90% Confidence Interval Calculator

Calculator

This tool computes 90% intervals (α = 0.10). Choose an estimate type, fill inputs, then submit.

Estimate type

Confidence level

Two-sided interval uses α = 0.10.

Output precision

Affects display only, not computation.

Sample mean (x̄)

Sample standard deviation (s)

Population sigma (σ)

Sample size (n)

Successes (x)

Trials (n)

Mean 1 (x̄1)

SD 1 (s1)

n1

Mean 2 (x̄2)

SD 2 (s2)

n2

Assume equal variances (pooled)

Unchecked uses Welch’s method.

Successes x1

Trials n1

Successes x2

Trials n2

Example data table

Sample scenarios and typical outputs (rounded). Values may differ slightly due to rounding and method choice.

Scenario	Inputs	Center	90% CI	Method
Mean (unknown σ)	x̄=52.3, s=10.1, n=40	52.3	[~49.5, ~55.1]	t interval
Proportion	x=37, n=50	~0.73	[~0.62, ~0.82]	Wilson
Δ means	x̄1=14.2,s1=3.1,n1=25; x̄2=12.7,s2=2.9,n2=28	1.5	[~0.2, ~2.8]	Welch

Formula used

Mean, known σ: x̄ ± z_α/2 · (σ/√n)
Mean, unknown σ: x̄ ± t_{α/2, df} · (s/√n), with df = n − 1
Proportion (Wilson): robust for small samples and extreme rates; uses z_α/2
Δ means: pooled (equal variances) or Welch (unequal variances) using t critical values
Δ proportions: (p̂1 − p̂2) ± z_α/2 · √(p̂1(1−p̂1)/n1 + p̂2(1−p̂2)/n2)

How to use this calculator

Select the estimate type you need.
Enter the required summary statistics or counts.
Press Submit to view the 90% interval.
Use CSV or PDF buttons to save the result.
Switch methods to compare interval behavior.

FAQs

1) What does a 90% interval mean?

If you repeated sampling many times, about 90% of the constructed intervals would contain the true parameter. It is not a probability statement about one fixed interval.

2) Why is α equal to 0.10 here?

A two-sided 90% interval leaves 10% total error outside the bounds, split as 5% in each tail. The critical value is computed from that tail probability.

3) When should I use a t interval for the mean?

Use a t interval when the population standard deviation is unknown and you estimate variability with the sample standard deviation. It is especially appropriate for small to moderate sample sizes.

4) Why use Wilson for proportions?

Wilson intervals behave better than the basic normal method when n is small or the success rate is near 0 or 1. They reduce boundary issues and improve coverage.

5) What is Welch’s method for two means?

Welch’s method estimates the standard error without assuming equal variances. It also uses an adjusted degrees-of-freedom formula to compute a more reliable critical value.

6) Can the bounds be negative or above 1?

For means and differences, negative bounds are possible and often valid. For single proportions, this calculator clamps Wilson bounds to the [0,1] range for interpretability.

7) How should I report the result?

State the estimate, the 90% interval, and the method used (t, z, Wilson, Welch). Include sample sizes and units so others can interpret uncertainty correctly.