A proportion calculator divides the number of cases that have a feature by the total number of cases, so the answer always sits between 0 and 1.

If 33 of 60 patients improved, the proportion is 33 ÷ 60 = 0.550, or 55.0%. A proportion on its own is only half the answer, so this tool also returns the standard error, a Wilson score confidence interval, an exact interval, and a z test against any value you choose.

Rule of thumb: report the Wilson score interval, not the Wald interval, and only trust the normal approximation when the number of successes and the number of failures are both at least 10.

A proportion is a part divided by a whole, and a proportion calculator is simply a tool that does that division and then tells you how much to trust the answer. You count how many cases have the feature you care about, call that x, divide by the total number of cases n, and you get a number between 0 and 1. Multiply by 100 and you have a percentage. Nothing else about a proportion is complicated; everything that follows is about how much a proportion from a sample can be trusted.

The eleven quantities this calculator reports for every group:

• Successes (x), the count of cases that meet your rule.
• Sample size (n), the count of all valid cases.
• Sample proportion (p̂), x divided by n, and the same value as a percentage.
• Standard error, how much p̂ would bounce around from sample to sample.
• Margin of error, the half width of the Wald interval, the number opinion polls quote.
• Wald confidence interval, the textbook interval, shown for comparison only.
• Wilson score confidence interval, the recommended interval and the one to report.
• Agresti-Coull interval, the add-two-successes-and-two-failures shortcut.
• Clopper-Pearson exact interval, built from the binomial distribution, safe at any n.
• One sample z test against a hypothesised proportion you set.
• Odds and the log odds, for anyone moving on to logistic regression.

With two groups it adds the risk difference, risk ratio, odds ratio, a two sample proportion test and Cohen's h. With three or more it adds a chi-square test of homogeneity.

A worked one-liner: for the five values 52, 48, 55, 61, 47 with the rule "counts as a success at or above 50", three values qualify, so x = 3, n = 5 and p̂ = 0.600. The Wald interval is wildly optimistic at this sample size, which is exactly why the Wilson interval is the default here.

Who uses which: clinical trials report the proportion of patients who respond and the risk difference between arms. Pollsters report a proportion with a margin of error. Quality control reports the proportion defective against a target. Ecology reports the proportion of plots occupied. Education reports the proportion of students at or above a benchmark score, which is exactly the cutoff mode this tool opens with.

A note on wording: "proportionality" in mathematics means two quantities that scale together, as in a direct proportion a/b = c/d. That is a different idea from the statistical proportion on this page. Section 4.8 solves the mathematical kind so both meanings are covered.

Paste, upload or type your numbers, then press Run Analysis. The proportion calculator computes nothing until you do, so you can change the cutoff, the confidence level and the group names as many times as you like first.

A number from a proportion calculator is easy to compute and easy to misread. This section walks through every quantity the tool returns, says in plain words what it means, and shows the sentence you would actually write. Wherever you see live numbers, they come from the analysis you just ran; before you run it you will see underscores instead.

Each block below fills with your own numbers the moment you press Run Analysis.

4.1 The point estimate

The sample proportion p̂ is your single best guess at the population proportion. It is unbiased, which means that if you repeated the study many times the average of all the p̂ values would land on the true value. Being unbiased is not the same as being accurate in one study. A single p̂ from a small sample can sit a long way from the truth, and the confidence interval is the honest statement of how far.

Always report p̂ with the counts behind it. "55%" tells the reader nothing about whether it came from 11 of 20 or 550 of 1000. "0.550 (33/60)" tells them everything.

4.2 The standard error, and why 0.5 is the worst case

The standard error is the typical distance between a sample proportion and the population proportion. Its formula, √(p̂(1−p̂)/n), has two moving parts. The n in the denominator means precision improves with the square root of sample size, so to halve your error you need four times the data. The p̂(1−p̂) term peaks at p̂ = 0.5, where it equals 0.25, and shrinks towards zero as the proportion approaches 0 or 1.

That second point has a practical consequence. A rare outcome, say a 2% complication rate, has a small standard error in absolute terms but a huge one relative to the estimate itself. Two percent plus or minus one percentage point is a range that spans a threefold difference in risk. Never judge precision by the width of the interval alone; judge it against the size of the effect you care about.

4.3 Which confidence interval to report

This calculator gives you four intervals on purpose, because they disagree exactly where it matters most.

• Wald is the one in most textbooks: p̂ ± z·SE. It is simple, it is what a margin of error means, and it is the least accurate of the four. Its real coverage can fall to about 85% when it claims 95%, and it can produce limits below 0 or above 1, which are impossible values.
• Wilson score inverts the score test instead of the Wald test. It never leaves the 0 to 1 range, it stays close to nominal coverage even at n = 10, and it is asymmetric around p̂, pulled slightly towards 0.5. This is the interval to report.
• Agresti-Coull adds two successes and two failures, then applies the Wald formula to the adjusted counts. It is a hand-computable approximation to Wilson and is almost always close to it.
• Clopper-Pearson exact is built directly from the binomial distribution, so its coverage is guaranteed to be at least the stated level. The price is that it is conservative, meaning it is wider than it strictly needs to be. Use it when x or n−x is under 10, when a regulator asks for an exact method, or when you must never overstate precision.

A 95% confidence interval does not mean there is a 95% chance the true proportion is inside this particular interval. It means the procedure that built it captures the true value 95% of the time across repeated studies. In practice you report the interval and let the reader see the range of values your data are compatible with.

4.4 The one sample proportion test

The one sample test asks whether your observed proportion is far enough from a fixed value p₀ to be surprising. Note that the standard error in the test uses p₀, not p̂, because under the null hypothesis the true proportion is p₀. That is why the test and the Wald interval can occasionally disagree, one saying significant and the other including p₀. When they disagree, trust the score based results, which means the test statistic here and the Wilson interval, because they use the same standard error and are internally consistent.

Choose p₀ before you look at the data. Common choices are 0.5 for a coin or a two option preference, a published benchmark, a regulatory threshold, or last year's figure.

4.5 Comparing two proportions

With two groups the tool reports three different ways of expressing the same comparison, and they answer different questions.

• Risk difference (p̂₁ − p̂₂) is on the natural scale of the outcome. It is what tells you how many extra people are affected, and its reciprocal is the number needed to treat.
• Risk ratio (p̂₁ ÷ p̂₂) says how many times more likely the outcome is. It is easy to say aloud and easy to exaggerate: a doubling of a very rare risk is still a very small risk.
• Odds ratio compares odds rather than probabilities. It is the quantity logistic regression estimates and the only one that works in a case-control design, but it overstates the risk ratio whenever the outcome is common.

Report the risk difference with its confidence interval as your headline, because that is the number a decision rests on, and give the ratio alongside it.

4.6 Effect size: Cohen's h

A p-value tells you whether an effect is detectable. It does not tell you whether the effect is big. Cohen's h fixes that for proportions. It transforms each proportion with φ = 2·arcsin(√p) and takes the difference, which stabilises the variance so a given h means the same thing whether the proportions are near 0.5 or near 0.05.

Conventional bands are 0.20 small, 0.50 medium and 0.80 large. Treat them as a rough language, not a law. In a clinical setting an h of 0.15 that prevents strokes matters far more than an h of 0.8 in a preference survey about packaging colour.

4.7 Assumption checks and what a failure costs you

Four things have to hold. Each observation must be independent of every other, the outcome must be genuinely binary, the sample must be drawn from the population you want to talk about, and the counts must be large enough for the normal approximation. The tool checks the last one directly and flags the others as design questions only you can answer.

Failure has a direction. Clustered data, for example several measurements from the same school or the same patient, makes the standard error too small, the interval too narrow and the p-value too small, so you find effects that are not there. A biased sampling frame shifts p̂ itself and no amount of extra data will fix it. Small counts make the normal approximation unreliable in both directions, which is why the exact interval is reported alongside.

4.8 Direct proportion and proportionality, the other meaning

People searching for a proportionality calculator or a solve proportion calculator often want the algebraic kind: given a/b = c/d with one unknown, find it. The rule is cross multiplication, a·d = b·c, so the missing term is whichever product divided by the known partner. For 3/4 = x/20, cross multiply to get 4x = 60, so x = 15. A direct proportion means y = kx, and doubling x doubles y. An inverse proportion means y = k/x, and doubling x halves y.

The statistical proportion on this page is a special case of the first idea, where the whole is fixed at n and the part is x, so the answer is always between 0 and 1. If your question is about scaling a recipe, a map or a mixture, the cross multiplication rule is all you need. If it is about how confident you are in a share measured on a sample, this is the right page.

4.9 A decision guide

Find your situation in the left column. It tells you what this proportion calculator should report and which test belongs with it.

4.10 Common mistakes this proportion calculator prevents

• Reporting a percentage with no denominator. The tool always prints x and n beside p̂.
• Using the Wald interval at small n. The Wilson interval is shown first and the exact interval appears whenever the counts are small.
• An interval that runs below 0 or above 1. Wilson and exact cannot do this, and the tool flags it when Wald does.
• Comparing two proportions by checking whether their intervals overlap. That test is far too conservative. The tool gives you the proper two sample test instead.
• Treating paired data as independent. The assumption panel warns you and names McNemar's test.
• Quoting a risk ratio with no baseline risk. The tool prints the risk difference beside every ratio.
• Chasing significance with no effect size. Cohen's h is reported automatically for two groups.

When reporting proportion calculator results in APA format, give the counts, the proportion, the confidence interval and the test statistic in one sentence, in that order. Every card below fills itself with your own numbers after you run the analysis, and every card has a Copy button. Paste it into your manuscript and edit only the variable names.

Seven formulas do all the work in this proportion calculator. They are shown in the order the calculator applies them, from the point estimate outward to the test and the effect size.

Ten steps, start to finish. The worked example runs alongside: five test scores, 52, 48, 55, 61, 47, with a pass mark of 50.

Everything this proportion calculator does can be rebuilt in a plain Excel sheet with no add-ins. The worked example below is the same one used in sections 9, 10 and 12: Clinic A, 33 of 60 patients improved, compared later with Clinic B, 21 of 55. Every number you see in the screenshots is the number Excel actually returns.

8.1 The functions you need

8.2 Step by step

8.3 The awkward cases

• Text codes instead of numbers. If your column holds "Yes" and "No", use =COUNTIF(A2:A61,"Yes") and =COUNTA(A2:A61). Watch for trailing spaces, which make "Yes " fail to match.
• Blanks in the middle of the range. COUNT skips them, which is usually what you want, but state in your write-up how many cases were dropped.
• Percentages already entered. If a cell shows 55% Excel stores 0.55, so do not divide by 100 again.
• A proportion of exactly 0 or 1. The Wald interval collapses to zero width. Use the Wilson formula, which still returns a sensible one sided bound.
• Weighted survey data. Excel has no built-in weighted proportion. Use =SUMPRODUCT(B2:B61,W2:W61)/SUM(W2:W61) and treat the effective sample size as smaller than n.

8.4 Why Excel, R, Python and this page can disagree

They agree to six decimals if you set them up the same way, and they disagree in three specific places if you do not.

8.5 Charting it in Excel

Select the group names and the proportions, then insert a clustered column chart. To add the confidence interval, click the series, choose Add Chart Element, then Error Bars, then More Error Bar Options, and set Custom values pointing at two helper columns holding the distance from p̂ down to the lower limit and up to the upper limit. Because the Wilson interval is asymmetric, the plus and minus values are genuinely different, so do not use the single "Fixed value" option.

Set the vertical axis maximum to 1 rather than letting Excel auto-scale, otherwise a difference of two percentage points can be drawn to look enormous.

8.6 Error messages and what they mean

You do not need this proportion calculator to do any of this. Base R has everything you need: prop.test for the score interval and test, and binom.test for the exact interval. The script below is complete, runs top to bottom with no packages installed, prints a plain English verdict and saves one 300 dpi figure. Change only the data block at the top.

9.1 The complete script

# ============================================================
# Proportion calculator in R
# Sample proportion, Wilson / exact confidence intervals,
# one sample and two sample proportion tests, Cohen's h.
# Base R only. Saves proportion_plot.png at 300 dpi.
# ============================================================

set.seed(42)

# ---- 1. YOUR DATA -------------------------------------------------
# Change only this block.
# Option A: raw values plus a cutoff rule (comma separated)
scores <- c(52, 48, 55, 61, 47, 50, 58, 53)
cutoff <- 50
successes_A <- sum(scores >= cutoff)
n_A         <- length(scores)

# Option B: you already have the counts. This is the worked example.
successes_A <- 33 ; n_A <- 60    # Clinic A
successes_B <- 21 ; n_B <- 55    # Clinic B

# From a CSV instead:
# df <- read.csv("clinic.csv")
# df <- df[!is.na(df$improved), ]          # drop missing values
# successes_A <- sum(df$improved == 1) ; n_A <- nrow(df)

conf  <- 0.95
p0    <- 0.50

# ---- 2. POINT ESTIMATE AND STANDARD ERROR -------------------------
p_A <- successes_A / n_A
p_B <- successes_B / n_B
se_A <- sqrt(p_A * (1 - p_A) / n_A)
z    <- qnorm(1 - (1 - conf) / 2)

# ---- 3. CONFIDENCE INTERVALS --------------------------------------
wald <- c(p_A - z * se_A, p_A + z * se_A)

wilson <- local({
  d  <- 1 + z^2 / n_A
  ct <- (p_A + z^2 / (2 * n_A)) / d
  hw <- z * sqrt(p_A * (1 - p_A) / n_A + z^2 / (4 * n_A^2)) / d
  c(ct - hw, ct + hw)
})

agresti <- local({
  nt <- n_A + z^2
  pt <- (successes_A + z^2 / 2) / nt
  pt + c(-1, 1) * z * sqrt(pt * (1 - pt) / nt)
})

exact <- binom.test(successes_A, n_A, conf.level = conf)$conf.int

# ---- 4. ONE SAMPLE TEST -------------------------------------------
se0  <- sqrt(p0 * (1 - p0) / n_A)
z1   <- (p_A - p0) / se0
pv1  <- 2 * (1 - pnorm(abs(z1)))

# ---- 5. TWO SAMPLE TEST -------------------------------------------
p_pool <- (successes_A + successes_B) / (n_A + n_B)
z2     <- (p_A - p_B) / sqrt(p_pool * (1 - p_pool) * (1 / n_A + 1 / n_B))
pv2    <- 2 * (1 - pnorm(abs(z2)))
rd     <- p_A - p_B
se_rd  <- sqrt(p_A * (1 - p_A) / n_A + p_B * (1 - p_B) / n_B)
rd_ci  <- rd + c(-1, 1) * z * se_rd
h      <- abs(2 * asin(sqrt(p_A)) - 2 * asin(sqrt(p_B)))

# ---- 6. PLAIN ENGLISH VERDICT -------------------------------------
cat("PROPORTION RESULTS\n")
cat("------------------\n")
cat(sprintf("Clinic A: %d of %d, p-hat = %.6f (%.1f%%)\n", successes_A, n_A, p_A, 100 * p_A))
cat(sprintf("Standard error       : %.6f\n", se_A))
cat(sprintf("Wald      95%% CI     : %.4f to %.4f\n", wald[1], wald[2]))
cat(sprintf("Wilson    95%% CI     : %.4f to %.4f   <-- report this one\n", wilson[1], wilson[2]))
cat(sprintf("Agresti-Coull 95%% CI : %.4f to %.4f\n", agresti[1], agresti[2]))
cat(sprintf("Exact     95%% CI     : %.4f to %.4f\n", exact[1], exact[2]))
cat(sprintf("One sample z vs %.2f  : z = %.6f, p = %.6f\n", p0, z1, pv1))
cat(sprintf("Clinic B: %d of %d, p-hat = %.6f\n", successes_B, n_B, p_B))
cat(sprintf("Two sample z         : z = %.6f, p = %.6f\n", z2, pv2))
cat(sprintf("Risk difference      : %.6f, 95%% CI %.4f to %.4f\n", rd, rd_ci[1], rd_ci[2]))
cat(sprintf("Cohen's h            : %.6f\n", h))
cat("\nVERDICT: ")
if (pv2 < 0.05) {
  cat(sprintf("the two proportions differ significantly (p = %.4f).\n", pv2))
} else {
  cat(sprintf("no significant difference between the two proportions (p = %.4f).\n", pv2))
}
cat("The assumption check ")
if (min(successes_A, n_A - successes_A, successes_B, n_B - successes_B) >= 10) {
  cat("passed: every count is at least 10.\n")
} else {
  cat("FAILED: use the exact interval and Fisher's exact test.\n")
}

# Built-in check. correct = FALSE is required to match the numbers above.
print(prop.test(c(successes_A, successes_B), c(n_A, n_B), correct = FALSE))

# ---- 7. FIGURE ----------------------------------------------------
png("proportion_plot.png", width = 2000, height = 1400, res = 300)
par(mar = c(4.5, 4.5, 3, 1))
est <- c(p_A, p_B)
lo  <- c(wilson[1], NA)
hi  <- c(wilson[2], NA)
wB  <- local({
  d  <- 1 + z^2 / n_B
  ct <- (p_B + z^2 / (2 * n_B)) / d
  hw <- z * sqrt(p_B * (1 - p_B) / n_B + z^2 / (4 * n_B^2)) / d
  c(ct - hw, ct + hw)
})
lo[2] <- wB[1] ; hi[2] <- wB[2]
bp <- barplot(est, names.arg = c("Clinic A", "Clinic B"), ylim = c(0, 1),
              col = c("#2563eb", "#f59e0b"), border = NA,
              ylab = "Proportion improved",
              main = "Proportion with 95% Wilson score interval")
arrows(bp, lo, bp, hi, angle = 90, code = 3, length = 0.08, lwd = 2, col = "#1f2937")
abline(h = p0, lty = 2, col = "#b91c1c")
text(bp, est + 0.05, sprintf("%.3f", est), font = 2)
dev.off()
cat("\nFigure saved as proportion_plot.png (300 dpi).\n")

9.2 What each part does

9.3 What the figure shows

Two bars, one per clinic, drawn on a fixed 0 to 1 axis so the eye is not misled by auto-scaling. Each bar carries a whisker running from the lower to the upper Wilson limit, and the printed proportion sits above it. The dashed red line marks the hypothesised value of 0.50. If a whisker crosses that line, the group is not detectably different from 50%.

9.4 Problems and fixes

9.5 One-liners worth remembering

SciPy gives you the exact interval and the normal distribution, statsmodels gives you the standard proportion tests, and matplotlib draws the figure. The script below builds the Wilson and Agresti-Coull intervals by hand as well, so you can see exactly what statsmodels is doing when you call it.

Install with pip install numpy scipy statsmodels matplotlib. Only numpy, scipy and matplotlib are needed to run the script as written; statsmodels appears in the commented cross-check at the end.

10.1 The complete script

# ============================================================
# Proportion calculator in Python
# Sample proportion, Wilson / exact confidence intervals,
# one sample and two sample proportion tests, Cohen's h.
# Saves proportion_plot.png at 300 dpi.
# pip install numpy scipy statsmodels matplotlib
# ============================================================

import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from scipy import stats

np.random.seed(42)

# ---- 1. YOUR DATA -------------------------------------------------
# Change only this block.
# Option A: raw values plus a cutoff rule (comma separated)
scores = np.array([52, 48, 55, 61, 47, 50, 58, 53])
cutoff = 50
x_A, n_A = int((scores >= cutoff).sum()), scores.size

# Option B: you already have the counts. This is the worked example.
x_A, n_A = 33, 60      # Clinic A
x_B, n_B = 21, 55      # Clinic B

# From a CSV instead:
# import pandas as pd
# df = pd.read_csv("clinic.csv").dropna(subset=["improved"])   # drop missing values
# x_A, n_A = int(df["improved"].sum()), len(df)

conf = 0.95
p0 = 0.50

# ---- 2. POINT ESTIMATE AND STANDARD ERROR -------------------------
p_A, p_B = x_A / n_A, x_B / n_B
se_A = np.sqrt(p_A * (1 - p_A) / n_A)
z = stats.norm.ppf(1 - (1 - conf) / 2)

# ---- 3. CONFIDENCE INTERVALS --------------------------------------
def wilson(x, n, z):
    p = x / n
    d = 1 + z**2 / n
    centre = (p + z**2 / (2 * n)) / d
    half = z * np.sqrt(p * (1 - p) / n + z**2 / (4 * n**2)) / d
    return centre - half, centre + half

def agresti(x, n, z):
    nt = n + z**2
    pt = (x + z**2 / 2) / nt
    se = np.sqrt(pt * (1 - pt) / nt)
    return pt - z * se, pt + z * se

def exact(x, n, conf):
    a = 1 - conf
    lo = 0.0 if x == 0 else stats.beta.ppf(a / 2, x, n - x + 1)
    hi = 1.0 if x == n else stats.beta.ppf(1 - a / 2, x + 1, n - x)
    return lo, hi

wald = (p_A - z * se_A, p_A + z * se_A)
wil_A = wilson(x_A, n_A, z)
wil_B = wilson(x_B, n_B, z)
ac = agresti(x_A, n_A, z)
cp = exact(x_A, n_A, conf)

# ---- 4. ONE SAMPLE TEST -------------------------------------------
se0 = np.sqrt(p0 * (1 - p0) / n_A)
z1 = (p_A - p0) / se0
pv1 = 2 * (1 - stats.norm.cdf(abs(z1)))

# ---- 5. TWO SAMPLE TEST -------------------------------------------
p_pool = (x_A + x_B) / (n_A + n_B)
z2 = (p_A - p_B) / np.sqrt(p_pool * (1 - p_pool) * (1 / n_A + 1 / n_B))
pv2 = 2 * (1 - stats.norm.cdf(abs(z2)))
rd = p_A - p_B
se_rd = np.sqrt(p_A * (1 - p_A) / n_A + p_B * (1 - p_B) / n_B)
rd_ci = (rd - z * se_rd, rd + z * se_rd)
h = abs(2 * np.arcsin(np.sqrt(p_A)) - 2 * np.arcsin(np.sqrt(p_B)))

# ---- 6. PLAIN ENGLISH VERDICT -------------------------------------
print("PROPORTION RESULTS")
print("------------------")
print(f"Clinic A: {x_A} of {n_A}, p-hat = {p_A:.6f} ({100*p_A:.1f}%)")
print(f"Standard error       : {se_A:.6f}")
print(f"Wald      95% CI     : {wald[0]:.4f} to {wald[1]:.4f}")
print(f"Wilson    95% CI     : {wil_A[0]:.4f} to {wil_A[1]:.4f}   <-- report this one")
print(f"Agresti-Coull 95% CI : {ac[0]:.4f} to {ac[1]:.4f}")
print(f"Exact     95% CI     : {cp[0]:.4f} to {cp[1]:.4f}")
print(f"One sample z vs {p0:.2f}  : z = {z1:.6f}, p = {pv1:.6f}")
print(f"Clinic B: {x_B} of {n_B}, p-hat = {p_B:.6f}")
print(f"Two sample z         : z = {z2:.6f}, p = {pv2:.6f}")
print(f"Risk difference      : {rd:.6f}, 95% CI {rd_ci[0]:.4f} to {rd_ci[1]:.4f}")
print(f"Cohen's h            : {h:.6f}")

verdict = "differ significantly" if pv2 < 0.05 else "do not differ significantly"
print(f"\nVERDICT: the two proportions {verdict} (p = {pv2:.4f}).")
ok = min(x_A, n_A - x_A, x_B, n_B - x_B) >= 10
print("Assumption check " + ("passed: every count is at least 10."
      if ok else "FAILED: use the exact interval and Fisher's exact test."))

# Built-in check with statsmodels:
# from statsmodels.stats.proportion import proportions_ztest, proportion_confint
# print(proportions_ztest([x_A, x_B], [n_A, n_B]))
# print(proportion_confint(x_A, n_A, alpha=0.05, method="wilson"))

# ---- 7. FIGURE ----------------------------------------------------
fig, ax = plt.subplots(figsize=(7, 4.6), dpi=300)
labels = ["Clinic A", "Clinic B"]
est = [p_A, p_B]
lows = [est[0] - wil_A[0], est[1] - wil_B[0]]
highs = [wil_A[1] - est[0], wil_B[1] - est[1]]
bars = ax.bar(labels, est, color=["#2563eb", "#f59e0b"], width=0.55, zorder=2)
ax.errorbar(labels, est, yerr=[lows, highs], fmt="none",
            ecolor="#1f2937", elinewidth=2, capsize=7, zorder=3)
ax.axhline(p0, ls="--", color="#b91c1c", lw=1.2, zorder=1)
for b, v in zip(bars, est):
    ax.text(b.get_x() + b.get_width() / 2, v + 0.05, f"{v:.3f}",
            ha="center", fontweight="bold")
ax.set_ylim(0, 1)
ax.set_ylabel("Proportion improved")
ax.set_title("Proportion with 95% Wilson score interval")
ax.grid(axis="y", alpha=0.25, zorder=0)
fig.tight_layout()
fig.savefig("proportion_plot.png", dpi=300)
print("\nFigure saved as proportion_plot.png (300 dpi).")

10.2 What each part does

10.3 What the figure shows

The same picture as the R script: two bars on a 0 to 1 axis, asymmetric Wilson whiskers, the value printed above each bar, and a dashed red reference line at the hypothesised proportion. Because the Wilson interval is not symmetric around p̂, the lower and upper whisker lengths differ slightly, which is exactly why yerr is given as two separate lists.

10.4 Problems and fixes

10.5 One-liners worth remembering

Seven lookup tables you can use without running anything. Every value was computed with the same formulas this proportion calculator uses, so the page and the tables can never disagree.

Table 11.1 Critical z values

What to conclude: use the two tailed column unless you decided the direction of the effect before collecting data. At 95% the value is 1.9600, not 2.

Table 11.2 Margin of error at 95% confidence, worst case p = 0.50

What to conclude: precision improves with the square root of n. Going from 1000 to 2000 respondents only cuts the margin from 3.1 to 2.2 percentage points.

Table 11.3 Sample size needed for a given margin of error at 95% confidence

What to conclude: plan with p = 0.50 unless you have solid prior information, because it is the value that needs the largest sample. The familiar figure of about 1067 for a 3 point margin comes from this column.

Table 11.4 How the four intervals differ at small n (95% confidence)

What to conclude: the red Wald limits are impossible values, below 0 or above 1. At 0 out of 10 the Wald interval has zero width, which claims perfect certainty from ten observations. Wilson and exact never do this.

Table 11.5 Cohen's h and what it means

What to conclude: the sample size column is per group, two tailed, at alpha = 0.05, and comes from n = (1.96 + 0.84)² ÷ h² rounded up. The 0.34 row is the worked example on this page, which explains why p = 0.071 with only 60 and 55 patients.

Table 11.6 Minimum n for the normal approximation, by expected proportion

What to conclude: rare outcomes need very large samples. A 1% complication rate needs 1000 patients before the normal approximation is safe, which is why exact methods dominate in rare-event research.

Table 11.7 Proportion, percentage, odds and logit

What to conclude: odds and proportions are close for small p and diverge fast above about 0.20. That is why an odds ratio badly overstates a risk ratio when the outcome is common.

Eight worked examples run through the proportion calculator, each with a figure, the numbers, what they mean and the sentence you would write. They deliberately cover the good cases and the bad ones, because knowing what a broken result looks like is more useful than another clean one.

Design statement. Before you open a proportion calculator you need data worth putting in it. A proportion study measures one binary outcome on a defined set of units drawn from a defined population, and reports the share of units with the outcome together with an interval that reflects the sampling uncertainty. Everything below exists to make sure the denominator is honest, because a wrong denominator is the only error a proportion cannot recover from.

1. Define the population. Write down exactly who or what could have been sampled. "Adults in the district aged 18 to 65 registered with a clinic in 2025" is a definition; "patients" is not.
2. Define the sampling frame. Name the list you actually drew from and note, in writing, who is on the list but not in the population, and who is in the population but missing from the list.
3. Choose the sample size before collecting. Use Table 11.3. If you expect p near 0.5 and want a 5 point margin, plan for 385.
4. Define the outcome as a rule, not a judgement. "Score of 50 or more on the standard test" is a rule. "Doing well" is a judgement, and two raters will disagree on it.
5. Decide the cutoff and p₀ in advance. Record both in a pre-registration or a dated protocol file before you look at any data.
6. Record one row per unit. Never record a group summary if you can record individuals, because you cannot go back the other way.
7. Record missing values explicitly. Use a blank cell or NA, never a zero, since a zero silently becomes a failure.
8. Check independence. If two rows can come from the same person, the same household, the same school or the same litter, the observations are clustered and a plain proportion interval will be too narrow.
9. Log every exclusion with a reason. Your reported n must equal the number recruited minus the number excluded, and both figures belong in the paper.
10. Freeze the dataset before analysis. Save a read-only copy, then analyse a working copy.
11. Run the analysis once. Changing the cutoff after seeing the result invalidates the p-value.
12. Report the counts, not only the percentage. x and n go in the text or the table, always.

How to lay the data out

One row per unit, one column for the group, one for the raw measurement and one for the derived binary outcome. Keep the raw column even when you already have the binary one, so a reviewer can check your cutoff.

Common mistakes at the collection stage

• Counting the same person twice because they appear in two clinics.
• Using the number of responses as n when the denominator should be the number invited.
• Dropping non-responders silently, which turns a response rate question into a different study.
• Letting the cutoff drift between data collectors.
• Recording percentages instead of counts, which makes the interval impossible to compute later.

This free proportion calculator is designed for anyone who has a count out of a total and needs to say how confident they are in the share it represents.

The short version: use a proportion calculator when your outcome has exactly two states and every observation is independent of every other.

Use it when

• Your outcome has exactly two categories, such as pass or fail, improved or not, defective or good.
• Every observation is independent of every other.
• You want a confidence interval around a percentage, not just the percentage.
• You want to compare an observed rate with a benchmark, a target or a published figure.
• You are comparing two or more independent groups on a yes or no outcome.
• You are checking whether a survey result of, say, 55% is really different from half.

Do not use it when

• The same people are measured twice, before and after. Use McNemar's test.
• Your data are clustered inside schools, clinics, households or litters. Use a mixed effects logistic model.
• Your outcome has three or more unordered categories. Use a chi-square test of independence on the full table.
• You want to adjust for other variables such as age or sex. Use logistic regression.
• Your denominator is time or exposure rather than a fixed count. That is a rate, so use Poisson methods.
• The sample was not drawn from the population you want to describe. No calculator repairs a biased frame.

Four worked situations

Four everyday jobs and the proportion calculator setting each one needs.

1. Clinical. 33 of 60 patients improve. You want the response rate with an interval and a test against a published 50% benchmark. This tool, one sample mode.
2. Education. Two classes sit the same test and you want the proportion at or above the pass mark in each. This tool, cutoff mode with two groups.
3. Manufacturing. 2 of 40 units are defective and you must state the plausible defect rate. This tool, exact interval, no z test.
4. Marketing. An A/B test with 10,000 users per arm. This tool, two sample mode, and read Cohen's h before celebrating the p-value.

Decision tree

Twelve things that go wrong with a proportion calculator, and the fix for each.

A proportion calculator will return a number for almost any input. These are the conditions under which that number means what you think it means.

Assumptions

1. Binary outcome. Every unit is exactly one of success or failure, with no third state and no partial credit. Check: read the coding rule. If it fails: the denominator is ambiguous and the proportion is meaningless; use a categorical method instead.
2. Independence. Knowing one outcome tells you nothing about another. Check: ask whether any two rows share a person, household, clinic or classroom. If it fails: the standard error is too small, so intervals are too narrow and p-values too small. You will find effects that are not there.
3. Constant probability. Every unit has the same underlying chance of success. Check: look for obvious subgroups with different rates. If it fails: the single proportion hides real heterogeneity; report by subgroup or fit a model.
4. Random sample from the target population. Check: compare the sampling frame with the population definition. If it fails: p̂ itself is biased and no interval width fixes it. This is the only failure that gets worse with more data.
5. Large enough counts for the normal approximation. x ≥ 10 and n − x ≥ 10. Check: the tool does this automatically. If it fails: the z test and the Wald interval are unreliable in both directions; use the exact interval and an exact test.
6. Fixed n. The sample size was decided before the data were seen. Check: your protocol. If it fails: stopping when the result looks good inflates the false positive rate badly.

Limitations

• No covariate adjustment. This tool compares raw proportions only. If groups differ in age, severity or exposure, use logistic regression.
• No paired analysis. Before and after data on the same people need McNemar's test or a paired interval, neither of which this tool provides.
• No clustering correction. For data nested in schools or clinics, use a generalised linear mixed model, or at minimum inflate the variance by the design effect.
• No survey weights. Complex survey designs need a design based variance estimator, available in R's survey package.
• Omnibus chi-square only. With three or more groups the tool tells you a difference exists but not where; run pairwise tests with a Holm adjustment.
• Finite population correction is optional and simple. It assumes simple random sampling without replacement, which is not the same as a stratified or cluster design.
• Frequentist only. If you need a probability that the proportion lies in a range, that is a Bayesian credible interval, not a confidence interval.

A proportion calculator answers a question that looks trivial and is not. Dividing x by n takes a second. Deciding what belongs in x, what belongs in n, and how much a single sample can tell you about the population behind it is the part that takes judgement, and it is the part that decides whether your conclusion survives review.

Three habits carry most of the value. Always publish the counts beside the percentage, so a reader can rebuild your interval. Always report the Wilson score interval rather than the textbook Wald interval, because the Wald interval quietly overstates precision exactly where you are least able to afford it. And always read the effect size beside the p-value, because with a large enough sample a difference of two percentage points becomes statistically significant without becoming important.

Where the counts are small, or the outcome is rare, switch to the exact Clopper-Pearson interval and stop running z tests. It is wider, and that width is the honest answer rather than a defect. Where the design is paired, clustered or in need of adjustment, this page will point you at the right method rather than give you a comfortable wrong one.

Used carefully, the humble proportion calculator is one of the most reliable tools in applied statistics: transparent arithmetic, a well understood interval, and a test whose assumptions you can check on the same screen.

Twenty questions people actually search before and after using a proportion calculator, answered in one or two sentences each.

APA 7th edition

BibTeX

Methods paragraph

Six calculators on statsunlock.com that pick up exactly where this proportion calculator stops. Each one handles a question this proportion calculator points you towards but does not answer itself.

Every term this proportion calculator puts on screen, in plain English, alphabetically. No definition here uses a word that is not itself defined.

Twenty peer-reviewed journal articles behind the methods in this proportion calculator, in APA 7th edition, each with a resolvable DOI link. Every interval and test on the page traces back to one of them.

1. 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
2. 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
3. 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
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. 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
6. Newcombe, R. G. (1998). Interval estimation for the difference between independent proportions: Comparison of eleven methods. Statistics in Medicine, 17(8), 873–890. https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I
7. 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
8. Vollset, S. E. (1993). Confidence intervals for a binomial proportion. Statistics in Medicine, 12(9), 809–824. https://doi.org/10.1002/sim.4780120902
9. Fleiss, J. L. (1971). Measuring nominal scale agreement among many raters. Psychological Bulletin, 76(5), 378–382. https://doi.org/10.1037/h0031619
10. Yates, F. (1934). Contingency tables involving small numbers and the χ² test. Supplement to the Journal of the Royal Statistical Society, 1(2), 217–235. https://doi.org/10.2307/2983604
11. Cochran, W. G. (1954). Some methods for strengthening the common χ² tests. Biometrics, 10(4), 417–451. https://doi.org/10.2307/3001616
12. Armitage, P. (1955). Tests for linear trends in proportions and frequencies. Biometrics, 11(3), 375–386. https://doi.org/10.2307/3001775
13. McNemar, Q. (1947). Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika, 12(2), 153–157. https://doi.org/10.1007/BF02295996
14. Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159. https://doi.org/10.1037/0033-2909.112.1.155
15. Sterne, T. E. (1954). Some remarks on confidence or fiducial limits. Biometrika, 41(1–2), 275–278. https://doi.org/10.1093/biomet/41.1-2.275
16. Blyth, C. R., & Still, H. A. (1983). Binomial confidence intervals. Journal of the American Statistical Association, 78(381), 108–116. https://doi.org/10.1080/01621459.1983.10477938
17. 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
18. 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
19. 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
20. Altman, D. G., & Bland, J. M. (2011). How to obtain the confidence interval from a P value. BMJ, 343, d2090. https://doi.org/10.1136/bmj.d2090