Calculator
Example Data Table
| Case | Distribution | Alpha | Degrees | Expected Critical Values |
|---|---|---|---|---|
| Large sample mean test | Normal z | 0.05 | Not needed | -1.959964, 1.959964 |
| Small sample mean test | Student t | 0.05 | 10 | About -2.228139, 2.228139 |
| Variance test | Chi-square | 0.05 | 12 | Lower and upper chi-square limits |
| Variance ratio test | F distribution | 0.05 | 5, 20 | Lower and upper F limits |
Formula Used
For a two tailed test, alpha is split into two equal parts.
Left tail probability = α / 2
Right tail probability = 1 - α / 2
Normal critical values = Φ⁻¹(α / 2) and Φ⁻¹(1 - α / 2)
Student t critical values = t⁻¹(α / 2, df) and t⁻¹(1 - α / 2, df)
Chi-square critical values = χ²⁻¹(α / 2, df) and χ²⁻¹(1 - α / 2, df)
F critical values = F⁻¹(α / 2, df1, df2) and F⁻¹(1 - α / 2, df1, df2)
Two tailed p value = 2 × smaller tail probability.
How to Use This Calculator
Select the distribution that matches your hypothesis test.
Enter alpha, such as 0.05, or enter confidence level.
Enter degrees of freedom when the selected distribution needs them.
For an F test, enter numerator and denominator degrees.
Add the observed test statistic if you want a decision.
Choose decimal places for the displayed output.
Press Calculate to show results above the form.
Use the CSV or PDF buttons to save the result.
About the Critical Value Two Tailed Test Calculator
A two tailed test checks both ends of a distribution. It is used when a result can be too small or too large. This calculator finds the lower and upper critical values for common statistical tests. It supports z, t, chi-square, and F distributions.
Why critical values matter
A critical value is a boundary for decision making. The area beyond each boundary is the rejection region. In a two tailed test, alpha is split equally between the left tail and the right tail. For example, alpha 0.05 gives 0.025 in each tail.
Supported distributions
Use the normal option when population variation is known, or when a large sample justifies a z test. Use the t option when the sample standard deviation estimates variation. Use chi-square for variance tests. Use F for comparing two variances or model mean squares.
Interpreting results
The calculator reports the left critical value and the right critical value. A test statistic outside these values is statistically significant at the chosen alpha. A statistic inside the interval is not significant. The decision depends on the distribution, alpha level, and degrees of freedom.
Advanced inputs
You can enter alpha directly, or enter confidence level. The tool then converts confidence to alpha. Degrees of freedom are required for t, chi-square, and F tests. F tests need numerator and denominator degrees of freedom. Decimal control keeps outputs readable.
Practical use
This calculator is useful for classwork, research checks, lab reports, and quality studies. It also helps compare manual textbook values with computed values. Use the example table to understand typical settings. Then enter your own test details and download the result report.
Good reporting habits
Report the distribution, alpha, degrees of freedom, and both critical values. Also report the observed statistic. This makes the decision easy to audit. Never round too early in the calculation. Rounding can move a statistic across a boundary. Use the same alpha that was chosen before looking at data. This protects the test from bias. When assumptions are weak, treat the result with care. Critical values support judgment, but they do not replace study design. Clear notes help readers repeat the same method later accurately.
FAQs
What is a two tailed critical value?
It is a cutoff used on both sides of a sampling distribution. Values beyond either cutoff fall in the rejection region for the selected alpha level.
How is alpha split in a two tailed test?
Alpha is divided equally between both tails. If alpha is 0.05, the left tail gets 0.025 and the right tail gets 0.025.
When should I use the normal z option?
Use it for z tests. It fits large samples or cases where the population standard deviation is known and normal assumptions are reasonable.
When should I use the t option?
Use it when working with sample standard deviation and degrees of freedom. It is common for small sample mean tests.
Why does chi-square have two positive critical values?
The chi-square distribution starts at zero. A two tailed variance test still has lower and upper rejection limits, but both are nonnegative.
Why does the F test need two degrees of freedom?
The F distribution compares two variance estimates. It needs numerator degrees of freedom and denominator degrees of freedom.
Can I enter confidence instead of alpha?
Yes. Enter a confidence percentage. The calculator converts it to alpha using alpha equals one minus confidence as a decimal.
What does the decision mean?
If the statistic is outside the critical interval, reject the null hypothesis. If it is inside, do not reject the null hypothesis.