outcomes, estimate p from observations, produce confidence intervals, or test whether your data are binomial. If p comes from an estimate, uncertainty in that estimate is separate from the variance calculated here and may require additional analysis.
What you can do with it
Plan a repeated-trial experiment
Find the expected success count and its spread before collecting results from independent trials.
Check statistics coursework
Verify the three standard binomial moments from a trial count and success probability.
Summarize operational outcomes
Estimate the center and variability of pass-or-fail counts under a stable probability assumption.
FAQ
What formulas does the calculator use?
It uses mean = n × p, variance = n × p × (1 − p), and standard deviation = the square root of the variance.
Can p be entered as a percentage?
Enter p as a decimal probability. For example, enter 0.25 for 25 percent.
Why must p be between 0 and 1?
A probability cannot be negative or exceed certainty. Values outside that inclusive range return an invalid-input error.
Can n be zero?
Yes. With zero trials, the mean, variance, and standard deviation are all zero.
Does this calculate the probability of exactly k successes?
No. It returns distribution moments only; an exact-event calculation requires the binomial probability mass function.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/stat/binomial-mean-variance \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":20,"p":0.35}'const res = await fetch("https://api.kit.forhosting.com/stat/binomial-mean-variance", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 20,
"p": 0.35
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/stat/binomial-mean-variance",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 20,
"p": 0.35
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/stat/binomial-mean-variance", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":20,"p":0.35}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":20,"p":0.35}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/stat/binomial-mean-variance", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": 20,
"p": 0.35
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "stat.binomial_mean_variance",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Enter trials and probability correctly
Probability may be written as a decimal from 0 through 1, including both endpoints.
Technical questions
What does an API calculation cost?
Each API request costs $0.002. The browser calculator is available without an API request.