Confidence Interval Calculator
Enter your dataset to get mean and SD instantly, then compute the confidence interval CI = mean ± z×(SD/√n). Results show 90%, 95%, and 99% CIs with interpretation.
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How It Works
Enter all values in the Descriptive Stats tab, separated by commas. Press Calculate. Find: Mean (x̄), Sample SD (s), and SEM = s/√n in the results panel. These three values are the inputs to the CI formula. The n (count) also shows in the results.
SEM = SD / √n | find these three values in resultsn=25, mean=50, SD=10 → SEM=10/5=2.0Using mean, SEM, and z-score: CI = mean ± z × SEM. 90% CI: z=1.645. 95% CI: z=1.96. 99% CI: z=2.576. For mean=50, SEM=2, n=25: 95% CI = 50 ± 1.96×2 = 50 ± 3.92 = [46.08, 53.92]. The margin of error = z × SEM.
CI = mean ± z × SEM | z: 90%→1.645, 95%→1.96, 99%→2.576mean=50, SEM=2: 95% CI = [46.08, 53.92] (±3.92)For n < 30, replace z with t (from t-distribution with n−1 degrees of freedom). For n=10: t₀.₀₅ = 2.262 (wider than z=1.96). For n ≥ 30, z and t converge and z is appropriate. This calculator uses z — for small samples, note that the CI should use t-values from a t-table.
n≥30: use z | n<30: use t with df=n−1n=10: use t=2.262 (not z=1.96) for 95% CIMargin of error (ME) = z × SEM = z × SD/√n. To achieve a target ME: n = (z × SD / ME)². For ME=2, SD=10, 95% CI: n = (1.96×10/2)² = 9.8² = 96.04 → need n≥97. Larger SD or smaller ME requires more observations.
Required n = (z × SD / ME)²Target ME=2, SD=10, 95%: n=(1.96×10/2)²=96.04 → n≥97A 95% CI [46, 54] means: this interval was constructed using a method that captures the true mean 95% of the time. It does NOT mean there is a 95% probability the true mean is in [46,54] — the true mean either is or is not in that interval. Wider CI = less precision. Narrow CI + large n = high precision.
95% CI: the procedure works 95% of the timeCI [46,54] means: constructed from a method with 95% long-run coverageQuick Reference
Verify these in the calculator above.
z-score
z for 90% CI
1.645
z-score
z for 95% CI
1.96
z-score
z for 99% CI
2.576
SEM
SEM with SD=10, n=25
2.0
95% CI
95% CI: mean=50, SEM=2
[46.08, 53.92]
Sample size
ME doubles when n
halves (×4 reduces ME by ½)
Width
n=100 vs n=25: CI width
n=100 is half as wide
Small n
Use t instead of z when
n < 30
Tips & Shortcuts
The SEM appears in the results panel — divide it by SD if you want to verify: SEM = SD/√n. A lower SEM means better precision in your mean estimate.
For quick 95% CI by hand: mean ± 2×SEM (using z≈2 instead of 1.96). Close enough for a rough estimate.
The CV% (Coefficient of Variation) in the results helps assess data quality before computing CI — high CV means the data is very spread out relative to the mean.
Report both the CI and the sample size (n). A CI of [46,54] from n=100 is much more trustworthy than the same CI from n=10.
Outliers inflate SD and SEM, widening the CI. Check the outlier list in results before computing — removing genuine errors tightens the interval.
Common Mistakes
Saying "95% probability the true mean is in the CI"
The 95% refers to the procedure, not a specific interval. The true mean either is or is not in [46,54] — probability doesn't apply to a fixed (but unknown) constant. Correct: "we are 95% confident the true mean is between 46 and 54."
Using population SD for the CI formula
Use Sample SD (s, divides by n−1) for CIs based on sample data. Population SD underestimates true spread for samples. The results panel labels both — use the Sample SD row.
Using z when n < 30
For small samples (n < 30), the t-distribution gives wider, more accurate CIs. Use t with n−1 degrees of freedom. The calculator uses z — for n < 30, look up the appropriate t-value (e.g. t=2.262 for n=10, 95% CI).
Confusing SEM with SD
SD describes spread of your data. SEM = SD/√n describes precision of your mean estimate. SEM is always smaller than SD. Report SD when describing data variability; use SEM (or CI) when reporting precision of a mean estimate.
Assuming a wide CI means bad data
Wide CIs come from small n, large SD, or high confidence level. They indicate imprecision, not errors. Solution: increase sample size — doubling n reduces width by √2.
Frequently Asked Questions
A confidence interval gives a range of plausible values for a population parameter. A 95% CI means: if you repeated the study 100 times, about 95 of the resulting intervals would contain the true population mean. It is not a 95% chance the true mean falls in this specific interval.
CI = mean ± z × (SD / √n). For 95% CI: z = 1.96. Enter your data → get mean and SD → apply: lower = mean − 1.96×(SD/√n), upper = mean + 1.96×(SD/√n). The SEM (Standard Error of Mean) = SD/√n shows in the results panel.
SEM = SD / √n. It measures uncertainty in the sample mean itself — how much the mean would vary if you repeated sampling. A larger sample (bigger n) gives smaller SEM and tighter confidence intervals.
SD measures spread within your data — how much individual values vary from the mean. SEM measures uncertainty of the mean estimate itself. SEM = SD/√n, always smaller than SD for n > 1. Use SD to describe data variability; use SEM for confidence intervals.
z = 1.96 for 95% CI (two-tailed). z = 1.645 for 90%. z = 2.576 for 99%. These come from the standard normal distribution — 95% of values lie within ±1.96 standard deviations of the mean.
Larger n → smaller SEM → narrower CI. Doubling sample size reduces margin of error by √2 ≈ 1.41. For n=25: SEM=SD/5. For n=100: SEM=SD/10. A 95% CI with n=100 is half the width of one with n=25 (same SD).
If you repeated the sampling procedure 100 times and computed a 95% CI each time, about 95 of those intervals would contain the true population parameter. The "95%" refers to the procedure's long-run success rate, not the probability for any single interval.
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