90 Confidence Interval Two Samples Calculator

Calculator Inputs

Interval Method

Critical Value

Custom Critical Value

Group 1 Label

Group 2 Label

Decimal Places

Mean Difference Inputs

Sample 1 Mean

Sample 1 Standard Deviation

Sample 1 Size

Sample 2 Mean

Sample 2 Standard Deviation

Sample 2 Size

Known Sigma 1

Known Sigma 2

Proportion Difference Inputs

Sample 1 Successes

Sample 1 Size

Sample 2 Successes

Sample 2 Size

Reset

Example Data Table

Scenario	Method	Sample 1	Sample 2	Approximate 90% Interval
Average score comparison	Welch means	Mean 82.4, SD 10.5, n 42	Mean 78.6, SD 9.8, n 39	0.05 to 7.55
Known process variation	Known sigma	Mean 76, sigma 8.5, n 50	Mean 72, sigma 8.0, n 45	1.21 to 6.79
Survey conversion rate	Two proportions	56 successes from 120	47 successes from 110	-0.07 to 0.15

Formula Used

Welch means: CI = (x̄1 - x̄2) ± t0.95,df × √(s1²/n1 + s2²/n2).

Welch degrees of freedom: df = (a + b)² / [a²/(n1 - 1) + b²/(n2 - 1)], where a = s1²/n1 and b = s2²/n2.

Pooled means: CI = (x̄1 - x̄2) ± t0.95,df × sp × √(1/n1 + 1/n2).

Known sigma means: CI = (x̄1 - x̄2) ± z0.95 × √(σ1²/n1 + σ2²/n2).

Two proportions: CI = (p1 - p2) ± z0.95 × √[p1(1-p1)/n1 + p2(1-p2)/n2].

How to Use This Calculator

Choose the interval method that matches your data type.
Enter means, standard deviations, and sizes for average comparisons.
Enter successes and sample sizes for proportion comparisons.
Keep the automatic 90% critical value or enter your own.
Press the calculate button to view the interval above the form.
Download the result as CSV or PDF for reporting.

Understanding the 90% Two Sample Interval

A 90% confidence interval for two samples estimates a difference between two populations. The difference may compare two averages or two proportions. This calculator keeps the focus on the interval, the standard error, and the margin of error. It helps you see whether the difference is practically small, large, or uncertain.

What It Estimates

For average data, enter each sample mean, standard deviation, and sample size. Welch's method is usually the safer default. It does not assume equal population variances. The pooled method is useful when equal variance is reasonable. The known sigma method applies when population standard deviations are known from trusted process data.

For proportion data, enter successes and sample sizes. The tool calculates each sample rate, their difference, and a 90% interval using the normal approach. This works best when both groups have enough successes and failures. Very small counts may need exact or adjusted methods.

Method Choice

The confidence level affects the critical value. A 90% interval uses less width than a 95% interval. It gives a narrower range, but it also accepts more long run risk. In repeated sampling, about ninety of one hundred intervals would capture the true difference, when assumptions hold.

Reading Bounds

The lower and upper bounds give the estimated range for group one minus group two. A positive interval suggests group one is higher. A negative interval suggests group two is higher. An interval crossing zero means the data do not clearly separate the groups at this confidence level.

Reporting Tips

Use the report carefully. Check that samples are independent. Verify that measurements are consistent. Watch for extreme outliers. For proportions, ensure the events have the same definition in both groups.

The calculator is useful for classroom problems, quality checks, experiments, surveys, and business comparisons. It can document the formula path and export the main result clearly.

When reporting results, include the method name, confidence level, sample sizes, standard error, and margin of error. Avoid saying the true value has a ninety percent chance of being inside one finished interval. The interval either contains it or not. The confidence statement describes repeated sampling performance, not probability for a fixed parameter under the stated sampling model.

FAQs

What does a 90% confidence interval mean?

It means the method would capture the true population difference in about 90% of repeated samples, if assumptions hold. It does not mean one finished interval has a 90% probability of containing the fixed true value.

When should I use Welch's method?

Use Welch's method when comparing two means and you do not want to assume equal population variances. It is a strong default for independent samples with different standard deviations or different sample sizes.

When is the pooled method suitable?

The pooled method is suitable when both groups are independent and equal population variance is reasonable. It combines sample variation into one pooled standard deviation before calculating the standard error.

Can this calculator compare proportions?

Yes. Choose the two proportion method. Enter successes and total sample sizes for both groups. The calculator estimates p1 minus p2 and builds a 90% interval with the normal approach.

Why does my interval cross zero?

An interval crossing zero means zero difference is plausible at the 90% confidence level. The sample difference may still exist, but the interval does not clearly separate the groups.

What is the margin of error?

The margin of error is the critical value multiplied by the standard error. It is added and subtracted from the estimated difference to create the lower and upper interval bounds.

Should I enter a custom critical value?

Use the automatic setting for most work. Enter a custom value only when your class, report, table, or software standard requires a specific critical value.

Can I use this for paired samples?

This tool is designed for independent two sample comparisons. For paired data, calculate the within-pair differences first, then use a one sample interval on those differences.