📊 Select Interval Type
📌 Use when population standard deviation σ is known. Critical value z* = 1.960 for 95% CI. Formula: x̅ ± 1.960 × (σ / √n)
x̅
Enter a valid sample mean.
σ
Enter σ > 0.
n
Enter n ≥ 1.
📌 Use when σ is unknown. Uses the t-distribution with df = n − 1. For large n (≥ 30) the t critical value approaches z* = 1.960.
x̅
Enter a valid sample mean.
s
Enter s > 0.
n
Enter n ≥ 2.
📌 For proportions (success/failure). Formula: p̂ ± 1.960 × √(p̂(1−p̂)/n). Valid when np̂ ≥ 5 and n(1−p̂) ≥ 5.
x
Enter a valid count (0 to n).
n
Enter n ≥ 1.
📌 95% CI for the difference μ⊂1;−μ⊂2; between two independent group means. If zero is outside the CI, the difference is statistically significant at α = 0.05.
x̅
Enter mean 1.
s
Enter s⊂1; > 0.
n
Enter n⊂1; ≥ 2.
x̅
Enter mean 2.
s
Enter s⊂2; > 0.
n
Enter n⊂2; ≥ 2.
95% Confidence Interval
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⚠️ Disclaimer: Results are for educational and informational purposes. Verify critical statistical decisions with a qualified statistician.
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Sources & Methodology
Critical values and formulas verified against NIST Statistical Tables, ASA guidelines, and standard research methodology textbooks.
NIST/SEMATECH e-Handbook of Statistical Methods (Section 7.2)
Interval estimation for means and proportions. Reference for z-interval vs t-interval selection, conditions for normal approximation of proportions, and two-sample confidence interval methods.
American Statistical Association (2016) — Statement on P-Values and Confidence Intervals
ASA recommends reporting confidence intervals alongside or instead of p-values. The 95% CI communicates both statistical significance and effect size magnitude, addressing key limitations of p-value-only reporting.
Formulas:
95% CI for mean (known σ): x̅ ± 1.960 × (σ / √n) 95% CI for mean (unknown σ): x̅ ± t*(df=n−1, 95%) × (s / √n) 95% CI for proportion: p̂ ± 1.960 × √(p̂(1−p̂) / n) 95% CI for diff. of means: (x̅⊂1;−x̅⊂2;) ± 1.960 × √(s⊂1;²/n⊂1; + s⊂2;²/n⊂2;) Two-means uses Welch’s method (unequal variances). z* = 1.95996 (exact). T critical values via lookup table.
95% CI for mean (known σ): x̅ ± 1.960 × (σ / √n) 95% CI for mean (unknown σ): x̅ ± t*(df=n−1, 95%) × (s / √n) 95% CI for proportion: p̂ ± 1.960 × √(p̂(1−p̂) / n) 95% CI for diff. of means: (x̅⊂1;−x̅⊂2;) ± 1.960 × √(s⊂1;²/n⊂1; + s⊂2;²/n⊂2;) Two-means uses Welch’s method (unequal variances). z* = 1.95996 (exact). T critical values via lookup table.
95% Confidence Interval — The Standard in Scientific Research
The 95% confidence interval is the most widely used and reported confidence interval in scientific research. It is the default for clinical trials, psychological studies, economics, ecology, epidemiology, and most peer-reviewed journals. Its dominance traces back to Ronald Fisher’s establishment of α = 0.05 as the conventional significance threshold in the 1920s — the 95% CI is the direct complement of this convention.
The Critical Value: z* = 1.960
For a 95% CI, the critical value from the standard normal distribution is z* = 1.960 (exact value: 1.959964). The middle 95% of the standard normal distribution lies between −1.960 and +1.960, with 2.5% in each tail (α/2 = 0.025).
95% CI for mean (known σ): x̅ ± 1.960 × (σ / √n)
Example: Survey of n=400 adults measuring daily steps. Sample mean x̅=7,842. Population σ=2,100.
SE = 2100/√400 = 2100/20 = 105
E = 1.960 × 105 = 205.8
95% CI = (7842 − 206, 7842 + 206) = (7,636, 8,048)
Interpretation: We are 95% confident the true mean daily steps is between 7,636 and 8,048.
SE = 2100/√400 = 2100/20 = 105
E = 1.960 × 105 = 205.8
95% CI = (7842 − 206, 7842 + 206) = (7,636, 8,048)
Interpretation: We are 95% confident the true mean daily steps is between 7,636 and 8,048.
95% CI for a Mean When σ is Unknown (t-distribution)
In practice, the population standard deviation σ is almost always unknown. When you estimate σ from the sample (using s), use the t-distribution with degrees of freedom df = n − 1. The t critical value is always > 1.960 and approaches 1.960 as n increases.
95% CI for mean (unknown σ): x̅ ± t*(df, 95%) × (s / √n)
T critical values for 95% CI:
df = 9 (n=10): t* = 2.262 | df = 19 (n=20): t* = 2.093 | df = 29 (n=30): t* = 2.045
df = 49 (n=50): t* = 2.010 | df = 99 (n=100): t* = 1.984 | df = ∞: t* = 1.960
df = 9 (n=10): t* = 2.262 | df = 19 (n=20): t* = 2.093 | df = 29 (n=30): t* = 2.045
df = 49 (n=50): t* = 2.010 | df = 99 (n=100): t* = 1.984 | df = ∞: t* = 1.960
95% CI for a Proportion — Survey & Poll Calculations
For proportions, the 95% CI formula uses z* = 1.960. This is the formula behind every published poll margin of error: a survey of 1,000 people gives a worst-case margin of error of about ±3.1 percentage points at 95% confidence.
95% CI for proportion: p̂ ± 1.960 × √(p̂(1−p̂) / n)
Poll example: 520 of 1000 voters support Candidate A (p̂ = 0.52).
SE = √(0.52 × 0.48 / 1000) = √0.0002496 = 0.01580
E = 1.960 × 0.01580 = 0.031 = 3.1 percentage points
95% CI = (48.9%, 55.1%) — this includes 50%, so the race is within the margin of error.
SE = √(0.52 × 0.48 / 1000) = √0.0002496 = 0.01580
E = 1.960 × 0.01580 = 0.031 = 3.1 percentage points
95% CI = (48.9%, 55.1%) — this includes 50%, so the race is within the margin of error.
95% CI for the Difference Between Two Means
When comparing two independent groups, the 95% CI for the difference (μ⊂1;−μ⊂2;) tells you whether the two groups are statistically different. If the CI excludes zero, the difference is significant at α = 0.05. If zero is inside the CI, there is insufficient evidence of a real difference.
95% CI for (μ⊂1;−μ⊂2;): (x̅⊂1;−x̅⊂2;) ± 1.960 × √(s⊂1;²/n⊂1; + s⊂2;²/n⊂2;)
This uses Welch’s method for unequal variances — appropriate in most practical situations. If the CI contains zero: groups are not significantly different (p > 0.05). If CI excludes zero: difference is significant (p < 0.05).
Why 95% CI? The Statistical Convention
The 95% CI corresponds exactly to testing at the 5% significance level (α = 0.05). This threshold was informally established by Ronald Fisher in his 1925 “Statistical Methods for Research Workers” as a convenient benchmark, not a mathematically optimal threshold. Fisher himself cautioned against treating it as absolute. Despite this, it became entrenched as the dominant standard across scientific disciplines.
| Sample Size (n) | SE (for mean, σ=10) | 95% CI Half-Width | Required n to halve width |
|---|---|---|---|
| 25 | 2.000 | 3.920 | 100 |
| 100 | 1.000 | 1.960 | 400 |
| 400 | 0.500 | 0.980 | 1,600 |
| 1,000 | 0.316 | 0.619 | 4,000 |
💡 CI vs p-value: A 95% CI provides more information than a p-value alone. It shows both whether the effect is significant (does the CI include zero?) AND the magnitude and precision of the estimate (how wide is the interval?). The ASA Statement on P-Values (2016) explicitly recommends reporting CIs. Most modern research journals require CIs alongside or instead of p-values.
Frequently Asked Questions
A 95% CI is a range of values that would contain the true population parameter in 95 out of 100 repeated studies. The 5% error rate means 1 in 20 CIs built this way will miss the true value. It does NOT mean “there is a 95% chance the true value is in this specific interval.” The true value is fixed; once computed, the interval either contains it or does not.
z* = 1.960 (exact: 1.959964) is the standard normal z-score with 95% of the distribution between −z* and +z*, and 2.5% in each tail. It comes from the standard normal CDF: Φ(1.960) = 0.975, meaning 97.5% of the distribution lies below z=1.960. Often rounded to z=1.96, but 1.960 is more precise. Compare: 90% CI uses z*=1.645, 99% uses z*=2.576.
Known σ: x̅ ± 1.960 × (σ/√n). Unknown σ (t-distribution): x̅ ± t*(df=n−1, 95%) × (s/√n). For n≥30, using z*=1.960 instead of t* introduces minimal error. For small samples, always use the t-distribution. Margin of error E = critical value × standard error. CI = (x̅−E, x̅+E).
The 95% CI corresponds to α=0.05, which has been the conventional significance threshold since Ronald Fisher popularized it in the 1920s. It provides a balance: narrow enough to be practically useful but reliable enough that only 1 in 20 intervals misses the true value. Most scientific journals expect 95% CIs by default. Using higher confidence (99%) makes intervals wider; lower (90%) makes them narrower but less certain.
p̂ ± 1.960 × √(p̂(1−p̂)/n). Check: np̂≥5 and n(1−p̂)≥5. Example: 300 out of 500 (p̂=0.60). SE=√(0.60×0.40/500)=0.02191. E=1.960×0.02191=0.043. 95% CI=(55.7%, 64.3%). This is the Wald interval — easy to compute but slightly anti-conservative for small n or extreme proportions. Use Wilson interval for p̂ near 0 or 1.
When the 95% CI for a difference (group1 mean minus group2 mean) does not include zero, the difference is statistically significant at α=0.05 (two-tailed). This is exactly equivalent to rejecting H⊂0;: μ⊂1;=μ⊂2; at the 5% significance level. When zero IS inside the CI, the difference is not significant. This CI/p-value duality is fundamental: the CI simultaneously shows both significance and the practical magnitude of the effect.
Use z*=1.960 when σ is known, or n≥30 (CLT approximation). Use t*(df=n−1) when σ is unknown and n<30. T values at 95%: n=10 gives t*=2.262, n=20 gives t*=2.093, n=30 gives t*=2.045, n=100 gives t*=1.984. The difference between t* and z*=1.960 becomes negligible for n≥100. Always use t when σ is unknown — it gives wider, more honest intervals for small samples.
For a proportion at worst case (p̂=0.5): n=100 → ME=±9.8%, n=400 → ME=±4.9%, n=1000 → ME=±3.1%, n=2500 → ME=±2.0%. Halving the margin of error requires quadrupling the sample size. This is why most national polls use n=1000 (3% ME) and large studies use n=1600+ (2.5% ME). The standard “±3 points” margin of error in political polling assumes n≈1000 at 95% CI.
APA 7th edition: M = 74.3, 95% CI [71.2, 77.4]. For a regression coefficient: B = 0.42, 95% CI [0.18, 0.66]. For an odds ratio: OR = 2.14, 95% CI [1.45, 3.16]. Always use square brackets, no “to” between bounds. Report both bounds. Include n and SD or SE. The CI format is now preferred over p-values alone in most APA-compliant research reporting.
CI for (p̂⊂1;−p̂⊂2;): (p̂⊂1;−p̂⊂2;) ± 1.960 × √(p̂⊂1;(1−p̂⊂1;)/n⊂1; + p̂⊂2;(1−p̂⊂2;)/n⊂2;). If the CI excludes zero, the difference is significant at 5%. Example: Group 1: 60/100=0.60. Group 2: 45/100=0.45. Diff=0.15. SE=√(0.60×0.40/100+0.45×0.55/100)=0.0706. E=1.960×0.0706=0.138. 95% CI=(0.012, 0.288). Excludes zero → significant.
A 95% CI estimates where the population mean is (precision of the mean estimate). A 95% prediction interval (PI) estimates where a single new observation will fall (range for individual values). PI is always wider: PI = x̅ ± t* × s × √(1+1/n). For large n, PI ≈ mean ± 1.96σ (the empirical rule). Example: CI might be ±2 points for the mean, while PI is ±20 points for individual observations. Use CI for estimating the population mean; use PI for predicting a new individual value.
There is a direct equivalence: if the null hypothesis value falls outside the 95% CI, you reject H⊂0; at α=0.05 (two-tailed). If the null value is inside the CI, fail to reject H⊂0;. This duality means: (1) The CI tells you everything the p-value tells you AND more. (2) If 95% CI for a mean excludes μ⊂0;=0, then p<0.05. (3) If 95% CI for a difference excludes 0, then p<0.05 for the t-test. CIs provide a richer summary by showing the plausible effect size range, not just binary significance.
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