📊 Select Interval Type
📌 Use when population standard deviation σ is known. Critical value z* = 1.645 for 90% CI. Formula: x̅ ± 1.645 × (σ / √n)
x̅
Enter a valid sample mean.
σ
Enter σ > 0.
n
Enter n ≥ 1.
📌 Use when population std dev is unknown. Uses the t-distribution with df = n − 1. For large n (≥ 30) the t critical value approaches z* = 1.645.
x̅
Enter a valid sample mean.
s
Enter s > 0.
n
Enter n ≥ 2.
📌 Use for proportions (success/failure counts). Formula: p̂ ± 1.645 × √(p̂(1−p̂)/n). Valid when np̂ ≥ 5 and n(1−p̂) ≥ 5.
x
Or enter proportion directly below
Enter a valid count (0 to n).
n
Enter n ≥ 1.
90% Confidence Interval
—
⚠️ Disclaimer: Results are for educational and informational purposes. Verify critical statistical decisions with a qualified statistician.
Was this calculator helpful?
Sources & Methodology
Critical values and formulas verified against NIST Statistical Tables and standard introductory statistics textbooks. T critical values computed via accurate approximation.
NIST/SEMATECH e-Handbook of Statistical Methods
Section 7.2 — Interval Estimation. Reference for one-sample confidence interval formulas, critical value selection, and appropriate use of z vs t distributions.
Moore, D.S. — The Basic Practice of Statistics (8th ed.)
Standard reference for confidence interval interpretation, the distinction between z-intervals and t-intervals, and the conditions required for valid confidence interval construction.
Formulas:
90% CI for mean (known σ): x̅ ± 1.645 × (σ / √n) 90% CI for mean (unknown σ): x̅ ± t*(df=n−1, 90%) × (s / √n) 90% CI for proportion: p̂ ± 1.645 × √(p̂(1−p̂) / n) z* = 1.645 exact (invNorm(0.95)). T critical values via rational approximation. Margin of error E = half the interval width.
90% CI for mean (known σ): x̅ ± 1.645 × (σ / √n) 90% CI for mean (unknown σ): x̅ ± t*(df=n−1, 90%) × (s / √n) 90% CI for proportion: p̂ ± 1.645 × √(p̂(1−p̂) / n) z* = 1.645 exact (invNorm(0.95)). T critical values via rational approximation. Margin of error E = half the interval width.
90% Confidence Interval — Formula, Examples, and When to Use It
A 90% confidence interval is a range of values constructed from sample data that, if the procedure were repeated many times, would contain the true population parameter 90% of the time. The 90% CI is narrower than a 95% or 99% CI, offering more precision at the cost of slightly less confidence.
The Critical Value: z* = 1.645
The defining characteristic of a 90% CI is its critical value. The middle 90% of the standard normal distribution lies between z = −1.645 and z = +1.645. This means 5% of the distribution falls below −1.645 and 5% falls above +1.645 (α/2 = 0.05 in each tail).
90% CI for mean (known σ): x̅ ± z* × SE = x̅ ± 1.645 × (σ / √n)
Example: A sample of n=25 IQ scores has mean x̅=105. Population σ=15.
SE = 15/√25 = 15/5 = 3.0
Margin of error E = 1.645 × 3.0 = 4.935
90% CI = (105 − 4.935, 105 + 4.935) = (100.07, 109.94)
Interpretation: We are 90% confident the true mean IQ lies between 100.1 and 109.9.
SE = 15/√25 = 15/5 = 3.0
Margin of error E = 1.645 × 3.0 = 4.935
90% CI = (105 − 4.935, 105 + 4.935) = (100.07, 109.94)
Interpretation: We are 90% confident the true mean IQ lies between 100.1 and 109.9.
90% CI When Population Std Dev Is Unknown (t-distribution)
When σ is unknown and you estimate it using the sample standard deviation s, replace z* = 1.645 with the t critical value from the t-distribution with df = n − 1. The t critical value is always larger than 1.645, reflecting additional uncertainty from estimating σ.
90% CI for mean (unknown σ): x̅ ± t*(df, 90%) × (s / √n)
T critical values at 90% confidence level:
df = 9 (n=10): t* = 1.833 | df = 14 (n=15): t* = 1.761
df = 19 (n=20): t* = 1.729 | df = 29 (n=30): t* = 1.699
df = 49 (n=50): t* = 1.677 | df = 99 (n=100): t* = 1.660
df = ∞ (large n): t* → 1.645 (converges to z*)
df = 9 (n=10): t* = 1.833 | df = 14 (n=15): t* = 1.761
df = 19 (n=20): t* = 1.729 | df = 29 (n=30): t* = 1.699
df = 49 (n=50): t* = 1.677 | df = 99 (n=100): t* = 1.660
df = ∞ (large n): t* → 1.645 (converges to z*)
90% CI for a Proportion
When estimating a population proportion (e.g., the percentage of voters supporting a candidate), use the normal approximation with z* = 1.645. The conditions for validity: np̂ ≥ 5 and n(1−p̂) ≥ 5. If either condition fails, use exact (Clopper-Pearson) methods.
90% CI for proportion: p̂ ± 1.645 × √(p̂(1−p̂) / n)
Example: Survey of n=400 voters, 216 support a candidate (p̂ = 216/400 = 0.54).
SE = √(0.54 × 0.46 / 400) = √(0.000621) = 0.02492
E = 1.645 × 0.02492 = 0.041
90% CI = (0.54 − 0.041, 0.54 + 0.041) = (0.499, 0.581)
Interpretation: We are 90% confident between 49.9% and 58.1% of voters support the candidate.
SE = √(0.54 × 0.46 / 400) = √(0.000621) = 0.02492
E = 1.645 × 0.02492 = 0.041
90% CI = (0.54 − 0.041, 0.54 + 0.041) = (0.499, 0.581)
Interpretation: We are 90% confident between 49.9% and 58.1% of voters support the candidate.
Comparing Confidence Levels: 90% vs 95% vs 99%
| Confidence Level | Critical Value z* | Relative Width | Alpha (α) | Best For |
|---|---|---|---|---|
| 90% | 1.645 | Narrowest | 0.10 | Exploratory, limited n, economics |
| 95% | 1.960 | Middle | 0.05 | Default in most research |
| 99% | 2.576 | Widest | 0.01 | Safety-critical, regulatory |
A 90% CI is approximately 16% narrower than a 95% CI (ratio 1.645/1.960 = 0.839). This means if your 95% CI has margin of error 10 units, your 90% CI has margin of error about 8.4 units with the same data. The tradeoff: 10% of 90% CIs constructed this way will miss the true value, versus 5% of 95% CIs.
What a 90% Confidence Interval Does NOT Mean
- Wrong: “There is a 90% probability the true mean is between 100.1 and 109.9.” The true mean is a fixed (unknown) number, not random. Probability does not apply.
- Wrong: “90% of data values fall in this interval.” That would describe a prediction interval, not a confidence interval.
- Correct: “This interval was constructed by a procedure that captures the true mean 90% of the time across repeated samples.”
- Correct: “We are 90% confident [in our procedure] that the interval (100.1, 109.9) contains the true mean.”
💡 Sample size and precision: The margin of error of a 90% CI is proportional to 1/√n. To halve the margin of error, you must quadruple the sample size. If n=100 gives a margin of error of 8 units, you need n=400 to get 4 units. This is why collecting four times as much data only doubles precision — a common surprise in research planning.
Frequently Asked Questions
A 90% confidence interval is a range of values that would contain the true population parameter in 90% of repeated samples. The 90% refers to the long-run success rate of the procedure, not the probability for any single interval. The true value is either in the interval or it is not — for any single computed interval, we say we are “90% confident” it contains the true value.
z* = 1.645 for a 90% CI. This is the exact value of invNorm(0.95): the z-score with 95% of the standard normal distribution to its left and 5% in the right tail. With 5% in each tail (α/2 = 0.05), the middle 90% lies between −1.645 and +1.645. For comparison: 95% CI uses z*=1.960, 99% CI uses z*=2.576.
Known σ: x̅ ± 1.645 × (σ/√n). Unknown σ (use t): x̅ ± t*(df=n−1) × (s/√n). The standard error SE = σ/√n or s/√n. The margin of error E = z* or t* times the SE. The interval is (x̅−E, x̅+E). For n≥30, z*=1.645 is a good approximation even when σ is unknown.
p̂ ± 1.645 × √(p̂(1−p̂)/n). Compute p̂ = x/n (successes/total). Check validity: n×p̂ ≥ 5 AND n×(1−p̂) ≥ 5. Calculate SE = √(p̂(1−p̂)/n). Margin of error E = 1.645 × SE. CI = (p̂−E, p̂+E). Example: x=180, n=300 → p̂=0.60, SE=0.02828, E=0.0465, 90% CI=(0.554, 0.647).
Because less confidence requires a smaller critical value. 90% CI uses z*=1.645; 95% uses z*=1.960. The ratio is 1.645/1.960 = 0.839, so the 90% CI is about 16% narrower for the same data and same n. The tradeoff: the narrower 90% CI misses the true value 10% of the time (vs 5% for 95%). Narrower CI = more precise estimate = more uncertainty about coverage.
Use 90% CI when: (1) you need a more precise (narrower) interval and can accept 10% error rate instead of 5%, (2) the field convention uses 90% (common in economics, some engineering applications), (3) sample size is limited and a 95% CI would be too wide to be useful, or (4) exploratory research where a preliminary estimate is needed. The 95% CI remains the default for most published scientific research.
E = z* × SE = 1.645 × (σ/√n) for a mean with known σ. E = 1.645 × √(p̂(1−p̂)/n) for a proportion. The full interval width = 2E. To reduce the margin of error by half, quadruple the sample size. Worst-case proportion margin of error at 90%: E = 1.645/2 × (1/√n) = 0.8225/√n. For n=1000: worst-case E = 0.026 = 2.6%.
For a proportion: n = (z*/E)² × p̂(1−p̂). Worst case (p̂=0.5): n = (1.645/E)² × 0.25. For E=5%: n = (32.9)² × 0.25 = 271. For E=3%: n = (54.8)² × 0.25 = 750. For E=2%: n = (82.25)² × 0.25 = 1690. For a mean: n = (z* × σ / E)² = (1.645 × σ / E)².
Correct: “We are 90% confident the true population mean lies between [lower] and [upper].” Or: “This interval was constructed by a procedure that successfully captures the true parameter 90% of the time.” Incorrect: “There is a 90% probability the true value is in this interval.” The true value is fixed; once computed, the interval either contains it or does not. The confidence refers to the method’s long-run reliability.
Use z* = 1.645 when population σ is known, or when n ≥ 30 (CLT makes the normal approximation valid). Use t* from the t-distribution with df = n−1 when σ is unknown and n < 30. The t* for a 90% CI is always > 1.645. For n=10: t*=1.833. For n=20: t*=1.729. For n=30: t*=1.699. For n=100: t*=1.660. As n → ∞, t* → 1.645.
Alpha (α) is the significance level = 1 − confidence level = 1 − 0.90 = 0.10 = 10%. This is the error rate: 10% of 90% CIs constructed this way will miss the true parameter. In a two-tailed construction, α/2 = 5% falls in each tail. A 90% CI corresponds to a two-tailed hypothesis test at the α = 10% significance level: if the null value lies outside the 90% CI, the test is significant at the 10% level.
Yes, if the estimated parameter (mean or proportion) minus the margin of error goes below zero. For a mean, negative lower bounds are perfectly valid (e.g., change scores, temperature differences). For a proportion, the CI should be bounded at 0 and 1 since proportions cannot fall outside [0,1]. If the calculated lower bound falls below 0 or the upper bound exceeds 1, apply a boundary correction or use exact Clopper-Pearson methods for small samples.
Related Statistics Calculators