Confidence Interval Calculator: 95% CI Steps | StatsUnlock

A confidence interval calculator finds a range of plausible values for a population value: the mean plus or minus the margin of error, where the margin is the critical value times the standard error.

For 16 leaf lengths with mean 52.44 mm and SD 4.19 mm, the standard error is 4.19 ÷ √16 = 1.049, t* for 95% with 15 degrees of freedom is 2.131, so the margin of error is 2.235 and the 95% confidence interval is 50.20 to 54.67 mm. More data makes it narrower. A higher confidence level makes it wider.

Rule of thumb: use z = 1.645 for 90%, 1.960 for 95% and 2.576 for 99% when n is large. With n under 30 and an unknown SD, use t instead. For proportions, use the Wilson interval, not the textbook Wald formula, whenever np or n(1 − p) is below 10.

A confidence interval (CI) is a range of values, worked out from a sample, that is likely to contain the true value for the whole population. You almost never measure every member of a population, so the sample mean or sample proportion is only an estimate. The confidence interval puts honest error bars around that estimate.

What it measures: precision. A narrow interval says the sample pins the population value down tightly. A wide interval says the data are too few or too variable to say much. Every interval has the same three parts:

• Point estimate, your best single guess, such as the sample mean x̄ or the sample proportion p̂.
• Standard error, how much that estimate would wobble from sample to sample.
• Critical value, a multiplier from the t, z, chi-square or beta distribution chosen by your confidence level.

The margin of error is the critical value times the standard error. The interval is the estimate minus the margin to the estimate plus the margin.

A worked one-liner: 42 of 120 survey respondents said yes, so p̂ = 0.35. The Wilson 95% interval is 0.271 to 0.439, so the true share of yes answers in the population is plausibly anywhere from about 27% to 44%.

Who uses it: doctors report the 95% CI for a treatment effect, pollsters call the margin of error "plus or minus 3 points", ecologists put intervals around population density, engineers use them for tolerance limits, and every major journal style guide, including APA 7th, now asks for intervals beside p-values.

4.5 The correct way to read "95% confidence"

The 95% belongs to the method, not to your single interval. If you repeated your study many times and built an interval each time in the same way, about 95% of those intervals would contain the true population value. Your interval either contains it or it does not. You just do not know which.

So the safe sentence is: "We are 95% confident that the population mean lies between 50.2 and 54.7 mm." The risky sentence is: "There is a 95% probability that the true mean is between 50.2 and 54.7 mm." The second one is a Bayesian credible interval statement. Most reviewers will accept the first and query the second.

4.6 What the interval does not say

A 95% interval for the mean is not the range that holds 95% of the individual values. That range, called a prediction or reference interval, is far wider, roughly the mean plus or minus 2 standard deviations. For the leaf data, the 95% prediction interval for one new leaf runs from about 43.2 to 61.7 mm, while the 95% confidence interval for the mean is only 50.2 to 54.7 mm. Mixing the two up is the most common mistake in student reports.

It also does not say that values in the middle are much more likely than values near the ends in any strict sense, and it says nothing about bias. A biased sample gives a narrow interval around the wrong value.

4.7 Margin of error, width and precision

The margin of error is half the width of a two-sided interval. Width depends on three things only: the spread of the data (s or p(1 − p)), the sample size n, and the confidence level. Because the standard error divides by √n, doubling the sample size only cuts the width by about 29%. Halving the width needs four times as many observations. Chart 4 shows this curve for your own data.

A handy way to judge precision is the relative margin: margin divided by the estimate. Under 5% is usually called high precision, 5% to 10% moderate, and above 10% low. The calculator gives this as a verdict badge.

4.8 Choosing the confidence level

95% is a convention, not a law. A 90% interval is narrower and is used when a one-sided 5% test is the real question, as in bioequivalence work. A 99% interval is wider and is used when a wrong claim would be costly, as in safety limits. Whatever you choose, choose it before you see the data and report it. Changing from 95% to 90% after seeing that the 95% interval crossed zero is a form of p-hacking.

4.9 Using the interval as a test

A two-sided 95% interval contains exactly the null values that a two-sided test at α = 0.05 would not reject. So if the interval for a difference in means excludes 0, the t-test gives p < 0.05. If the interval for a proportion excludes 0.5, a test against 50% would be significant. The interval is more useful than the p-value because it also shows the size of the effect and every value the data are compatible with.

4.10 Overlapping intervals do not mean "no difference"

Two group intervals can overlap a little while the difference between the groups is still significant. When two 95% intervals overlap by less than about half of one margin of error, the difference is usually significant at about p < 0.05. The only reliable check is the interval for the difference itself, which this calculator gives whenever you enter two or more clusters.

4.11 Why the method matters for proportions

The textbook Wald formula, p̂ ± z√[p̂(1 − p̂)/n], gives intervals that are too short when n is small or p is near 0 or 1. Its real coverage can fall well below 95%, and with 0 events it gives a zero-width interval, which is plainly wrong. Brown, Cai and DasGupta (2001) showed this in detail and recommend the Wilson or Jeffreys interval. Clopper-Pearson is called "exact" because it guarantees at least 95% coverage, which also makes it wider than necessary. This tool defaults to Wilson for one proportion and Newcombe's method for a difference.

4.12 Small samples and skewed data

The t interval for a mean assumes the sampling distribution of the mean is close to normal. With n of 30 or more this holds for most data, thanks to the central limit theorem. With small samples, check the Shapiro-Wilk result and Chart 2. If the data are strongly skewed, the calculator also gives a distribution-free interval for the median, which needs no normality at all, and you may prefer to report that instead.

4.13 One-sided bounds

A one-sided lower bound answers "at least how large?". For example, "we are 95% confident the mean strength is at least 210 MPa". It uses t at 1 − α instead of 1 − α/2, so for the same data it sits closer to the estimate than the lower end of the two-sided interval. Only use it when the direction was fixed before the data were seen.

4.14 Practical judgement

Look first at where the whole interval sits, not just at the estimate. If every value in the interval would lead to the same practical decision, the result is conclusive even if it is wide. If the lower end means "no real effect" and the upper end means "a large effect", the study is inconclusive and the honest next step is more data, which the sample size option in Section 2 can plan for you.

When reporting confidence interval calculator results in APA format, give the estimate first, then the confidence level, then the lower and upper limits in square brackets with the units. APA 7th writes it as M = 52.44, 95% CI [50.20, 54.67].

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

5.1 Methods wording

Name the estimate, the interval method, the confidence level and the software: "Means are reported with two-sided 95% confidence intervals based on the t distribution with n − 1 degrees of freedom. Proportions are reported with Wilson score intervals. Analyses used the StatsUnlock Confidence Interval Calculator and were checked in R 4.4."

5.2 Results sentence

Lead with the estimate and the interval together, never the interval alone: "Mean leaf length was 52.44 mm, 95% CI [50.20, 54.67]." For a difference: "Treatment lowered systolic pressure by 9.67 mmHg relative to placebo, 95% CI [7.15, 12.19]." That is the Drug Trial sample dataset in the calculator, with the Welch interval.

5.3 Tables and figures

In a table, add a column headed "95% CI" with entries like "50.20 to 54.67". In a figure, draw the interval as an error bar and write in the caption that the bars are 95% confidence intervals, not standard deviations or standard errors. Unlabelled error bars are one of the most common reasons for revision (Cumming, Fidler and Vaux, 2007).

5.4 Discussion sentence

Interpret both ends of the interval. For a wider interval of 3 to 13 mmHg you might write: "The data are compatible with a reduction as small as 3 mmHg, which is of little clinical value, or as large as 13 mmHg, which would be important. A larger trial is needed to narrow this range."

5.5 Reporting checklist

• Estimate, confidence level and both limits are given together.
• Interval method named (t, z, Wilson, Clopper-Pearson, Welch, Newcombe).
• n, and for means the SD, are reported.
• One-sided bounds are labelled as one-sided.
• Error bars in figures are labelled as confidence intervals.
• For a group comparison, the interval for the difference is given.

This flowchart shows how to calculate a 95% confidence interval for a mean, step by step, from a raw column of numbers to the final lower and upper limits. Every box uses the same worked example as the Excel, R and Python sections: 16 leaf lengths measured to the nearest millimetre in a botany practical.

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 a mean of 52.4375 and a 95% confidence interval of 50.2023 to 54.6727 for this dataset.

1. Choose what you need an interval for. Pick a mean from raw data, a mean from summary numbers, one proportion, two proportions, or the sample size for a target margin of error.
2. Enter your data. In the default tab type comma-separated numbers exactly as the placeholder shows: 52, 48, 55, 61, 47, .... Spaces, tabs, semicolons and new lines also work, so a column pasted from Excel is fine.
3. Name each group. Click the group name above each box and type, for example Placebo or Site A. The name flows into the tables, charts and report.
4. Add more groups if you want to compare. Press Add cluster. Every extra cluster also gets an interval for its difference from the first cluster.
5. Or upload a file. On the Upload tab choose a CSV or Excel file, then click each column you want. Every column you select becomes its own cluster.
6. Pick the confidence level. 95% is the usual choice. 90%, 99% and any custom level such as 93% are also available.
7. Choose two-sided or one-sided. Two-sided gives a lower and an upper limit. One-sided gives only an "at least" or "at most" bound.
8. Press Calculate Confidence Interval. Nothing is computed before you do, and changing any input clears old results so you never read stale numbers.
9. Read the result and the checks. The top card gives the interval, the cards give the margin of error, standard error and critical value, and the checks table tells you if normality or the success-failure rule is a problem.
10. Copy or download. Section 5 fills five ready-to-paste report styles. The buttons save a text report or a print-ready PDF with the full working.

8.1 How to calculate a confidence interval on a TI-84

1. Press STAT, choose EDIT, and type your data into list L1.
2. Press STAT, move right to TESTS.
3. For a mean with unknown σ choose 8:TInterval. For a known σ choose 7:ZInterval. For a proportion choose A:1-PropZInt. For two proportions choose B:2-PropZInt.
4. Select Data if your values are in L1, or Stats to type x̄, Sx and n directly.
5. Set C-Level to .95, highlight Calculate and press ENTER.

For the leaf data the TI-84 TInterval shows (50.202, 54.673), the same as this calculator. Note that the TI-84 1-PropZInt uses the Wald formula, so for 42 of 120 it returns (0.2647, 0.4353), which matches the "Wald" row of this tool, not the recommended Wilson row.

Excel can calculate a 95% confidence interval for a mean in one formula, but the function returns only the margin of error, not the interval. That trips up nearly everyone the first time. You also need to pick between CONFIDENCE.T and CONFIDENCE.NORM, and Excel has no built-in function for a proportion interval at all. This section shows the whole sheet step by step.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

A one-sided bound. Use the one-tailed t: =C2-T.INV(0.95,C4-1)*C5 gives the 95% lower bound 50.5991. People often reuse T.INV.2T(0.05,...) here, which gives a 97.5% bound instead.

The Analysis ToolPak. Enable it first under File, Options, Add-ins, Manage Excel Add-ins, Analysis ToolPak. Then Data, Data Analysis, Descriptive Statistics, tick Confidence Level for Mean. The row it prints, "Confidence Level(95.0%)", is again the margin of error (2.2352), not the interval.

Difference between two means (Welch). With group 1 in A and group 2 in B: margin =T.INV.2T(0.05,df)*SQRT(VAR.S(A2:A11)/COUNT(A2:A11)+VAR.S(B2:B11)/COUNT(B2:B11)), where df is the Welch formula from card 6 in Section 6. Excel does not compute Welch df for you; T.TEST(...,2,3) gives only the p-value.

Exact Clopper-Pearson limits. =BETA.INV(0.025,42,79) returns 0.2652 and =BETA.INV(0.975,43,78) returns 0.4424. Wrap each in IF to return 0 when x = 0 and 1 when x = n.

9.4 Why your Excel answer might differ from R or Python

9.5 Charting the interval in Excel

1. Put group names in one column and means in the next. Select both.
2. Insert, Charts, Clustered Column (or a scatter if you prefer dots).
3. Click the chart, press the + button, tick Error Bars, then More Options, Custom, Specify Value, and select your margin-of-error cells for both the positive and negative values.
4. Write "Error bars show 95% confidence intervals" in the caption. Unlabelled bars are a common reason for journal revision.
5. For two or more groups, a dot-and-whisker plot (scatter with error bars) is clearer than bars, because a bar suggests the data start at zero.

9.6 Excel errors you will probably hit

R gives you a 95% confidence interval for a mean in one line, t.test(x)$conf.int, but that line hides the margin of error, the critical value and every check a reviewer will ask about. The script below is complete and runnable. It builds the interval by hand and confirms it with t.test(), adds the intervals for the SD, the median and a proportion, 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 confidence_interval.png

# ============================================================
# Confidence interval calculator: mean, SD, median and proportion
# StatsUnlock - https://statsunlock.com/confidence-interval-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your data ------------------------------------------
# Replace with your own values, or read a CSV:
#   x <- read.csv("data.csv")$leaf_length
x <- c(52, 48, 55, 61, 47, 50, 58, 53,
       49, 56, 51, 54, 46, 57, 50, 52)
x <- x[!is.na(x)]                      # drop missing values
conf  <- 0.95
label <- "Leaf length"
units <- "mm"

# ---- 2. t interval for the mean ----------------------------
n     <- length(x)
df    <- n - 1
m     <- mean(x)
s     <- sd(x)                         # sd() always uses n - 1
se    <- s / sqrt(n)
tcrit <- qt(1 - (1 - conf) / 2, df)
me    <- tcrit * se
ci    <- c(m - me, m + me)
tt    <- t.test(x, conf.level = conf)  # same interval in one line

# ---- 3. SD interval (chi-square) and median interval -------
chi   <- qchisq(c(1 - (1 - conf) / 2, (1 - conf) / 2), df)
sd_ci <- sqrt(df * s^2 / chi)

xs <- sort(x)
ks <- 1:(n %/% 2)
ok <- ks[pbinom(ks - 1, n, 0.5) <= (1 - conf) / 2]
if (length(ok)) {
  k      <- max(ok)
  med_ci <- c(xs[k], xs[n - k + 1])
  cover  <- 1 - 2 * pbinom(k - 1, n, 0.5)
}
sw <- shapiro.test(x)

# ---- 4. Proportion interval (Wilson and exact) -------------
succ <- 42; trials <- 120
wil  <- prop.test(succ, trials, conf.level = conf, correct = FALSE)$conf.int
cp   <- binom.test(succ, trials, conf.level = conf)$conf.int

# ---- 5. Print the results ----------------------------------
rel <- me / abs(m) * 100
cat(sprintf("n = %d, df = %d\n", n, df))
cat(sprintf("mean = %.4f, SD = %.4f, SE = %.4f\n", m, s, se))
cat(sprintf("t* = %.4f, margin of error = %.4f\n", tcrit, me))
cat(sprintf("%.0f%% CI for the mean: %.4f to %.4f\n", conf * 100, ci[1], ci[2]))
cat(sprintf("t.test() agrees:      %.4f to %.4f\n", tt$conf.int[1], tt$conf.int[2]))
cat(sprintf("%.0f%% CI for the SD:   %.4f to %.4f\n", conf * 100, sd_ci[1], sd_ci[2]))
if (length(ok)) cat(sprintf("CI for the median: %g to %g (exact coverage %.1f%%)\n",
                            med_ci[1], med_ci[2], cover * 100))
cat(sprintf("Shapiro-Wilk W = %.4f, p = %.4f\n", sw$statistic, sw$p.value))
cat(sprintf("Proportion %d/%d = %.4f\n", succ, trials, succ / trials))
cat(sprintf("  Wilson %.4f to %.4f; exact %.4f to %.4f\n", wil[1], wil[2], cp[1], cp[2]))
verdict <- if (rel < 5) "high" else if (rel < 10) "moderate" else "low"
cat(sprintf("Verdict: the margin of error is %.1f%% of the mean, so precision is %s.\n",
            rel, verdict))
if (n < 30 && sw$p.value < 0.05)
  cat("Warning: small sample and non-normal data, prefer the median interval.\n")

# ---- 6. One figure, two panels -----------------------------
png("confidence_interval.png", width = 3000, height = 1500, res = 300)
par(mfrow = c(1, 2), mar = c(4.5, 4.8, 3.2, 1), family = "sans")
set.seed(42)                           # seed = reproducible jitter
plot(runif(n, 0.75, 1.25), x, xlim = c(0.4, 1.6), xaxt = "n", xlab = "",
     ylab = sprintf("%s (%s)", label, units), pch = 21, cex = 1.4, las = 1,
     bg = adjustcolor("#16a34a", alpha.f = 0.6), col = "white",
     main = "Data, mean and 95% CI", cex.main = 1)
arrows(1.45, ci[1], 1.45, ci[2], angle = 90, code = 3, length = 0.08,
       lwd = 2.5, col = "#2563eb")
points(1.45, m, pch = 19, cex = 1.6, col = "#2563eb")

levels <- c(0.90, 0.95, 0.99)
cis <- t(sapply(levels, function(L) m + c(-1, 1) * qt(1 - (1 - L) / 2, df) * se))
plot(NA, xlim = range(cis), ylim = c(0.5, 3.6), yaxt = "n",
     xlab = sprintf("%s (%s)", label, units), ylab = "",
     main = "Wider confidence, wider interval", cex.main = 1)
axis(2, at = 1:3, labels = paste0(levels * 100, "%"), las = 1)
segments(cis[, 1], 1:3, cis[, 2], 1:3, lwd = 5,
         col = c("#0d9488", "#2563eb", "#9333ea"))
points(rep(m, 3), 1:3, pch = 19, cex = 1.3)
abline(v = m, lty = 2, col = "grey50")
text(cis[, 2], 1:3 + 0.2, sprintf("%.2f to %.2f", cis[, 1], cis[, 2]),
     adj = 1, cex = 0.75)
dev.off()
cat("Saved confidence_interval.png\n")

Run it with Rscript ci.R from a terminal, or paste it into the RStudio console. It prints the mean 52.4375, t* 2.1314, margin 2.2352 and the 95% interval 50.2023 to 54.6727, then the Wilson interval 0.2705 to 0.4388 for 42 of 120.

10.2 What each part does

10.3 What the figure shows

The left panel shows every leaf as a dot, with the mean and its 95% confidence interval as a blue error bar on the right. Notice how much shorter the error bar is than the spread of the dots: the interval describes the mean, not the individual leaves. The right panel draws the 90%, 95% and 99% intervals on the same axis so you can see that asking for more confidence always buys a wider interval.

10.4 Common R problems and fixes

10.5 Useful one-liners

In Python, scipy.stats.t.interval(0.95, df, loc, scale) gives the interval in one call, but only if you pass the standard error as scale, which is the most common mistake. The script below computes every part by hand, checks it, adds the SD, median and proportion intervals, prints a verdict and saves one 300 dpi figure.

Install once with pip install numpy scipy matplotlib. SciPy 1.10 or newer is needed for binomtest(...).proportion_ci.

11.1 The complete script

Python 3, saves confidence_interval.png

# ============================================================
# Confidence interval calculator: mean, SD, median and proportion
# StatsUnlock - https://statsunlock.com/confidence-interval-calculator/
# Needs numpy, scipy and matplotlib.
# ============================================================
import numpy as np
from scipy import stats
import matplotlib
matplotlib.use("Agg")                 # draw to file, no window needed
import matplotlib.pyplot as plt

# ---- 1. Your data ------------------------------------------
# Replace with your own values, or read a CSV:
#   import pandas as pd
#   x = pd.read_csv("data.csv")["leaf_length"].dropna().to_numpy()
x = np.array([52, 48, 55, 61, 47, 50, 58, 53,
              49, 56, 51, 54, 46, 57, 50, 52], dtype=float)
x = x[~np.isnan(x)]                   # drop missing values
conf = 0.95
label, units = "Leaf length", "mm"

# ---- 2. t interval for the mean ----------------------------
n = x.size
df = n - 1
m = x.mean()
s = x.std(ddof=1)                     # ddof=1 matches Excel STDEV.S and R sd()
se = s / np.sqrt(n)
tcrit = stats.t.ppf(1 - (1 - conf) / 2, df)
me = tcrit * se
lo, hi = m - me, m + me
lo2, hi2 = stats.t.interval(conf, df, loc=m, scale=se)   # same answer, one line

# ---- 3. SD interval (chi-square) and median interval -------
chi_hi, chi_lo = stats.chi2.ppf([1 - (1 - conf) / 2, (1 - conf) / 2], df)
sd_lo, sd_hi = np.sqrt(df * s**2 / chi_hi), np.sqrt(df * s**2 / chi_lo)

xs = np.sort(x)
ks = np.arange(1, n // 2 + 1)
ok = ks[stats.binom.cdf(ks - 1, n, 0.5) <= (1 - conf) / 2]
if ok.size:
    k = ok.max()
    med_lo, med_hi = xs[k - 1], xs[n - k]
    cover = 1 - 2 * stats.binom.cdf(k - 1, n, 0.5)

sw = stats.shapiro(x)

# ---- 4. Proportion interval (Wilson and exact) -------------
succ, trials = 42, 120
bt = stats.binomtest(succ, trials)
wil = bt.proportion_ci(confidence_level=conf, method="wilson")
cp = bt.proportion_ci(confidence_level=conf, method="exact")

# ---- 5. Print the results ----------------------------------
rel = me / abs(m) * 100
print(f"n = {n}, df = {df}")
print(f"mean = {m:.4f}, SD = {s:.4f}, SE = {se:.4f}")
print(f"t* = {tcrit:.4f}, margin of error = {me:.4f}")
print(f"{conf:.0%} CI for the mean: {lo:.4f} to {hi:.4f}")
print(f"{conf:.0%} CI for the SD:   {sd_lo:.4f} to {sd_hi:.4f}")
if ok.size:
    print(f"CI for the median: {med_lo:g} to {med_hi:g} (exact coverage {cover:.1%})")
print(f"Shapiro-Wilk W = {sw.statistic:.4f}, p = {sw.pvalue:.4f}")
print(f"Proportion {succ}/{trials} = {succ/trials:.4f}")
print(f"  Wilson {wil.low:.4f} to {wil.high:.4f}; exact {cp.low:.4f} to {cp.high:.4f}")
verdict = "high" if rel < 5 else "moderate" if rel < 10 else "low"
print(f"Verdict: the margin of error is {rel:.1f}% of the mean, so precision is {verdict}.")
if n < 30 and sw.pvalue < 0.05:
    print("Warning: small sample and non-normal data, prefer the median interval.")

# ---- 6. One figure, two panels -----------------------------
rng = np.random.default_rng(42)       # seeded so the jitter is reproducible
fig, (a1, a2) = plt.subplots(1, 2, figsize=(10, 5))
a1.scatter(rng.uniform(0.75, 1.25, n), x, s=60, color="#16a34a",
           alpha=0.6, edgecolor="white")
a1.errorbar(1.45, m, yerr=me, fmt="o", color="#2563eb", capsize=8,
            lw=2.5, ms=9, label=f"Mean and {conf:.0%} CI")
a1.set_xlim(0.4, 1.6)
a1.set_xticks([])
a1.set_ylabel(f"{label} ({units})")
a1.set_title("Data, mean and 95% CI", fontweight="bold")
a1.legend(frameon=False, loc="upper left")

levels = [0.90, 0.95, 0.99]
cols = ["#0d9488", "#2563eb", "#9333ea"]
for i, (L, c) in enumerate(zip(levels, cols), start=1):
    h = stats.t.ppf(1 - (1 - L) / 2, df) * se
    a2.hlines(i, m - h, m + h, color=c, lw=5)
    a2.plot(m, i, "o", color="black")
    a2.text(m + h, i + 0.15, f"{m - h:.2f} to {m + h:.2f}", ha="right", fontsize=9)
a2.axvline(m, ls="--", color="grey")
a2.set_yticks([1, 2, 3], [f"{L:.0%}" for L in levels])
a2.set_ylim(0.5, 3.6)
a2.set_xlabel(f"{label} ({units})")
a2.set_title("Wider confidence, wider interval", fontweight="bold")
fig.tight_layout()
fig.savefig("confidence_interval.png", dpi=300, bbox_inches="tight")
print("Saved confidence_interval.png")

11.2 What each part does

11.3 What the figure shows

Exactly the same two panels as the R figure: the raw data with the mean and its 95% interval, and the 90%, 95% and 99% intervals stacked so you can see the trade-off between confidence and width. The labels give the limits, 50.20 to 54.67 for 95%.

11.4 Common Python problems and fixes

11.5 Useful one-liners

Run it with python ci.py from a terminal. In Jupyter, drop the matplotlib.use("Agg") line and end with plt.show().

12.1 z critical values for common confidence levels

What to conclude: 1.645, 1.960 and 2.576 are the numbers to remember for 90%, 95% and 99%. Use them only when σ is known or n is large; otherwise use t from 12.2. Values from the standard normal quantile function (Neyman, 1937).

12.2 t critical values (two-sided) by degrees of freedom

What to conclude: with df = 15 the 95% value is 2.131, about 9% bigger than z = 1.960. The gap matters below about 30 df and almost vanishes after 120 (Student, 1908).

12.3 Margin of error for a proportion near 50%

What to conclude: a poll of 1,000 people has a 95% margin of about ±3.1 points. Going from 1,000 to 2,000 only shrinks it to ±2.2. Values use z√(0.25/n), the worst case p = 0.5.

12.4 Sample size for a proportion (p = 0.5)

What to conclude: 385 respondents give ±5 points at 95%, and 1,068 give ±3 points. Halving the margin quadruples n (Lenth, 2001).

12.5 Which interval method to use

What to conclude: the method matters most for proportions with small counts and for skewed small samples (Brown, Cai and DasGupta, 2001; Newcombe, 1998).

12.6 Wald, Wilson and exact intervals side by side (95%)

What to conclude: with 0 of 20 the Wald interval collapses to 0 to 0, while Wilson and the exact method give an honest upper limit near 0.16. With large counts, as in 42 of 120, all three agree to about one percentage point.

Study design: a confidence interval assumes a random (or at least representative) sample of independent units from one clearly defined population, measured the same way each time. For a mean the outcome is numeric on an interval or ratio scale. For a proportion each unit is a yes or no outcome.

1. Decide what you will put a confidence interval around before collecting. A mean, a proportion, a difference between groups, or a standard deviation. Each needs a different data layout and a different formula.
2. Define the population and the unit. One leaf, one patient, one household. Repeat readings on the same unit are not independent and make the interval falsely narrow.
3. Plan the sample size for your target margin of error. Use the sample size option in Section 2 with a pilot SD or an expected proportion. To estimate a proportion to ±5 points at 95% you need 385 units; to ±3 points, 1,068.
4. Choose the confidence level in advance. 95% unless your field or regulator says otherwise. Write it down.
5. Randomise selection. A confidence interval only covers random sampling error. It cannot correct for a convenience sample or for non-response.
6. Fix the measuring instrument, units and resolution. Changing scales or observers part-way adds hidden variation that widens the interval.
7. Record yes or no outcomes as counts. For a proportion keep the number of successes x and the number of trials n, not just the percentage, because the interval depends on n.
8. Record missing data as blanks, never zeros. A zero is a real value and shifts the mean and the SD.
9. Note the group of every unit if you plan to compare groups, so that each group becomes its own column (cluster).
10. Write the outlier rule before you see the data. Dropping awkward values afterwards to narrow a confidence interval is a form of bias.

One column per group, one row per unit, numbers only below the header row. Blank cells are ignored, so groups can be different sizes. Upload this layout and click each column to load it as a cluster.

Minimum sample: a t interval can be computed with n = 2, but it is very wide (t* = 12.71 at 95%). Aim for at least 10 per group for a usable interval and 30 or more if the data may be skewed. For a proportion the Wilson interval works at any n, but aim for at least 10 successes and 10 failures if you plan to report the Wald formula.

Pre-registration: state the estimate, the interval method (t, Wilson, Newcombe), the confidence level, the target margin of error and the planned n before data collection.

Common collection mistakes:

• Counting repeat measurements of one subject as separate units.
• Keeping only the percentage and losing the sample size.
• Pooling two different populations into one column.
• Stopping data collection as soon as the interval excludes zero.
• Recording "not measured" as 0.

This free confidence interval calculator is designed for students, researchers, analysts and survey teams who need to state how precisely a sample estimates a population value.

Use it when:

• ✓ You want a range for a population mean, from raw data or from a reported mean, SD and n.
• ✓ You want a range for a percentage or proportion, including rare events with 0 successes.
• ✓ You want the interval for the difference between two group means or two proportions.
• ✓ You need the margin of error for a survey or the sample size to reach a target margin.
• ✓ A journal asks for effect sizes with 95% confidence intervals instead of p-values alone.

Use something else when:

• ✗ The two groups are paired (before and after on the same people). Use a paired interval on the differences: paste the differences as one cluster.
• ✗ You want a range for one future observation. Use a prediction interval.
• ✗ You need intervals for regression slopes, odds ratios or hazard ratios. Use the matching model tool.
• ✗ The data come from a stratified or clustered survey design. Use design-based standard errors.
• ✗ You have prior information you want to include. Use a Bayesian credible interval.

Real-world examples:

1. Clinical trial: 95% CI for the mean drop in blood pressure under a new drug.
2. Opinion poll: the margin of error around 52% support from 1,000 voters.
3. Ecology: 95% CI for mean seedling height on three restoration plots.
4. Quality control: a one-sided 95% upper bound for the defect rate when 0 of 300 items failed.

Decision tree:

Assumptions

1. Random sampling. Check how units were chosen. If the sample is biased, the interval is centred on the wrong value and its coverage can be near zero; it does not widen to warn you.
2. Independence. Check the design for repeats, clusters or time series. Positive dependence makes the true SE larger, so the interval is too narrow and coverage falls below the nominal level.
3. Approximate normality of the mean (t and z intervals). Check Shapiro-Wilk and Chart 2 when n < 30. Strong right skew makes the interval too short on the long-tail side, so coverage drops, worst for one-sided bounds.
4. Normality of the data (SD interval). The chi-square interval for σ needs normal data at any n. Heavy tails make it too narrow, sometimes badly.
5. Fixed n and a binary outcome (proportions). Each unit must be a success or failure with the same chance. Overdispersion (clustered yes or no outcomes) makes every binomial interval too narrow.
6. Success-failure condition (Wald only). Check np̂ ≥ 10 and n(1 − p̂) ≥ 10. When it fails, Wald coverage falls below 95%, sometimes far below.
7. Known σ (z interval only). If the σ you enter is a guess and too small, the interval is too narrow.

Limitations

• The interval reflects random sampling error only, not measurement bias, selection bias or confounding. Use sensitivity analysis or a better design for those.
• It is about the mean or proportion, not individuals. For a range of individual values use a prediction or tolerance interval.
• Overlap of group intervals is not a test. Use the difference interval or the two-sample t-test.
• The tool treats data as a simple random sample. For weighted, stratified or cluster samples use survey software (R survey, Stata svy).
• The median interval is conservative: its exact coverage is at or above the requested level because order statistics jump in steps. A bootstrap BCa interval is an alternative.
• Multiple intervals at 95% each do not all hold together with 95% confidence. For many comparisons use a Bonferroni or Tukey adjustment.
• Sample size results depend entirely on the guessed σ or p. A guess that is too small gives a study that is too small.

18.1 What was found

A confidence interval calculator turns a single sample estimate into a range that honestly shows how much that estimate could move if you sampled again. For the worked leaf data, a mean of 52.44 mm became 50.20 to 54.67 mm at 95% confidence, a margin of error of 2.24 mm, or about 4% of the mean. That is a precise estimate from only 16 leaves because the leaves vary little.

18.2 What it means in practice

The interval is the result to report, not an extra. It tells the reader the effect size and the precision in the units they care about, and it makes the link to significance testing obvious: if the interval for a difference excludes zero, the difference is significant at the matching level. For proportions, the choice of method matters more than most people expect. Wilson and Jeffreys intervals keep close to their promised coverage, while the textbook Wald interval can fail badly with small samples or rare events.

18.3 Cautions

A confidence interval only covers random sampling error. It assumes independent units from the population you care about, and for small samples it assumes roughly normal data. It is not the range of individual values, and the 95% belongs to the method, not to one particular interval. Overlapping group intervals do not prove "no difference".

18.4 Next steps

Report the estimate, the confidence level and both limits together, name the method, and label error bars as confidence intervals. If the interval is too wide to support a decision, use the sample size option to plan the next study around a target margin of error. Then come back to this confidence interval calculator whenever you need a mean, proportion, difference or sample size interval checked against Excel, R and Python.

The following peer-reviewed references support the methods used in this confidence interval calculator: the t interval, binomial and score intervals for proportions, intervals for differences, normality checking, sample size planning and the correct interpretation of confidence intervals.

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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. 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
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8. 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
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10. Hoekstra, R., Morey, R. D., Rouder, J. N., & Wagenmakers, E.-J. (2014). Robust misinterpretation of confidence intervals. Psychonomic Bulletin & Review, 21(5), 1157–1164. https://doi.org/10.3758/s13423-013-0572-3
11. Lenth, R. V. (2001). Some practical guidelines for effective sample size determination. The American Statistician, 55(3), 187–193. https://doi.org/10.1198/000313001317098149
12. Newcombe, R. G. (1998a). 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
13. Newcombe, R. G. (1998b). 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
14. Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Philosophical Transactions of the Royal Society of London, Series A, 236(767), 333–380. https://doi.org/10.1098/rsta.1937.0005
15. Royston, P. (1995). Remark AS R94: A remark on algorithm AS 181: The W-test for normality. Journal of the Royal Statistical Society, Series C (Applied Statistics), 44(4), 547–551. https://doi.org/10.2307/2986146
16. Schenker, N., & Gentleman, J. F. (2001). On judging the significance of differences by examining the overlap between confidence intervals. The American Statistician, 55(3), 182–186. https://doi.org/10.1198/000313001317097960
17. Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3–4), 591–611. https://doi.org/10.1093/biomet/52.3-4.591
18. Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25. https://doi.org/10.1093/biomet/6.1.1
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