Binomial Distribution Calculator with Steps, PMF & CDF

A binomial distribution calculator gives the probability of getting exactly x successes in n independent trials when each trial succeeds with the same probability p.

For example, the chance of exactly 5 heads in 10 fair coin tosses is 0.2461. "At least" and "at most" questions add up the individual probabilities. The mean is np and the standard deviation is √(np(1 − p)).

Rule of thumb: always use the exact binomial value. Use the normal approximation only as a check, and only when both np and n(1 − p) are at least 10.

The binomial distribution describes how many successes you get when you repeat the same yes-or-no trial a fixed number of times. A trial is one attempt: one coin toss, one seed, one patient, one free throw. A success is simply the outcome you are counting. It does not have to be a good thing. A defective item can be a "success" if defects are what you count.

Four conditions make a count binomial. People remember them as BINS:

• B, Binary outcome. Each trial has only two results: success or failure.
• I, Independent trials. One result does not change the chance of the next.
• N, Number of trials is fixed. You decide n before you start.
• S, Same probability. The chance of success p is the same on every trial.

If all four hold, the number of successes X follows a binomial distribution, written X ~ Binomial(n, p) or B(n, p). The distribution tells you the probability of every possible count from 0 to n.

A worked one-liner: a seed supplier says 60% of seeds germinate. You sow 80 seeds in a tray. You expect np = 80 × 0.60 = 48 to germinate, with a standard deviation of √(80 × 0.60 × 0.40) = 4.38. The chance that 55 or more germinate is only 0.0675, so a tray of 55 is unusual but not impossible.

Who uses it: doctors count patients who respond to a drug. Quality engineers count defective items per batch. Ecologists count eggs that hatch per nest. Geneticists count offspring with a trait. Pollsters count yes answers. Sports analysts count made shots. Marketers count visitors who buy. Any time the question is "how many out of n?", the binomial distribution is the starting point.

4.1 What the headline probability means

The big number at the top of the results is the exact probability for the question you picked. If it says P(X ≥ 55) = 0.0675, it means: if you repeated the whole experiment of 80 trials many times, about 6.75% of those repeats would give 55 or more successes. It is a long-run frequency, not a promise about one run.

Small probabilities do not mean "impossible". A probability of 0.01 still happens once in every hundred repeats. What a small probability does tell you is that, if your values of n and p are correct, the outcome is surprising. When a surprising outcome actually happens, the first thing to doubt is usually the value of p, not your luck.

4.2 Exactly, at most, at least, and between: choosing the right question

Most wrong answers come from asking the wrong question, not from bad arithmetic. Read the words in your problem carefully and match them to the table below.

Because X is a whole number, the difference between "<" and "≤" is a whole bar of the distribution. With continuous distributions such as the normal it makes no difference; with the binomial it can change the answer a lot.

4.3 Mean, variance and standard deviation

The mean μ = np is the number of successes you expect on average. The variance np(1 − p) measures how much the count moves around that average, and the standard deviation is its square root. A handy rule: about 95% of outcomes fall within two standard deviations of the mean when np and n(1 − p) are both reasonably large.

The spread is largest when p = 0.5 and shrinks towards zero as p nears 0 or 1. That is why a 50/50 coin gives much more variable results than a process that almost always succeeds. It is also why the standard error of a proportion is widest at 50%, which is the value survey planners use when they do not know p.

4.4 The mode and the shape

The mode is the single most likely count. It is the whole number part of (n + 1)p. When (n + 1)p is a whole number, there are two modes of equal height, for example n = 9 and p = 0.5 gives modes of 4 and 5.

Skewness tells you which way the tail points. It is zero when p = 0.5, positive when p is below 0.5 (a long tail of large counts) and negative when p is above 0.5. The skew fades as n grows, which is why large binomials look bell-shaped.

4.5 When the normal approximation is safe

Older textbooks approximate the binomial with a normal curve because exact sums were hard to compute by hand. This calculator shows the approximation next to the exact value so you can see the error. The approximation uses a continuity correction: P(X ≥ 55) becomes the normal area above 54.5, because the bar for 55 covers 54.5 to 55.5.

The usual rule is that both np and n(1 − p) should be at least 10, although some books say 5. Even when the rule passes, the approximation is weakest in the tails, which is exactly where p-values live. For reporting, always use the exact value. The approximation is a teaching and checking tool.

4.6 Reading p̂ and its confidence interval

When you enter observed counts, the calculator adds them up and divides by the total number of trials. That pooled proportion is p̂ ("p-hat"), the best single estimate of the true success probability. The confidence interval gives a range of values for the true p that are consistent with your data.

A 95% interval does not mean there is a 95% chance the true p is inside this particular interval. It means that if you repeated the study many times, 95% of the intervals built this way would contain the true p. In practice you can read it as "the plausible range for p". A narrow interval means a precise estimate; it gets narrower roughly with the square root of the number of trials.

4.7 Clopper-Pearson versus Wilson intervals

The Clopper-Pearson interval is called "exact" because it inverts the exact binomial test. It is guaranteed to cover the true p at least 95% of the time, which makes it conservative: it is often a little wider than it needs to be. Regulators and many clinical journals prefer it for exactly that guarantee.

The Wilson score interval is usually a little narrower and has coverage very close to 95% on average. Brown, Cai and DasGupta (2001) recommend it for general use. The simple Wald interval, p̂ ± 1.96·SE, is the one to avoid: it can have coverage far below 95% for small n or for p near 0 or 1, and it can even run below 0. This calculator reports both good intervals so you can choose, and states which one it used.

4.8 The exact binomial test

The exact binomial test asks whether your observed number of successes is consistent with a hypothesised probability p₀. The p-value is the probability, if p₀ were true, of a result at least as extreme as yours. For a two-sided test this calculator uses the same rule as R’s binom.test and SciPy’s binomtest: it adds the probabilities of every count that is no more likely than the one you observed.

A p-value below α (usually 0.05) is called statistically significant. It says the data are hard to explain if p₀ were true. It does not say how big or important the difference is. For that, look at the difference p̂ − p₀, the confidence interval, and Cohen’s h, an effect size for proportions where 0.2 is small, 0.5 medium and 0.8 large.

4.9 The dispersion index and binomial goodness of fit

If your counts are truly binomial, their variance should be close to n·p̂·(1 − p̂). The dispersion index is the observed variance divided by that value. Near 1 means the binomial fits. Clearly above 1 means overdispersion: the counts vary more than a binomial allows, usually because p differs between trays, clinics or litters, or because trials within a group are linked.

The calculator tests this with the chi-square dispersion test described by Tarone (1979). A small p-value for overdispersion is a real warning. The binomial test and intervals will then look more certain than they should, so switch to a beta-binomial or quasi-binomial model. An index well below 1 (underdispersion) is rarer and often means the counts were constrained or recorded with rounding.

Chart 4 shows the same check visually. The bars are how often each count actually occurred; the purple line is how often a binomial with your p̂ says it should occur. Bars that spill wider than the line on both sides are the picture of overdispersion.

4.10 Decision guide

4.11 Common mistakes this calculator prevents

• Off-by-one in "at least". P(X ≥ x) is 1 − P(X ≤ x − 1), not 1 − P(X ≤ x). The calculator does this for you and shows the step.
• Entering p as a percentage. Typing 60 instead of 0.60 is caught with a clear message.
• Counts larger than n. Any observed value above n or below 0 is flagged, because it cannot come from a binomial.
• Trusting the normal approximation in the tails. The exact value is always the headline; the approximation is shown only as a check with its error.
• Using the Wald interval. It is not reported at all. You get Clopper-Pearson and Wilson.
• Ignoring overdispersion. The fit check runs automatically for every group with two or more observations.

4.12 What the binomial cannot tell you

The binomial model assumes you already know n and that p stays constant. It cannot tell you why the success rate is what it is, and it cannot fix a biased sample: 60 successes from a hand-picked set of 100 trials say nothing reliable about all trials. It also cannot compare two groups formally. For that, use a two-proportion test, Fisher’s exact test or logistic regression, which model the difference directly.

▶ Run the analysis above to auto-fill all five examples with your results.

A good binomial result states four things: the count and the total (x out of N), the proportion with a confidence interval, the test and its p-value against a named p₀, and an effect size or plain difference. If you used observed counts, add one sentence on the fit check. Say which interval method you used, because Clopper-Pearson and Wilson give slightly different limits.

This flowchart shows how to calculate a binomial distribution probability step by step, and then how to test real counts against it. Every box uses the same worked example as the Excel, R and Python sections: a supplier claims 60% germination, each tray holds n = 80 seeds, and 12 trays were counted.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return P(X ≥ 55) = 0.0675, p̂ = 0.6604 and an exact p-value of 0.00013 for this dataset.

1. Enter the number of trials n. Type n, the fixed number of yes-or-no trials in one experiment, for example 80 seeds per tray.
2. Enter the probability of success p. Type p as a decimal from 0 to 1. For 60% type 0.6. If you are testing a claim, this is the hypothesised value p0.
3. Pick the probability to find. Choose exactly, at most, fewer than, at least, more than, or between. For between, a second box appears for the upper value.
4. Enter x. Type the number of successes you care about, for example 55. The calculator handles the x minus 1 in "at least" for you.
5. Add observed counts if you have them. Paste comma-separated success counts such as 52, 48, 55, 61, 47 into a group box. Each value is successes out of n in one observation. Rename the group by typing in its name field.
6. Or upload a CSV or Excel file. On the Upload tab choose a file, click every column you want, and press Load. Each selected column becomes its own group.
7. Try a sample dataset. Ten built-in datasets cover seeds, clinical trials, free throws, defects, surveys, nests, quizzes, 0/1 conversions, coin tosses and an overdispersed failure case.
8. Choose the confidence level and test direction. Pick 90, 95 or 99% and a two-sided or one-sided alternative for the exact binomial test.
9. Press Calculate Binomial Probability. Nothing is computed until you press the button, and any change clears old results so you never read stale numbers.
10. Read, chart and export. Read the plain-English interpretation, check the four charts and the fit check, copy a reporting template from Section 5, and export a text report or PDF.

Excel has a full set of binomial functions, so you can calculate any binomial probability without add-ins. The hard parts are knowing which argument makes the result cumulative, getting "at least" right, and building the exact test and confidence interval, which Excel does not provide as single functions. This section walks through all of it with the spreadsheet shown at each stage.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases: the exact test and the dispersion check

Excel has no BINOM.TEST function. For a one-sided test that the rate is higher than 60%, the p-value is simply P(X ≥ 634) with N = 960:

That returns 0.000066. For a two-sided test, the quick spreadsheet method doubles the smaller tail:

This "doubling" rule gives 0.000132, very slightly different from the 0.000130 that R, SciPy and this calculator report, because they add up every outcome that is no more likely than the one observed. With symmetric distributions (p = 0.5) the two rules agree exactly. See 9.4.

The dispersion statistic needs only built-in functions. With p̂ in K1:

DEVSQ gives the sum of squared deviations directly. The chi-square is 10.79 on 11 degrees of freedom, p = 0.461, so there is no sign of overdispersion.

9.4 Why your Excel answer might differ from R or Python

9.5 Charting the binomial distribution in Excel

1. Select the x column (D) and the P(X = x) column (E) from the table in step 4.
2. Insert → Charts → Clustered Column.
3. Right-click a bar, choose Format Data Series, and set Gap Width to about 10% so the bars nearly touch, the usual look for a discrete distribution.
4. To shade your question region, add a third column with =IF(D2>=$B$3,E2,0), add it as a second series, and set Series Overlap to 100% with an orange fill.
5. Label the axes "Number of successes" and "Probability", and put n and p in the title.

Chart 1 in Section 3 of this page draws the same graph with your question shaded automatically.

9.6 Excel errors you will probably hit

R has the four binomial functions built in: dbinom for the pdf, pbinom for the cdf, qbinom for quantiles and rbinom for random draws, plus binom.test for the exact test. The script below is complete: it answers the probability question, estimates p from the tray counts, runs the exact test, computes both confidence intervals, checks for overdispersion, prints a plain-English verdict and saves one publication-ready figure at 300 dpi. It uses base R only.

10.1 The complete script

R, base only, saves binomial_plot.png

# ============================================================
# Binomial distribution: probabilities, exact test, CI and fit
# StatsUnlock - https://statsunlock.com/binomial-distribution-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. The model and your question ------------------------
n  <- 80       # trials per observation (seeds per tray)
p0 <- 0.60     # claimed probability of success
x  <- 55       # we want P(X >= 55)

# ---- 2. Your observed counts (one value per tray) ----------
# Or read a CSV:
#   counts <- read.csv("data.csv")$germinated
counts <- c(52, 48, 55, 61, 47, 50, 58, 53, 49, 56, 51, 54)
counts <- counts[!is.na(counts)]           # drop missing values

# ---- 3. Probabilities --------------------------------------
cat(sprintf("P(X = %d)  = %.4f\n", x, dbinom(x, n, p0)))
cat(sprintf("P(X <= %d) = %.4f\n", x, pbinom(x, n, p0)))
cat(sprintf("P(X >= %d) = %.4f   # 1 - pbinom(x - 1)\n", x,
            pbinom(x - 1, n, p0, lower.tail = FALSE)))
cat(sprintf("mean = %.4f  variance = %.4f  SD = %.4f\n",
            n * p0, n * p0 * (1 - p0), sqrt(n * p0 * (1 - p0))))

# ---- 4. Estimate p, exact test, confidence intervals -------
k  <- length(counts)
xs <- sum(counts)
N  <- k * n
bt <- binom.test(xs, N, p = p0, alternative = "two.sided", conf.level = 0.95)
ph <- xs / N
z  <- qnorm(0.975)
wil <- (ph + z^2 / (2 * N) + c(-1, 1) * z * sqrt(ph * (1 - ph) / N + z^2 / (4 * N^2))) /
       (1 + z^2 / N)
cat(sprintf("\n%d successes of %d trials, p-hat = %.4f\n", xs, N, ph))
cat(sprintf("95%% CI exact  = [%.4f, %.4f]\n", bt$conf.int[1], bt$conf.int[2]))
cat(sprintf("95%% CI Wilson = [%.4f, %.4f]\n", wil[1], wil[2]))
cat(sprintf("exact binomial test p-value = %.5f\n", bt$p.value))

# ---- 5. Dispersion check (Tarone chi-square) ---------------
D      <- sum((counts - mean(counts))^2) / (n * ph * (1 - ph))
p_disp <- pchisq(D, df = k - 1, lower.tail = FALSE)
cat(sprintf("dispersion chi2(%d) = %.2f, p = %.3f, index = %.2f\n",
            k - 1, D, p_disp, D / (k - 1)))

verdict <- if (bt$p.value < 0.05) "differs from" else "is consistent with"
fit     <- if (p_disp < 0.05) "overdispersed: use a beta-binomial model" else "binomial fit OK"
cat(sprintf("\nVerdict: the germination rate %s %.0f%% (%s).\n", verdict, p0 * 100, fit))

# ---- 6. One 300 dpi figure: PMF with the question shaded ---
set.seed(42)                         # nothing random here, kept for habit
mu <- n * p0; s <- sqrt(n * p0 * (1 - p0))
ks <- max(0, floor(mu - 5 * s)):min(n, ceiling(mu + 5 * s))
png("binomial_plot.png", width = 2700, height = 1650, res = 300)
par(mar = c(5, 5, 5, 2))
bp <- barplot(dbinom(ks, n, p0), names.arg = ks,
              col = ifelse(ks >= x, "#ea580c", "#86efac"), border = "#15803d",
              xlab = "Seeds germinated out of 80", ylab = "Probability", las = 1)
abline(v = bp[ks == round(mu)], col = "#2563eb", lty = 2, lwd = 2)
points(bp[match(counts, ks)], rep(0, k), pch = 17, col = "#7c3aed", cex = 1.4, xpd = TRUE)
title(sprintf("Binomial(n = %d, p = %.2f):  P(X >= %d) = %.4f", n, p0, x,
              pbinom(x - 1, n, p0, lower.tail = FALSE)), adj = 0, line = 2.6)
title(sprintf("purple triangles = observed trays, p-hat = %.3f", ph),
      adj = 0, line = 1.2, font.main = 1, cex.main = 0.9)
legend("topright", bty = "n", fill = c("#ea580c", "#86efac"),
       legend = c(sprintf("X >= %d", x), sprintf("X < %d", x)))
dev.off()
cat("Saved binomial_plot.png\n")

Expected console output: P(X = 55) = 0.0257, P(X ≥ 55) = 0.0675, p-hat = 0.6604, exact CI [0.6295, 0.6904], p-value = 0.00013, dispersion chi2(11) = 10.79, p = 0.461.

10.2 What each part does

10.3 What the figure shows

The bars are the binomial probabilities for every count from about 26 to 70, the range where almost all the probability sits. Bars at 55 and above are orange: their total height is P(X ≥ 55) = 0.0675. The dashed blue line marks the mean, 48. The purple triangles along the bottom are the 12 real trays.

Reading it takes a few seconds. Most triangles sit to the right of the mean, and several are in or near the orange region. That picture already tells you the supplier’s 60% is probably too low, which the exact test then confirms with p = 0.00013.

10.4 Common R problems and fixes

10.5 Useful one-liners

In Python the binomial distribution lives in scipy.stats.binom, and the exact test is scipy.stats.binomtest, which also returns Clopper-Pearson and Wilson intervals. The script below reproduces every number from the R section, prints a plain-English verdict and saves one 300 dpi figure.

Install once with pip install numpy scipy matplotlib.

11.1 The complete script

Python 3, saves binomial_plot.png

# ============================================================
# Binomial distribution: probabilities, exact test, CI and fit
# StatsUnlock - https://statsunlock.com/binomial-distribution-calculator/
# Needs: numpy, scipy, matplotlib   (pip install numpy scipy matplotlib)
# ============================================================
import numpy as np
from scipy import stats
import matplotlib
matplotlib.use("Agg")                  # save to file, no window needed
import matplotlib.pyplot as plt

# ---- 1. The model and your question ------------------------
n  = 80        # trials per observation (seeds per tray)
p0 = 0.60      # claimed probability of success
x  = 55        # we want P(X >= 55)

# ---- 2. Your observed counts (one value per tray) ----------
# Or read a CSV:
#   import pandas as pd
#   counts = pd.read_csv("data.csv")["germinated"].dropna().astype(int).to_numpy()
counts = np.array([52, 48, 55, 61, 47, 50, 58, 53, 49, 56, 51, 54])
counts = counts[~np.isnan(counts)].astype(int)   # drop missing values

# ---- 3. Probabilities --------------------------------------
B = stats.binom(n, p0)
print(f"P(X = {x})  = {B.pmf(x):.4f}")
print(f"P(X <= {x}) = {B.cdf(x):.4f}")
print(f"P(X >= {x}) = {B.sf(x - 1):.4f}   # sf(x-1), NOT sf(x)")
print(f"mean = {B.mean():.4f}  variance = {B.var():.4f}  SD = {B.std():.4f}")

# ---- 4. Estimate p, exact test, confidence intervals -------
k, xs = len(counts), int(counts.sum())
N = k * n
res = stats.binomtest(xs, N, p0, alternative="two-sided")
cp = res.proportion_ci(confidence_level=0.95, method="exact")    # Clopper-Pearson
wi = res.proportion_ci(confidence_level=0.95, method="wilson")
print(f"\n{xs} successes of {N} trials, p-hat = {xs / N:.4f}")
print(f"95% CI exact  = [{cp.low:.4f}, {cp.high:.4f}]")
print(f"95% CI Wilson = [{wi.low:.4f}, {wi.high:.4f}]")
print(f"exact binomial test p-value = {res.pvalue:.5f}")

# ---- 5. Dispersion check (Tarone chi-square) ---------------
ph = xs / N
D = ((counts - counts.mean()) ** 2).sum() / (n * ph * (1 - ph))
p_disp = stats.chi2.sf(D, k - 1)
print(f"dispersion chi2({k - 1}) = {D:.2f}, p = {p_disp:.3f}, index = {D / (k - 1):.2f}")

verdict = "differs from" if res.pvalue < 0.05 else "is consistent with"
fit = "overdispersed: use a beta-binomial model" if p_disp < 0.05 else "binomial fit OK"
print(f"\nVerdict: the germination rate {verdict} {p0:.0%} ({fit}).")

# ---- 6. One 300 dpi figure: PMF with the question shaded ---
np.random.seed(42)                     # nothing random here, kept for habit
ks = np.arange(max(0, int(B.mean() - 5 * B.std())), min(n, int(B.mean() + 5 * B.std())) + 1)
fig, ax = plt.subplots(figsize=(9, 5.5))
ax.bar(ks, B.pmf(ks), color=np.where(ks >= x, "#ea580c", "#86efac"),
       edgecolor="#15803d", linewidth=0.6)
ax.axvline(B.mean(), color="#2563eb", ls="--", lw=2, label=f"mean = {B.mean():.0f}")
for c in counts:
    ax.plot(c, -0.002, marker="^", color="#7c3aed", ms=8, clip_on=False)
ax.set_xlabel("Seeds germinated out of 80")
ax.set_ylabel("Probability")
ax.set_title(f"Binomial(n = {n}, p = {p0}):  P(X >= {x}) = {B.sf(x - 1):.4f}\n"
             f"purple triangles = observed trays, p-hat = {ph:.3f}", loc="left")
ax.legend(frameon=False)
fig.tight_layout()
fig.savefig("binomial_plot.png", dpi=300)
print("Saved binomial_plot.png")

Expected output: P(X = 55) = 0.0257, P(X ≥ 55) = 0.0675, p-hat = 0.6604, exact CI [0.6295, 0.6904], Wilson [0.6299, 0.6897], p-value = 0.00013, dispersion chi2(11) = 10.79, p = 0.461.

11.2 What each part does

11.3 What the figure shows

It is the same picture as the R figure: binomial probabilities for Binomial(80, 0.6), the P(X ≥ 55) region in orange, the mean as a dashed blue line, and the 12 observed trays as purple triangles below the axis. The title prints P(X ≥ 55) = 0.0675 and p-hat = 0.660.

11.4 Common Python problems and fixes

11.5 Useful one-liners

12.1 Binomial probability table, P(X = x), n = 10

What to conclude: Each column sums to 1. For p above 0.5, read the table with x replaced by n − x and p by 1 − p, because the binomial is symmetric in that way.

12.2 Cumulative binomial table, P(X ≤ x), n = 10

What to conclude: For "at least x", take 1 minus the value in the row for x − 1. Example: n = 10, p = 0.5, P(X ≥ 8) = 1 − 0.9453 = 0.0547.

12.3 Mean, SD and the normal-approximation rule

What to conclude: The approximation needs both np and n(1 − p) of at least 10. With p near 0 or 1 you need a much larger n.

12.4 Three 95% confidence interval methods compared

What to conclude: With 0 successes the Wald interval collapses to [0, 0], which is plainly wrong. Wilson and Clopper-Pearson stay sensible; Clopper-Pearson is always the widest.

12.5 Cohen’s h effect size bands for proportions

What to conclude: Report h beside the p-value. With very large samples even negligible differences become significant.

12.6 Trials needed for "at least one success"

What to conclude: From P(X ≥ 1) = 1 − (1 − p)ⁿ. A 1% event needs 299 trials to be 95% sure of seeing it at least once.

12.7 Total trials for a target 95% margin of error

What to conclude: N = 1.96² p(1 − p) ÷ margin². Use p = 0.5 when unsure; it gives the largest N.

Eight worked examples covering the cases you will actually meet: a textbook problem, a clearly significant result, a borderline one, a null result, a tiny sample, a significant but trivial difference, a failed assumption, and a real applied decision. Every number was computed with the same engine as the calculator.

Study design: the binomial model assumes repeated observations of a fixed number of independent yes-or-no trials, each with the same probability of success. Collect data so that every one of those conditions is true by design, not by hope.

1. Define "success" in writing before you start. A germinated seed must mean something exact, for example a radicle at least 2 mm long after 7 days. Vague definitions let the rate drift between observers.
2. Fix n for every observation. Sow exactly 80 seeds per tray, give exactly 20 patients per clinic the treatment, take exactly 10 shots per session. If n must vary, record it per row and analyse proportions with a binomial GLM instead.
3. Make trials independent. Randomise the order, keep physical separation where contagion or crowding could link outcomes, and avoid scoring the same subject twice.
4. Keep p constant within a group. Use one seed lot, one supplier, one protocol. If you suspect p differs between sites or days, make those separate groups so the calculator tests each one.
5. Decide p₀ and where it comes from. A supplier claim, a historical rate, a regulatory limit, or chance level (0.5 for a coin, 0.25 for four-option guessing). Write it down with its source.
6. Choose one-sided or two-sided before seeing data. One-sided is only honest when a result in the other direction would lead to the same action as no result.
7. Plan the total number of trials. The width of a 95% interval is about 2 × 1.96 × √[p(1 − p) ÷ N]. For ±5 percentage points at p = 0.5 you need about 385 trials in total.
8. Record counts, not percentages. Store "52 of 80", never "65%". Percentages lose n and cannot be tested.
9. Record missing observations as blank, never as 0. A zero is a real outcome (no successes); a blank is no data.
10. Lay the data out one column per group, one row per observation, exactly as below, ready to paste or upload.

One column per group, whole-number counts only below the header row. Blank cells are ignored, so unequal numbers of observations per group are fine. Every value must lie between 0 and n.

Minimum sample: the exact test works for any N, but power is low when N is small. For the dispersion check you need at least 2 observations per group, and 8 or more to detect moderate overdispersion.

Pre-registration: record n, p₀, the test direction, α, the interval method (Clopper-Pearson or Wilson) and the fallback model for overdispersion before collecting data.

Common collection mistakes:

• Letting n change silently between observations, for example trays with 78 or 83 seeds.
• Scoring successes from the same litter, class or plot as if they were independent.
• Mixing seed lots or clinics with different true rates into one group.
• Recording percentages or rounding counts, which destroys the binomial structure.
• Choosing p₀ or a one-sided test after looking at the result.

Use a binomial distribution calculator whenever the question is "how many successes out of a fixed number of tries?"

Use it when:

• ✓ Each trial has exactly two outcomes.
• ✓ The number of trials n is fixed in advance.
• ✓ Trials are independent of each other.
• ✓ The probability of success is the same on every trial.
• ✓ You want an exact probability, a test against a known rate, or a confidence interval for one proportion.

Do not use it when:

• ✗ You sample without replacement from a small population (more than about 5% of it). Use the hypergeometric distribution.
• ✗ There is no fixed n, only a rate of events in time or space. Use the Poisson distribution.
• ✗ You count trials until the first success. Use the geometric distribution.
• ✗ There are three or more outcomes. Use the multinomial distribution.
• ✗ p varies between groups and you want to model why. Use logistic regression or a beta-binomial model.
• ✗ You compare two proportions. Use a two-proportion z-test or Fisher’s exact test.

Real-world examples:

1. Agriculture: is a seed lot’s germination rate really 60%?
2. Medicine: does a new drug beat the historical 40% response rate?
3. Manufacturing: is a batch with 5 defects out of 50 evidence the line is above its 4% target?
4. Genetics: do offspring follow a 3:1 Mendelian ratio (p = 0.75)?
5. Education: is a student scoring 10 of 20 on a four-option quiz above chance?
6. Ecology: what share of nests hatch all four eggs?
7. Marketing: what conversion rate is plausible after 400 visits with 38 sales?

Design guidance: plan the number of trials from the interval width you need, keep n identical across observations, and randomise the order of trials.

Decision tree: two outcomes and fixed n? → binomial. No fixed n, just a rate? → Poisson. Drawing without replacement from a small pool? → hypergeometric. Counts vary far more than binomial? → beta-binomial. Comparing two groups? → Fisher’s exact or two-proportion test. Explaining p with predictors? → logistic regression.

Assumptions

1. Binary outcomes. Check: every trial is scored success or failure. If it fails: forcing three outcomes into two throws information away; use the multinomial.
2. Fixed number of trials. Check: n is the same for every observation. If it fails: pooled p̂ is biased towards observations with more trials; use a binomial GLM with trials as weights.
3. Independent trials. Check: the design, and the dispersion index. If it fails: positive dependence inflates variance, so p-values are too small and intervals too narrow (anti-conservative).
4. Constant probability p. Check: the Tarone dispersion test and Chart 4. If it fails: overdispersion, with the same anti-conservative bias; use beta-binomial or a GLMM.
5. Correct p₀. Check: its source. If it fails: every probability and test is answered for the wrong model.
6. Representative sampling. Check: how observations were chosen. If it fails: p̂ is precisely wrong, biased in the direction of the selection.

Limitations

• The Clopper-Pearson interval is conservative: its coverage is above the nominal level, so it is wider than necessary. The Wilson interval is shown for a tighter alternative.
• The exact test is discrete, so its true type I error rate is at or below α, never exactly α. A mid-p value is a common, less conservative alternative.
• The dispersion test has low power with few observations; passing it with k = 3 proves little.
• The normal approximation is shown only as a check and can be badly wrong in the tails or when np < 10.
• The calculator tests one group at a time against p₀. It does not compare groups with each other; use Fisher’s exact test or logistic regression.
• It does not fit a beta-binomial model. When overdispersion is flagged, refit in R (aod, VGAM) or Python (statsmodels).
• n is limited to 100,000 per observation for speed in the browser. Beyond that, the normal approximation is extremely accurate anyway.

A binomial distribution calculator answers one of the most common questions in statistics: how likely is a given number of successes when you repeat the same yes-or-no trial a fixed number of times? Three inputs, n, p and x, are enough to give the exact probability for exactly, at most, at least or between, together with the mean np, the variance np(1 − p) and the standard deviation.

The arithmetic is not the hard part. The hard parts are asking the right question and checking the model. "At least 55" is 1 − P(X ≤ 54), not 1 − P(X ≤ 55), and that single off-by-one error causes more wrong homework answers and wrong Excel sheets than anything else in this topic. Always read the words of the problem and match them to the correct symbol before you calculate.

When you have real data, the binomial becomes a tool for inference. Pool the counts to estimate p̂, give it an exact Clopper-Pearson or Wilson confidence interval, and use the exact binomial test to compare it with a claimed rate. Report the count and total, the interval, the p-value and an effect size, so readers can judge both significance and importance. In the worked example, 634 germinations out of 960 seeds gave p̂ = 0.660, well above the claimed 60%, with p = 0.00013.

Finally, never skip the fit check. The binomial assumes one constant p and independent trials. If your counts vary much more than a binomial allows, the dispersion index will rise well above 1 and every p-value will look more convincing than it should. That is the signal to move to a beta-binomial or mixed model rather than to report a false certainty.

Enter your n, p and x above, add your counts if you have them, and read the plain-English interpretation before you write anything down. Then use the five reporting templates to put the result into your paper, thesis or report. Come back to this binomial distribution calculator whenever you need an exact binomial probability, a binomial test or a confidence interval for a proportion.

These peer-reviewed articles support the methods used in this binomial distribution calculator: exact and approximate confidence intervals, the exact binomial test, the normal approximation, overdispersion and the software used in the worked examples.

1. Agresti, A., & Caffo, B. (2000). Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures. The American Statistician, 54(4), 280–288. https://doi.org/10.1080/00031305.2000.10474560
2. Agresti, A., & Coull, B. A. (1998). Approximate is better than “exact” for interval estimation of binomial proportions. The American Statistician, 52(2), 119–126. https://doi.org/10.1080/00031305.1998.10480550
3. Blyth, C. R. (1986). Approximate binomial confidence limits. Journal of the American Statistical Association, 81(395), 843–855. https://doi.org/10.1080/01621459.1986.10478343
4. Brown, L. D., Cai, T. T., & DasGupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science, 16(2), 101–133. https://doi.org/10.1214/ss/1009213286
5. Clopper, C. J., & Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4), 404–413. https://doi.org/10.1093/biomet/26.4.404
6. Fagerland, M. W., Lydersen, S., & Laake, P. (2015). Recommended confidence intervals for two independent binomial proportions. Statistical Methods in Medical Research, 24(2), 224–254. https://doi.org/10.1177/0962280211415469
7. Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337–350. https://doi.org/10.1007/s10654-016-0149-3
8. Hinde, J., & Demétrio, C. G. B. (1998). Overdispersion: Models and estimation. Computational Statistics & Data Analysis, 27(2), 151–170. https://doi.org/10.1016/S0167-9473(98)00007-3
9. Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine, 17(8), 857–872. https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E
10. Schader, M., & Schmid, F. (1989). Two rules of thumb for the approximation of the binomial distribution by the normal distribution. The American Statistician, 43(1), 23–24. https://doi.org/10.1080/00031305.1989.10475601
11. Skellam, J. G. (1948). A probability distribution derived from the binomial distribution by regarding the probability of success as variable between the sets of trials. Journal of the Royal Statistical Society: Series B, 10(2), 257–261. https://doi.org/10.1111/j.2517-6161.1948.tb00014.x
12. Tarone, R. E. (1979). Testing the goodness of fit of the binomial distribution. Biometrika, 66(3), 585–590. https://doi.org/10.1093/biomet/66.3.585
13. Thulin, M. (2014). The cost of using exact confidence intervals for a binomial proportion. Electronic Journal of Statistics, 8(1), 817–840. https://doi.org/10.1214/14-EJS909
14. Virtanen, P., Gommers, R., Oliphant, T. E., et al. (2020). SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nature Methods, 17, 261–272. https://doi.org/10.1038/s41592-019-0686-2
15. Vollset, S. E. (1993). Confidence intervals for a binomial proportion. Statistics in Medicine, 12(9), 809–824. https://doi.org/10.1002/sim.4780120902
16. Wallis, S. (2013). Binomial confidence intervals and contingency tests: Mathematical fundamentals and the evaluation of alternative methods. Journal of Quantitative Linguistics, 20(3), 178–208. https://doi.org/10.1080/09296174.2013.799918
17. Warton, D. I., & Hui, F. K. C. (2011). The arcsine is asinine: The analysis of proportions in ecology. Ecology, 92(1), 3–10. https://doi.org/10.1890/10-0340.1
18. Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129–133. https://doi.org/10.1080/00031305.2016.1154108
19. Williams, D. A. (1982). Extra-binomial variation in logistic linear models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 31(2), 144–148. https://doi.org/10.2307/2347977
20. Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22(158), 209–212. https://doi.org/10.1080/01621459.1927.10502953