The margin of error is half the width of a confidence interval. It is the plus-or-minus figure you see after a poll result, and it answers one question: how far from the true value could this estimate plausibly be, given that you surveyed a sample rather than everybody?

1.1 The idea in one paragraph

You want to know what a whole population thinks, but you can only afford to ask some of them. Whatever number your sample returns will be a little off, purely because you happened to ask these particular people rather than a different set. The margin of error quantifies how far off it could plausibly be. Ask more people and it shrinks; ask fewer and it grows.

1.2 Its exact relationship to the confidence interval

The margin of error and the confidence interval are the same information written two ways:

So if a poll reports 52% with a margin of error of 3 points, the 95% confidence interval is 49% to 55%. Going the other way, if a paper reports an interval of 49% to 55%, the margin of error is (55 − 49) ÷ 2 = 3 points. Neither contains information the other lacks, which is why you can always convert between them.

1.3 What "95% confidence" actually means

It does not mean there is a 95% probability that the true value lies in this particular interval. The true value is a fixed number; it is either in there or it is not. What it means is that the procedure works 95% of the time: if you repeated the whole survey many times and built an interval each time, about 95 out of every 100 of those intervals would contain the true value.

In practice most people treat it as "we are fairly confident the truth is in this range", which is close enough for everyday use. The distinction matters when you start making probability statements about a specific interval, which is a Bayesian claim requiring different machinery.

1.4 The three levers

That last row is the counterintuitive one. Once the population is large relative to the sample, its exact size drops out of the arithmetic almost entirely. A sample of 1,000 gives a margin of about 3.1 points whether you are surveying a city of 50,000 or a country of 50 million. What matters is how many people you asked, not what fraction of the population they represent.

1.5 When population size does matter

If your sample is more than about 5% of the population, the finite population correction kicks in and shrinks the margin, because you are getting closer to a census. Surveying 1,000 people out of 5,000 gives ±2.77% rather than ±3.10%. Surveying 4,900 out of 5,000 gives a margin near zero, which is correct: you have nearly asked everybody.

This is why an internal survey of a 200-person company needs far fewer responses than a national poll to reach the same precision.

1.6 What the margin of error does not cover

This is the most important paragraph on the page. The quoted margin of error accounts for random sampling error only. It assumes you drew a genuine random sample from exactly the population you meant to study, that everyone responded, and that they answered honestly and understood the question.

In real surveys the larger errors are usually elsewhere: people who could not be reached, people who refused, a sampling frame that excluded part of the population, leading question wording, and the choices made when weighting the data. None of these appear in the ± figure. A poll with a stated margin of ±3 points can easily be off by considerably more, and historically most large polling misses have come from these sources rather than from sampling error.

4.1 What the plus-or-minus figure is claiming

When a poll reports 52% with a margin of error of 3 points, it is saying that the procedure used would capture the true population figure about 95 times out of 100, and that on this occasion it produced the range 49% to 55%. The true value is a fixed number, not a random one; what varies from survey to survey is the interval.

The practical reading is: the truth is plausibly anywhere between 49% and 55%, and values near the middle are more likely than values near the edges. That last point is often forgotten. The interval is not a uniform band where every value is equally credible; the estimate at the centre is the single most likely value and the plausibility falls away towards the limits.

4.2 The square root law, and why precision is expensive

The margin of error is proportional to one over the square root of n. This has consequences that surprise people who expect a linear relationship:

• Doubling your sample improves the margin by only about 29%, not 50%.
• To halve the margin you must quadruple the sample.
• Going from 1,000 to 2,000 respondents takes you from ±3.10% to ±2.19%. Going from 1,000 to 4,000 takes you to ±1.55%.
• The first few hundred responses buy you enormous precision; the next few thousand buy comparatively little.

This is why professional surveys cluster around 1,000 to 1,500 respondents. That is roughly where the curve flattens and additional fieldwork stops being worth the money.

4.3 Why the population size hardly matters

Almost everyone assumes a bigger population needs a bigger sample. It does not. A sample of 1,000 gives a margin of about ±3.1 points whether the population is 50,000 or 50 million, and the formula shows why: N does not appear in it at all unless the sample is a large fraction of the population.

The intuition is that you are measuring the composition of the population, not counting it. A well-stirred pot needs one spoonful to taste, whether the pot holds a litre or a hundred litres. What matters is that the stirring was thorough, which in survey terms means the sample was genuinely random.

The exception is when your sample is a meaningful share of the population, say above 5%. Then the finite population correction reduces the margin, because each person you sample removes some of the remaining uncertainty. Survey 1,000 out of 5,000 and the margin falls from 3.10% to 2.77%. Survey 4,900 out of 5,000 and it collapses to almost nothing, correctly: you have nearly conducted a census.

4.4 Why p = 0.5 is the worst case

The quantity p(1 − p) is largest at p = 0.5, where it equals 0.25, and it falls away symmetrically towards both extremes. At p = 0.1 it is 0.09, and the margin is about 40% smaller than at 50%.

This is why survey firms quote their headline margin at 50%: it is the widest the margin can be, so no result in the survey can be less precise than the quoted figure. It also means the margin reported for a whole survey applies exactly only to questions where the split is near even. A question where 90% agreed has a substantially tighter margin than the headline suggests.

4.5 Comparing two numbers, and the mistake journalists make

Suppose a poll with a margin of ±3 points shows Candidate A on 48% and Candidate B on 45%. Is A ahead? Almost certainly not in any statistically meaningful sense.

The margin applies to each estimate separately, and the difference between two estimates carries more uncertainty than either one alone. As a rough rule, the gap must exceed about twice the margin of error before you can be reasonably confident it is real. With a ±3 point margin, a 3-point lead is well inside the noise. Describing it as a lead, or worse as a "narrowing" lead compared with last month, is reading structure into randomness.

For proportions from the same survey the correct calculation is slightly more forgiving than doubling, because the two shares are negatively correlated, but doubling is a safe and simple heuristic that will keep you out of trouble.

4.6 When the standard formula breaks down

The usual formula is a normal approximation, and it fails when the percentage is close to 0% or 100%. Take 2 out of 100: the formula gives a margin of 2.74 points and an interval of −0.74% to 4.74%. A negative percentage is impossible, which tells you the method has broken.

Worse, at exactly 0 out of 50 the formula gives a margin of exactly zero, implying perfect certainty that nobody in the entire population holds that view, on the basis of 50 interviews. That is obviously absurd.

The Wilson score interval solves both problems. It is derived by inverting the test rather than approximating the estimate, cannot produce impossible values, and performs better than the standard interval at essentially all sample sizes. For 2 out of 100 it gives 0.55% to 7.00%, which is both possible and sensible. This calculator reports it alongside the standard interval every time, and where the two disagree you should use Wilson.

4.7 t or z, and why it matters for means

For a percentage the calculator uses z, the normal critical value. For a mean it uses t, because you almost always estimated the standard deviation from the same sample you are describing, and that estimate carries its own uncertainty.

The difference is largest at small n. At n = 5 the t critical value is 2.776 against the normal 1.960, so the margin is 42% wider. At n = 30 it is 2.045, only 4% wider. By n = 100 the gap is under 1%. Using z with a small sample understates your uncertainty, which is the more dangerous direction to err.

4.8 The design effect, for anything other than simple random sampling

The standard formula assumes every member of the population had an equal, independent chance of selection. Real surveys rarely work that way. They cluster by household, school, village or postcode, because sending an interviewer to one street is far cheaper than sending them to thirty scattered addresses.

Clustering costs precision, because people in the same cluster resemble each other, so each additional interview within a cluster adds less new information. The design effect quantifies that loss. A design effect of 2 means your survey carries the same sampling error as a simple random sample of half the size, so 1,000 clustered interviews behave like 500 independent ones. Typical values run from 1.5 to 3. Ignoring it will make your reported margin look considerably better than it is.

4.9 Reading the four charts

Chart 1 draws your interval with the estimate marked, which is the quickest way to see whether it crosses a threshold you care about. Chart 2 plots the margin against sample size and shows the square root law directly: the curve is steep at the left and almost flat by the right. Chart 3 shows how the margin varies with the percentage, peaking at 50%. Chart 4 shows what raising your confidence level costs you in precision.

Chart 2 is the one worth showing a client who is asking for a tighter number. It makes the cost of precision visible in a way that a table of figures does not.

4.10 The limitation that matters most

Everything on this page addresses random sampling error: the fact that you asked some people rather than all of them. It assumes you drew a genuine random sample from exactly the population you meant to study, that everybody responded, and that they understood and answered honestly.

Real surveys fail these assumptions routinely. Response rates for telephone polling have fallen into the single digits in many countries, and the people who still answer are not a random subset of those who do not. Online panels over-represent the engaged. Question wording nudges answers. Weighting corrects some of this and introduces assumptions of its own.

None of that appears in the ± figure. The margin of error is a statement about arithmetic under ideal conditions, and it should be read as a lower bound on your total uncertainty rather than a full accounting of it. When large surveys have been badly wrong, the failure has almost always been here rather than in the sampling error.

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

1. Paste your numbers into the data column. Mean from raw data opens first, with one column ready to go. Enter your values comma-separated exactly as the placeholder shows, and the calculator works out the mean, standard deviation, standard error and degrees of freedom, then applies the t critical value because your SD was estimated.
2. Or switch tab if you have something else. Survey percentage is the classic poll case. Mean from summary stats is for when you only have a published mean, SD and n. From a confidence interval works backwards from an interval you already have. Sample size I need is the planning calculation to run before collecting anything.
3. Load a sample dataset to see how it behaves. Eight are built in, including a skewed income set that triggers the normality warning and a set of very tight measurements where the margin is tiny. Selecting one replaces the data column immediately.
4. Add clusters to compare groups. Each gets its own margin of error and confidence interval. Every column has its own Clear and Remove button, and Remove is disabled on the last remaining column. You can also upload a CSV or Excel file and click the columns you want.
5. For a survey percentage, enter the result and the sample size. You can type 52 or 0.52, both are understood. If you do not yet know the percentage, leave it at 50, which is the worst case and gives the widest, safest margin.
6. Add clusters to compare groups. Each gets its own margin of error and interval. Every column has its own Clear and Remove button, and Remove is disabled on the last remaining column. You can also upload a CSV or Excel file and click the columns you want.
7. To reverse-engineer a published result, use the confidence interval tab. Enter the two limits and the calculator returns the margin, the point estimate, and, if you supply the sample size, the standard error and standard deviation behind them.
8. To plan a study, use the sample size tab. Tell it the margin you need and it tells you how many responses to collect, adjusted for the design effect, the population size and your expected response rate.
9. Choose your confidence level deliberately. 95% is the convention, but it is only a convention. Higher confidence always means a wider margin, and chart 4 shows exactly what it costs you.
10. Read the warnings before the numbers. They will tell you if your percentage is too close to 0 or 100 for the standard formula, if the interval has run outside the possible range, or if your sample is too small for the normal approximation. Where the Wilson interval is reported and differs, use it.

Excel handles every part of this, though the one-call function CONFIDENCE.T only works for means. For proportions you assemble it from three functions. Below is the whole workflow in ten steps, each with a picture of what your sheet should look like.

Step 1. Lay out the three inputs you actually have: the percentage as a decimal, the sample size, and the confidence level.

Step 2. NORM.S.INV converts the confidence level into the z critical value. At 95% it returns 1.959964, which is where the familiar 1.96 comes from.

Step 3. The standard error of a proportion. Note SQRT wraps the whole expression, not just n.

Step 4. Margin of error = z times the standard error. 0.030965 as a decimal, which is 3.0965 percentage points.

Step 5. The confidence interval is the estimate plus and minus the margin: 48.90% to 55.10%. Format these cells as percentages to read them directly.

Step 6. The finite population correction, for when your sample is a large share of the population. At n=1000 out of N=5000 it is 0.8945.

Step 7. The design effect widens the margin by its square root. A DEFF of 2 turns 3.10 points into 4.38.

Step 8. For a MEAN use T.INV.2T, not NORM.S.INV. At n=12 it gives 2.201 rather than 1.960, because the SD was estimated from the sample.

Step 9. CONFIDENCE.T does the whole thing in one call and returns 3.6027. Do NOT use CONFIDENCE.NORM, which assumes you know the population SD and gives too narrow a margin.

Step 10. Sample size for a target margin: =B4^2*B3*(1-B3)/B2^2 gives 1067.07, and ROUNDUP makes it 1068. Always round up, never down.

The formula cheat sheet

Six mistakes that catch people out

• Using CONFIDENCE.NORM instead of CONFIDENCE.T. The NORM version assumes the population standard deviation is known and gives a margin that is too narrow. At n = 12 it returns 3.208 rather than the correct 3.603, understating your uncertainty by 11%.
• Putting SQRT in the wrong place. It must wrap the whole fraction: SQRT(p*(1-p)/n), not SQRT(p*(1-p))/n. The second gives a completely different and much smaller number.
• Feeding NORM.S.INV the confidence level directly. NORM.S.INV(0.95) returns 1.645, the one-tailed value. For a two-sided margin you need NORM.S.INV(0.975).
• Entering the percentage as 52 instead of 0.52. The formula needs a decimal. Using 52 gives a nonsensical negative value under the square root.
• Rounding sample size down. 1067.07 becomes 1068, not 1067. Use ROUNDUP, not ROUND, or you will fall just short of your target precision.
• Applying the FPC when it does not apply. If your sample is under about 5% of the population the correction is nearly 1 and adds nothing but a chance to make an error.

Base R does all of this with qnorm(), qt() and arithmetic. No packages are needed. The script below covers proportions, means, the reverse calculation from a confidence interval, sample size planning and the Wilson interval, then draws one figure.

# Margin of Error Calculator in R (base R, no packages)

conf <- 0.95
z    <- qnorm(1 - (1 - conf)/2)          # 1.959964 at 95%

# ---- 1. Margin of error for a survey percentage -----------------------
p <- 0.52; n <- 1000
se_p  <- sqrt(p * (1 - p) / n)
moe_p <- z * se_p

# Finite population correction, only matters if n is a big share of N
N   <- 5000
fpc <- sqrt((N - n) / (N - 1))
moe_fpc <- moe_p * fpc

# Design effect, for cluster or multistage samples
deff <- 2
moe_deff <- moe_p * sqrt(deff)

# ---- 2. Margin of error for a mean ------------------------------------
x     <- c(52,48,55,61,47,58,50,63,45,56,54,49)
nm    <- length(x)
m     <- mean(x)
s     <- sd(x)                            # sd() always divides by n - 1
se_m  <- s / sqrt(nm)
tcrit <- qt(1 - (1 - conf)/2, nm - 1)     # t, not z: s was estimated
moe_m <- tcrit * se_m

# t.test() gives the same interval, which is a useful cross-check
ci <- t.test(x, conf.level = conf)$conf.int

# ---- 3. Margin of error FROM a confidence interval --------------------
lo <- 49.5640; hi <- 56.7694
moe_ci  <- (hi - lo) / 2
est_ci  <- (hi + lo) / 2
se_back <- moe_ci / tcrit                 # recover SE, if you know n
sd_back <- se_back * sqrt(nm)             # recover SD

# ---- 4. Sample size for a target margin -------------------------------
n_for_moe <- function(target, p = 0.5, conf = 0.95, N = NULL, deff = 1) {
  z <- qnorm(1 - (1 - conf)/2)
  n <- z^2 * p * (1 - p) / target^2
  n <- n * deff
  if (!is.null(N)) n <- n / (1 + (n - 1) / N)   # finite population
  ceiling(n)
}

# ---- 5. Wilson interval, for percentages near 0 or 100 ----------------
wilson <- function(k, n, conf = 0.95) {
  z <- qnorm(1 - (1 - conf)/2)
  p <- k / n
  d <- 1 + z^2 / n
  c <- (p + z^2 / (2*n)) / d
  h <- z * sqrt(p*(1-p)/n + z^2/(4*n^2)) / d
  c(max(0, c - h), min(1, c + h))
}

cat(sprintf("PROPORTION   %.0f%% of n=%d: MOE = +/- %.4f pp\n", p*100, n, moe_p*100))
cat(sprintf("             interval %.2f%% to %.2f%%\n", (p-moe_p)*100, (p+moe_p)*100))
cat(sprintf("  with FPC   N=%d: MOE = +/- %.4f pp  (fpc %.6f)\n", N, moe_fpc*100, fpc))
cat(sprintf("  with DEFF  deff=%g: MOE = +/- %.4f pp\n", deff, moe_deff*100))
cat(sprintf("MEAN         n=%d, mean=%.4f, s=%.4f, SE=%.4f\n", nm, m, s, se_m))
cat(sprintf("             t(%d) = %.6f, MOE = +/- %.4f\n", nm-1, tcrit, moe_m))
cat(sprintf("             interval %.4f to %.4f\n", m-moe_m, m+moe_m))
cat(sprintf("FROM CI      [%g, %g] -> MOE = +/- %.4f, estimate %.4f\n",
            lo, hi, moe_ci, est_ci))
cat(sprintf("             recovered SE %.6f, SD %.6f\n", se_back, sd_back))
cat(sprintf("SAMPLE SIZE  +/-5%%: %d   +/-3%%: %d   +/-1%%: %d\n",
            n_for_moe(0.05), n_for_moe(0.03), n_for_moe(0.01)))
cat(sprintf("             +/-3%% within a population of 5000: %d\n",
            n_for_moe(0.03, N = 5000)))
w <- wilson(2, 100)
cat(sprintf("WILSON       2/100: %.4f%% to %.4f%%\n", w[1]*100, w[2]*100))
cat(sprintf("  standard   2/100: %.4f%% to %.4f%%  <- impossible lower bound\n",
            (0.02 - z*sqrt(.02*.98/100))*100, (0.02 + z*sqrt(.02*.98/100))*100))

# ---- 6. One figure ----------------------------------------------------
ns <- seq(50, 5000, by = 10)
plot(ns, z*sqrt(0.5*0.5/ns)*100, type = "l", lwd = 2, col = "#0369a1",
     ylim = c(0, 8), bty = "n", xlab = "sample size",
     ylab = "margin of error (percentage points)",
     main = sprintf("Margin of Error vs Sample Size at %.0f%% Confidence", conf*100))
lines(ns, z*sqrt(0.3*0.7/ns)*100, lwd = 2, col = "#0f766e")
lines(ns, z*sqrt(0.1*0.9/ns)*100, lwd = 2, col = "#b45309")
abline(v = c(385, 1068), col = "#94a3b8", lty = 3)
text(385,  7.6, " n=385\n +/-5%", pos = 4, cex = 0.75, col = "#64748b")
text(1068, 7.6, " n=1068\n +/-3%", pos = 4, cex = 0.75, col = "#64748b")
points(n, moe_p*100, pch = 19, cex = 1.4, col = "#be185d")
legend("topright", bty = "n", cex = 0.85,
       legend = c("p = 50%", "p = 30%", "p = 10%",
                  sprintf("your survey: n=%d, +/-%.2f pp", n, moe_p*100)),
       col = c("#0369a1", "#0f766e", "#b45309", "#be185d"),
       lwd = c(2, 2, 2, NA), pch = c(NA, NA, NA, 19))

What the script prints

PROPORTION   52% of n=1000: MOE = +/- 3.0965 pp
             interval 48.90% to 55.10%
  with FPC   N=5000: MOE = +/- 2.7699 pp  (fpc 0.894517)
  with DEFF  deff=2: MOE = +/- 4.3791 pp
MEAN         n=12, mean=53.1667, s=5.6702, SE=1.6369
             t(11) = 2.200985, MOE = +/- 3.6027
             interval 49.5640 to 56.7694
FROM CI      [49.564, 56.7694] -> MOE = +/- 3.6027, estimate 53.1667
             recovered SE 1.636858, SD 5.670242
SAMPLE SIZE  +/-5%: 385   +/-3%: 1068   +/-1%: 9604
             +/-3% within a population of 5000: 880
WILSON       2/100: 0.5502% to 7.0012%
  standard   2/100: -0.7439% to 4.7439%  <- impossible lower bound

These are the same numbers the Python script produces and the same numbers the calculator at the top of this page produces.

Line-by-line explanation

• Block 1 computes the proportion margin from first principles, then shows both adjustments. Notice the finite population correction pulls the margin down while the design effect pushes it up: they work in opposite directions and both are commonly forgotten.
• Block 2 uses qt() rather than qnorm() because the standard deviation came from the sample. The t.test() call is there as a cross-check: its confidence interval should match your hand calculation exactly, and if it does not you have made an arithmetic error.
• Block 3 is the reverse direction, and it is two lines of arithmetic. The recovery of the standard deviation only works if you know n and the confidence level the original authors used.
• Block 4 is written as a reusable function because you will want it repeatedly. Note the ceiling(): sample sizes always round up.
• Block 5 implements Wilson directly rather than calling prop.test(), so you can see the arithmetic and control the continuity correction yourself.
• Block 6 draws the single most useful figure in survey planning: the margin against sample size for three different proportions, with the ±5% and ±3% landmarks marked. The flatness of the curve past about n = 1,500 is the whole argument for why polls are the size they are.

Useful one-liners

NumPy and SciPy cover everything here. The script below was run before being published, so the output shown underneath is the real console output, not typed by hand.

# Margin of Error Calculator in Python
# Proportions, means, reverse from a confidence interval, and sample size.
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from scipy import stats

conf  = 0.95
z     = stats.norm.ppf(1 - (1 - conf)/2)     # 1.959964 at 95%

# ---- 1. Margin of error for a survey percentage -----------------------
p, n = 0.52, 1000
se_p  = np.sqrt(p * (1 - p) / n)
moe_p = z * se_p

# Finite population correction, only matters if n is a big share of N
N     = 5000
fpc   = np.sqrt((N - n) / (N - 1))
moe_fpc = moe_p * fpc

# Design effect, for cluster or multistage samples
deff  = 2.0
moe_deff = moe_p * np.sqrt(deff)

# ---- 2. Margin of error for a mean ------------------------------------
x     = np.array([52,48,55,61,47,58,50,63,45,56,54,49], dtype=float)
nm    = x.size
mean  = x.mean()
s     = x.std(ddof=1)                         # ddof=1: sample SD
se_m  = s / np.sqrt(nm)
tcrit = stats.t.ppf(1 - (1 - conf)/2, nm - 1) # t, not z: s was estimated
moe_m = tcrit * se_m

# ---- 3. Margin of error FROM a confidence interval --------------------
lo, hi   = 49.5640, 56.7694
moe_ci   = (hi - lo) / 2
est_ci   = (hi + lo) / 2
se_back  = moe_ci / tcrit                     # recover SE, if you know n
sd_back  = se_back * np.sqrt(nm)              # recover SD

# ---- 4. Sample size for a target margin -------------------------------
def n_for_moe(target, p=0.5, conf=0.95, N=None, deff=1.0):
    z = stats.norm.ppf(1 - (1 - conf)/2)
    n = z**2 * p * (1 - p) / target**2
    n *= deff
    if N:                                     # finite population correction
        n = n / (1 + (n - 1) / N)
    return int(np.ceil(n))

# ---- 5. Wilson interval, for percentages near 0 or 100 ----------------
def wilson(k, n, conf=0.95):
    z = stats.norm.ppf(1 - (1 - conf)/2)
    p = k / n
    d = 1 + z**2 / n
    c = (p + z**2 / (2*n)) / d
    h = z * np.sqrt(p*(1-p)/n + z**2/(4*n**2)) / d
    return max(0.0, c - h), min(1.0, c + h)

print(f"PROPORTION   {p:.0%} of n={n}: MOE = +/- {moe_p*100:.4f} pp")
print(f"             interval {(p-moe_p)*100:.2f}% to {(p+moe_p)*100:.2f}%")
print(f"  with FPC   N={N}: MOE = +/- {moe_fpc*100:.4f} pp  (fpc {fpc:.6f})")
print(f"  with DEFF  deff={deff}: MOE = +/- {moe_deff*100:.4f} pp")
print(f"MEAN         n={nm}, mean={mean:.4f}, s={s:.4f}, SE={se_m:.4f}")
print(f"             t({nm-1}) = {tcrit:.6f}, MOE = +/- {moe_m:.4f}")
print(f"             interval {mean-moe_m:.4f} to {mean+moe_m:.4f}")
print(f"FROM CI      [{lo}, {hi}] -> MOE = +/- {moe_ci:.4f}, estimate {est_ci:.4f}")
print(f"             recovered SE {se_back:.6f}, SD {sd_back:.6f}")
print(f"SAMPLE SIZE  +/-5%: {n_for_moe(0.05)}   +/-3%: {n_for_moe(0.03)}   "
      f"+/-1%: {n_for_moe(0.01)}")
print(f"             +/-3% within a population of 5000: {n_for_moe(0.03, N=5000)}")
print(f"WILSON       2/100: {wilson(2,100)[0]*100:.4f}% to {wilson(2,100)[1]*100:.4f}%")
print(f"  standard   2/100: {(0.02-z*np.sqrt(.02*.98/100))*100:.4f}% to "
      f"{(0.02+z*np.sqrt(.02*.98/100))*100:.4f}%  <- impossible lower bound")

# ---- 6. One figure ----------------------------------------------------
fig, ax = plt.subplots(figsize=(9, 5))
ns = np.arange(50, 5001, 10)
for pp, col in [(0.5, "#0369a1"), (0.3, "#0f766e"), (0.1, "#b45309")]:
    ax.plot(ns, z*np.sqrt(pp*(1-pp)/ns)*100, color=col, lw=2,
            label=f"p = {pp:.0%}")
ax.scatter([n], [moe_p*100], s=90, color="#be185d", zorder=5,
           label=f"your survey: n={n}, +/-{moe_p*100:.2f} pp")
for mk, lbl in [(385, "+/-5%"), (1068, "+/-3%")]:
    ax.axvline(mk, color="#94a3b8", ls=":", lw=1.2)
    ax.text(mk, 7.6, f" n={mk}\n {lbl}", fontsize=8, color="#64748b")
ax.set_xlabel("sample size")
ax.set_ylabel("margin of error (percentage points)")
ax.set_title(f"Margin of Error vs Sample Size at {conf:.0%} Confidence")
ax.set_ylim(0, 8)
ax.legend(frameon=False, fontsize=9)
ax.spines[["top", "right"]].set_visible(False)
fig.tight_layout()
fig.savefig("margin_of_error.png", dpi=150)
print("saved margin_of_error.png")

Actual output from running the script

PROPORTION   52% of n=1000: MOE = +/- 3.0965 pp
             interval 48.90% to 55.10%
  with FPC   N=5000: MOE = +/- 2.7699 pp  (fpc 0.894517)
  with DEFF  deff=2.0: MOE = +/- 4.3791 pp
MEAN         n=12, mean=53.1667, s=5.6702, SE=1.6369
             t(11) = 2.200985, MOE = +/- 3.6027
             interval 49.5640 to 56.7694
FROM CI      [49.564, 56.7694] -> MOE = +/- 3.6027, estimate 53.1667
             recovered SE 1.636858, SD 5.670242
SAMPLE SIZE  +/-5%: 385   +/-3%: 1068   +/-1%: 9604
             +/-3% within a population of 5000: 880
WILSON       2/100: 0.5502% to 7.0012%
  standard   2/100: -0.7439% to 4.7439%  <- impossible lower bound

Line-by-line explanation

• Block 1 computes the proportion margin, then shows both adjustments separately. The finite population correction multiplies the margin by 0.8945 and the design effect multiplies it by the square root of 2. They pull in opposite directions, and real surveys often need both.
• Block 2 uses stats.t.ppf rather than stats.norm.ppf because the standard deviation was estimated. At n = 12 that is 2.201 rather than 1.960, a 12% wider margin.
• Block 3 is the reverse direction and it is two lines of arithmetic: half the width, and the midpoint. The recovery of the standard error and standard deviation requires knowing n and assuming the original authors used the same confidence level and distribution.
• Block 4 wraps the sample size calculation in a function with optional finite population and design effect arguments. Note np.ceil: always round up.
• Block 5 implements Wilson directly. The printed comparison makes the failure of the standard method obvious: for 2 out of 100 it produces a lower bound of −0.74%, a negative percentage.
• Block 6 draws the margin against sample size for three proportions, with the ±5% and ±3% landmarks marked. This is the figure to show anyone asking why the survey cannot simply be made more precise.

Useful one-liners

11.1 Margin of error by sample size and confidence level

Computed at p = 50%, the worst case, so no result in a survey of that size can be less precise than the figure shown. This is the table survey firms use to quote a headline margin.

11.2 Margin of error by sample size and the percentage itself

The margin is widest at 50% and narrows towards both extremes. A question where 90% agreed is considerably more precise than the survey's headline margin suggests.

11.3 Sample size needed for a target margin of error

The planning table. Read down to the precision you need and across to your confidence level. Assumes p = 50% and a very large population.

11.4 Sample size needed, by population size

This is the table that demonstrates how little population size matters. Look along the ±3% column: a population of 100,000 needs 1,056 responses and a population of 10 million needs 1,068. Twelve more people for a hundredfold larger population.

Population size only bites when it is small. A 500-person organisation needs 218 responses for ±5%, not 385, because the finite population correction is doing real work at that scale.

11.5 The square root law, made concrete

What you actually get for your money. All figures at p = 50% and 95% confidence.

Read the last row: to reach a quarter of the margin you need sixteen times the sample. This relationship is fixed by the mathematics and no amount of budget changes it.

11.6 When the finite population correction matters

A fixed sample of 200 drawn from populations of different sizes. The correction is substantial when your sample is a large share of the population and negligible otherwise.

Below about 5% sampling fraction the correction changes the margin by under 3% of its own value, which is why it is usually ignored for national surveys and always applied for organisational ones.

11.7 Where the standard formula fails, and Wilson rescues it

All at n = 100 except the first row, at 95% confidence. The highlighted rows are where the standard interval produces something impossible.

Note the first and last rows especially. At 0 out of 50 the standard formula reports a margin of exactly zero, claiming certainty that nobody in the population holds that view on the basis of 50 interviews. Wilson correctly returns 0% to 7.13%.

11.8 Critical values by confidence level

11.9 t versus z for means, and the small-sample penalty

Using z instead of t on a sample of 10 understates your margin by 15%. Above about n = 50 the difference stops mattering in practice, but there is never a penalty for using t.

11.10 Design effects you might encounter

A stratified sample can have a design effect below 1, which means it is more efficient than simple random sampling. Every other design listed here costs you precision.

11.11 The same calculation in five packages

Every number below was computed with the calculator on this page and cross-checked against SciPy. Each example has its own colour and its own figure. Together they cover the five things people actually need: a poll, a mean, a reverse calculation, the cases where the standard formula fails, and the planning calculation.

The margin of error is the last thing you calculate and the first thing people quote. Everything that determines whether it means anything happens before a single response arrives.

1. Define the target population precisely before anything else. "Adults in the region" is not a definition. Which adults, at what age boundary, resident how long, reachable by what means? The margin of error describes your uncertainty about this population, so if it is vague your result is vague.
2. Choose the sample size in advance and write it down. Use the sample size tab. Collecting until the result looks convincing is the survey equivalent of optional stopping and destroys the meaning of the margin.
3. Assume p = 0.5 when planning unless you have solid prior information. It is the worst case, so your realised margin can only be better than planned. Assuming 20% and finding 45% leaves you less precise than you promised.
4. Build the sampling frame and document what it excludes. A telephone frame misses people without phones. An email list misses people who never signed up. These exclusions do not appear in the margin of error and are usually the larger problem.
5. Randomise selection properly. The formula assumes every member of the population had a known, non-zero chance of selection. Convenience samples, self-selecting web polls and "whoever answered" have no defensible margin of error at all, however large they are.
6. Record the design effect if you cluster. Sampling by household, school or postcode is cheaper and less precise. Estimate the design effect from a pilot or from comparable published surveys, and apply it to both your planning and your reporting.
7. Plan for non-response and inflate the invitation list. A 30% response rate means inviting more than three times your target. Then record the actual rate, because it is a stronger signal of quality than the margin of error is.
8. Pilot the questionnaire. Ambiguous wording produces measurement error that no sample size can fix. A pilot of thirty people will catch most of it.
9. Decide the confidence level in advance and use the same one throughout. Reporting some results at 95% and others at 90% because it makes them look better is indefensible.
10. Work out the margin for your smallest subgroup, not just the total. A survey of 1,000 with a 3-point margin gives a 6-point margin on a quarter of the sample and a 10-point margin on a tenth of it. Subgroup findings are far less precise than the headline, and this is the single most common way survey results are oversold.
11. Keep the raw response file untouched. Do weighting and cleaning in a script so that every published figure can be traced back.
12. Report the margin alongside the response rate, the frame and the weighting. The AAPOR transparency standards ask for all of these. A margin of error quoted alone is close to meaningless.

Use it when

• You have a genuine probability sample and want to express how precise your estimate is.
• You are reporting a survey percentage or a mean to an audience who will otherwise read the point estimate as exact.
• You are planning a study and need to justify a sample size to a client, a funder or an ethics committee.
• You want to know whether two numbers differ meaningfully. Compare the gap against roughly twice the margin.
• You are checking somebody else's work. A published interval yields the margin, and with n it yields the standard deviation too.

Do not use it when

• The sample was not random. Opt-in web polls, convenience samples and self-selected respondents have no valid margin of error. Reputable pollsters explicitly decline to quote one for non-probability panels.
• You measured the entire population. A census has no sampling error. If you surveyed all 40 employees, the answer is the answer.
• The interval is asymmetric. Wilson intervals, bootstrap intervals, and intervals for ratios, odds ratios or skewed quantities are deliberately not centred on the estimate. Report the limits instead.
• Your outcome is a median, a maximum or a rank. These have sampling distributions that are not symmetric or normal. Use a bootstrap.
• The percentage is very close to 0% or 100%. The standard formula gives impossible values. Use Wilson or an exact Clopper-Pearson interval.
• The main problem is bias, not sampling error. A margin of ±3 points on a survey with a 4% response rate is a precise measurement of a possibly unrepresentative group.
• You are comparing more than two groups. Looking at whether many pairs of intervals overlap inflates your error rate. Use a proper multiple comparison procedure.

What to use instead

What the margin of error assumes, ranked by how much it matters

Limitations worth stating whenever you publish a margin

• It covers sampling error only. This is the limitation that matters most and the one most often omitted. Non-response, coverage gaps, question wording and weighting decisions are all outside it, and in modern surveys they are usually larger.
• It assumes a probability sample. Opt-in panels and web polls have no valid margin of error regardless of size. Some organisations report a "credibility interval" instead, which is honest about the modelling assumptions involved.
• It applies to the whole sample, not to subgroups. A survey of 1,000 with ±3 points gives about ±6 points on a subgroup of 250 and ±10 on a subgroup of 100. Reporting subgroup findings under the headline margin is the most common way survey results get oversold.
• The quoted figure is usually the worst case. Computed at p = 50%, it overstates the margin for questions with lopsided answers.
• Comparing two estimates needs a different, larger margin. A gap must exceed roughly twice the individual margin before it is worth discussing.
• The confidence level is a convention. 95% has no special mathematical status. It was a convenience Fisher suggested and it stuck.
• The interval is not uniform. Values near the centre are considerably more plausible than values at the edges, which the flat ± notation hides.
• Repeated polling guarantees apparent movement. If you run twenty polls at 95% confidence, one will fall outside its interval by chance. Tracking studies routinely report this noise as a trend.

The margin of error is half the width of a confidence interval, and it exists because you asked some people rather than all of them. For a percentage it is z times the square root of p(1 − p) over n. For a mean it is t times the standard error. If you already have an interval, it is simply half the width, and if you also know the sample size you can work back to the standard error and the standard deviation behind it. That is the entire arithmetic, and it fits in a paragraph.

Three properties of that arithmetic surprise almost everyone. The first is the square root law: the margin falls with the square root of n, so doubling your sample improves precision by only 29% and halving the margin costs four times the sample. This is why commercial polls cluster between 1,000 and 1,500 respondents, and why a client asking to go from ±3 points to ±1 point is asking for a survey nine times the size.

The second is that population size barely matters. A sample of 1,000 gives about ±3.1 points whether the population is 50,000 or 50 million; the reference table shows a hundredfold increase in population changing the required sample by twelve people. What you are measuring is the composition of the population, not counting it, and a well-stirred pot needs one spoonful whatever its size. The exception is when your sample is a large share of the population, which is precisely the situation an internal company survey is in, and there the finite population correction does real work.

The third is that p = 0.5 is the worst case. That is why survey firms quote their headline margin at fifty per cent before fieldwork begins: nothing in the survey can turn out less precise than the number they promised.

Two failure modes are worth memorising. When a percentage sits close to 0% or 100%, the standard formula produces impossible results, including a negative lower bound for 2 out of 100 and a margin of exactly zero for 0 out of 50. The Wilson score interval fixes both and is never worse, so this calculator reports it every time. And when a survey is not a simple random sample, the design effect widens the margin: a household cluster design with a design effect of 2 means your thousand interviews carry the same sampling error as five hundred independent ones. Omitting it is one of the commonest errors in published survey work.

The most consequential limitation is not mathematical at all. The margin of error accounts for random sampling error and nothing else. It assumes a genuine random sample from exactly the population you meant to study, that everybody responded, and that they understood and answered honestly. Real surveys fail these assumptions routinely, response rates for telephone polling have collapsed into single digits in many countries, and the people who still answer are not a random subset of those who do not. None of that appears in the plus-or-minus figure. Treat it as a lower bound on your total uncertainty rather than a full accounting of it, and report the response rate, the sampling frame and the weighting scheme alongside it.

Finally, resist the most common misreading of all. Two estimates whose intervals overlap are not distinguishable, and a gap must exceed roughly twice the individual margin before it is worth discussing. A three-point lead in a poll with a three-point margin is not a lead. It is noise with a headline attached.

1. Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22(158), 209-212. doi.org/10.1080/01621459.1927.10502953
2. Brown, L. D., Cai, T. T., & DasGupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science, 16(2), 101-133. doi.org/10.1214/ss/1009213286
3. Agresti, A., & Coull, B. A. (1998). Approximate is better than "exact" for interval estimation of binomial proportions. The American Statistician, 52(2), 119-126. doi.org/10.1080/00031305.1998.10480550
4. Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in Medicine, 17(8), 857-872. doi.org/10.1002/(SICI)1097-0258
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. doi.org/10.1093/biomet/26.4.404
6. Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Wiley. wiley.com
7. Kish, L. (1965). Survey Sampling. Wiley. wiley.com
8. Lohr, S. L. (2021). Sampling: Design and Analysis (3rd ed.). CRC Press. doi.org/10.1201/9780429298899
9. Groves, R. M., et al. (2009). Survey Methodology (2nd ed.). Wiley. wiley.com
10. Groves, R. M., & Peytcheva, E. (2008). The impact of nonresponse rates on nonresponse bias: A meta-analysis. Public Opinion Quarterly, 72(2), 167-189. doi.org/10.1093/poq/nfn011
11. American Association for Public Opinion Research (2023). Standard Definitions: Final Dispositions of Case Codes and Outcome Rates for Surveys (10th ed.). aapor.org
12. AAPOR Task Force (2013). Report on Non-Probability Sampling. American Association for Public Opinion Research. aapor.org
13. Kennedy, C., et al. (2018). An Evaluation of the 2016 Election Polls in the United States. Public Opinion Quarterly, 82(1), 1-33. doi.org/10.1093/poq/nfx047
14. Sturgis, P., et al. (2018). An assessment of the causes of the errors in the 2015 UK general election opinion polls. Journal of the Royal Statistical Society Series A, 181(3), 757-781. doi.org/10.1111/rssa.12329
15. Cumming, G., & Finch, S. (2005). Inference by eye: Confidence intervals and how to read pictures of data. American Psychologist, 60(2), 170-180. doi.org/10.1037/0003-066X.60.2.170
16. Greenland, S., et al. (2016). Statistical tests, p-values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31, 337-350. doi.org/10.1007/s10654-016-0149-3
17. Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Philosophical Transactions of the Royal Society A, 236(767), 333-380. doi.org/10.1098/rsta.1937.0005
18. Krejcie, R. V., & Morgan, D. W. (1970). Determining sample size for research activities. Educational and Psychological Measurement, 30(3), 607-610. doi.org/10.1177/001316447003000308
19. Virtanen, P., et al. (2020). SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nature Methods, 17, 261-272. doi.org/10.1038/s41592-019-0686-2
20. Lumley, T. (2010). Complex Surveys: A Guide to Analysis Using R. Wiley. doi.org/10.1002/9780470580066