Two Tailed Test Calculator

Calculator Inputs

Test Type

Sample Mean Or Mean Difference

Hypothesized Mean Or Difference

Sample Standard Deviation

Population Standard Deviation

Sample Size

Sample Variance

Hypothesized Variance

Alpha Level

Decimal Places

Example Data Table

Case	Test Type	Sample Value	Null Value	Spread	Sample Size	Alpha
Mean Study	T Test	102.4	100	8.5	36	0.05
Known Sigma	Z Test	49.2	50	Population SD 3.1	64	0.01
Variance Claim	Variance Test	Sample variance 72.25	64	df from n	25	0.05

Formula Used

Z test: z = (x̄ - μ₀) / (σ / √n). Two tailed p value = 2 × P(Z ≥ |z|).

T test: t = (x̄ - μ₀) / (s / √n). Degrees of freedom = n - 1.

Paired t test: t = (d̄ - d₀) / (s_d / √n).

Variance test: χ² = (n - 1)s² / σ₀². The smaller tail area is doubled.

How To Use This Calculator

Select the test type that matches your data and claim.
Enter the sample value, null value, spread measure, and sample size.
Use a common alpha level, such as 0.05, unless your study needs another value.
Press Calculate to view the result below the header and above the form.
Use the CSV or PDF button to save the same calculation details.

Two Tailed Testing Guide

A two tailed test checks both directions of a claim. It looks for a result that is either too high or too low. This matters when the research question is about change, difference, or mismatch. The null hypothesis says the parameter equals a chosen value. The alternative says it is not equal. The calculator uses that structure.

When The Test Is Useful

Use this tool when you have a sample mean, a claimed mean, and a measure of spread. Use the z option when the population standard deviation is known. Use the t option when only sample standard deviation is known. Use the paired option for mean differences from matched observations. Use the variance option when the claim is about variance.

How The Decision Works

The calculator finds a statistic first. It then finds a two sided p value. The p value measures how unusual the statistic is under the null hypothesis. A small value gives evidence against the claim. The alpha level sets the rejection rule. Common choices are 0.05, 0.01, and 0.10. The result also shows critical limits. These limits mark the rejection regions on both tails.

Why Confidence Limits Help

Confidence limits add practical meaning. For mean tests, the interval estimates the likely range for the true mean. For variance tests, it estimates the likely range for the true variance. If the null value falls outside the interval, the two tailed test usually rejects. This gives a useful cross check.

Reading The Output

A reject decision does not prove a claim is false. It means the sample is unlikely under the chosen model. A fail to reject decision also has limits. It does not prove the null value is true. It means the sample did not give enough evidence. Always check data quality, sample design, independence, and measurement units before trusting the result.

Practical Notes

Round only after calculation. Enter raw values when possible. Very small samples need careful review. Outliers can move the statistic sharply. The test also assumes the chosen distribution fits the situation. Use subject knowledge with the numeric answer. Keep the exported report with your data notes, because later readers need context. Review assumptions before use.

FAQs

What is a two tailed test?

It is a hypothesis test that checks for a difference in both directions. The result may be significantly higher or significantly lower than the null value.

When should I use a t test?

Use a t test when you are testing a mean and the population standard deviation is not known. The sample standard deviation is used instead.

When should I use a z test?

Use a z test when the population standard deviation is known and the sampling model is suitable. It is common with large, well defined studies.

What does the p value mean?

The p value shows how unusual the sample result is if the null hypothesis is true. Smaller values give stronger evidence against the null claim.

What does alpha mean?

Alpha is the chosen significance level. It is the cutoff used to decide whether the p value is small enough to reject the null hypothesis.

Can this test prove the null hypothesis?

No. A fail to reject decision means the sample did not provide enough evidence. It does not prove that the null value is exactly true.

Why are there two critical limits?

A two tailed test splits alpha between both tails. One limit checks unusually low results. The other limit checks unusually high results.

What should I check before trusting results?

Check sample quality, independence, measurement units, outliers, and whether the selected test matches your data. Bad inputs can give misleading decisions.