Finding a critical value starts with identifying the statistical test you are using. The significance level, whether the test is one-tailed or two-tailed, and the relevant degrees of freedom determine which value you need. Depending on the test, you may use a z critical value, t critical value, chi-square critical value, or F critical value.
For example, a two-tailed z-test at a 5% significance level uses critical values of approximately −1.96 and +1.96. A t-test at the same significance level can have different cutoffs because its critical value also depends on the degrees of freedom.
You can find a critical value using a statistical table, statistical software, or an appropriate calculator. The key is knowing which distribution and probability to use before looking up the number.
How to Calculate a Critical Value Step by Step (Quick)
Follow these steps to find the appropriate critical value for your statistics problem.
Step 1: Identify the statistical test. Determine whether the problem calls for a z-test, t-test, chi-square test, F-test, or another procedure. This determines which probability distribution to use.
Step 2: Find the significance level. Use the stated alpha value or calculate it from the confidence level. For example, a 95% confidence level corresponds to (\alpha=0.05).
Step 3: Determine the direction of the test. Check the alternative hypothesis to decide whether the test is left-tailed, right-tailed, or two-tailed.
Step 4: Calculate the degrees of freedom if required. Use the formula appropriate for the test. For a one-sample t-test, subtract one from the sample size.
Step 5: Determine the required probability. For a standard distribution expressed using cumulative probability, use the following rules:
- Left-tailed test: (\alpha)
- Right-tailed test: (1-\alpha)
- Two-tailed test: (\alpha/2) for the lower cutoff and (1-\alpha/2) for the upper cutoff
Step 6: Find the corresponding value. Use the correct statistical table, software function, or calculator with the distribution and inputs you identified.
Step 7: Check the result. Confirm that the table uses the probability convention you intended, that the degrees of freedom are correct, and that you have included any required negative critical value.
The important point is that there is no single numerical critical value formula that applies to every statistical test. The calculation depends on the distribution and the probability cutoff you need.
What Is a Critical Value in Statistics?
A critical value is a cutoff point in a probability distribution that helps determine whether a statistical test result falls in a rejection region. In hypothesis testing, it separates results considered sufficiently extreme under the null hypothesis from those that do not cross the chosen threshold.
Suppose you want to determine whether a new teaching method changes students’ average test scores. You collect sample data and calculate a test statistic. The critical value provides the reference point against which you compare that statistic.
If your test statistic crosses the appropriate critical boundary, you reject the null hypothesis at the selected significance level. If it does not cross the boundary, you fail to reject the null hypothesis.
A critical value is different from a test statistic:
- Critical value: The cutoff determined by the statistical distribution, significance level, test direction, and sometimes degrees of freedom.
- Test statistic: The value calculated from your sample data, such as a z-score, t-score, or chi-square statistic.
- P-value: The probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the observed result under the specified test.
These quantities work together, but they are not interchangeable. The critical-value method compares the test statistic with a cutoff, while the p-value method compares the p-value with the significance level.
What Information Do You Need to Calculate a Critical Value?
Before using a formula or statistical table, identify the information your particular test requires.
1. Significance level or confidence level
The significance level, represented by (\alpha), determines the probability assigned to the rejection region under the null hypothesis. Common choices are 0.10, 0.05, and 0.01.
If a problem gives you a confidence level instead, you can usually find the corresponding significance level by subtracting the confidence level from 1.
| Confidence level | Significance level ((\alpha)) |
|---|---|
| 90% | 0.10 |
| 95% | 0.05 |
| 99% | 0.01 |
For example, a 95% confidence level corresponds to (\alpha=0.05). For a corresponding two-sided confidence interval, the remaining 5% probability is split between the two tails of the distribution.
2. Whether the test is one-tailed or two-tailed
The direction of the alternative hypothesis determines where the rejection region lies.
- Right-tailed test: Used when the alternative hypothesis tests whether a parameter is greater than a specified value.
- Left-tailed test: Used when the alternative hypothesis tests whether a parameter is less than a specified value.
- Two-tailed test: Used when the alternative hypothesis tests whether a parameter differs from a specified value in either direction.
For a one-tailed test at (\alpha=0.05), the entire 5% rejection probability lies in one tail. For a two-tailed test at the same significance level, each tail receives 2.5%.
This distinction changes the critical value. For a standard normal distribution, a right-tailed test at (\alpha=0.05) has a critical value of approximately 1.645, while a two-tailed test has critical values of approximately −1.96 and +1.96.
3. Degrees of freedom
Degrees of freedom, often written as (df), are needed for distributions such as the t, chi-square, and F distributions. Their calculation depends on the statistical procedure.
For a one-sample t-test, the degrees of freedom are:
[
df=n-1
]
Here, (n) is the sample size.
If a one-sample t-test uses 15 observations, its degrees of freedom are (15-1=14). You would use 14 degrees of freedom when looking up the appropriate t critical value.
Do not automatically use this formula for every test. For example, a chi-square test of independence and a one-way ANOVA calculate degrees of freedom differently.
How to Calculate a Z Critical Value
A z critical value is a cutoff on the standard normal distribution. It is commonly used in normal-based statistical procedures, including appropriate tests and confidence intervals for population proportions or means when the required conditions are met.
For a two-tailed z-test, the critical values are:
[
z_{\text{critical}}=\pm z_{1-\alpha/2}
]
The notation (z_p) represents the standard normal value whose cumulative probability to the left is (p).
Example: Find the z critical value for a 5% two-tailed test
Suppose you are testing a hypothesis at a significance level of 0.05.
- Set (\alpha=0.05).
- Divide the significance level between the two tails: (0.05/2=0.025).
- Find the cumulative probability for the upper cutoff: (1-0.025=0.975).
- Look up the z-score corresponding to a cumulative probability of 0.975.
The result is approximately:
[
z_{\text{critical}}=\pm1.96
]
Therefore, the critical values are −1.96 and +1.96.
If the test statistic is 2.10, it exceeds the upper critical value, so you reject the null hypothesis at the 5% significance level. If the test statistic is 1.50, it falls between the two cutoffs, so you fail to reject the null hypothesis.
For comparison, a right-tailed z-test at the same significance level uses an upper critical value of approximately 1.645 because the entire 5% rejection probability is in the right tail.
How to Calculate a T Critical Value
A t critical value comes from Student’s t distribution. It is often used for inference about a population mean when the population standard deviation is unknown and the assumptions of the selected t procedure are appropriate.
Unlike the standard normal distribution, the t distribution depends on degrees of freedom. With fewer degrees of freedom, it has heavier tails, which generally means a more extreme critical value is needed for the same confidence level.
For a two-tailed t-test, the critical values can be written as:
[
t_{\text{critical}}=\pm t_{1-\alpha/2,,df}
]
Example: Find the t critical value for a sample of 10
Suppose you have a sample of 10 observations and need the critical value for a two-tailed t-test at (\alpha=0.05).
Step 1: Calculate the degrees of freedom.
[
df=n-1=10-1=9
]
Step 2: Split alpha between the two tails.
[
\alpha/2=0.05/2=0.025
]
Step 3: Find the upper-tail cutoff using a cumulative probability of (1-0.025=0.975) and 9 degrees of freedom.
The t-table gives an upper critical value of approximately 2.262.
Therefore, the critical values are:
[
t_{\text{critical}}=\pm2.262
]
If the calculated t-statistic is 2.50, it lies beyond the upper cutoff, so you reject the null hypothesis. If the t-statistic is 1.80, it does not cross either cutoff, so you fail to reject the null hypothesis.
The sample size matters here because it determines the degrees of freedom. With a different sample size, the critical value may change even if the significance level remains 0.05.
How to Calculate a Chi-Square Critical Value
Chi-square critical values are used in several procedures, including tests of independence, goodness-of-fit tests, and tests concerning a population variance. The appropriate test determines the degrees of freedom and whether one or both tails are relevant.
Unlike the standard normal and t distributions, the chi-square distribution is not symmetric around zero. Its values are nonnegative, and its shape depends on the degrees of freedom.
Example: Find a chi-square critical value
Suppose a chi-square test uses:
- Significance level: (\alpha=0.05)
- Degrees of freedom: (df=4)
- Test direction: right-tailed
Look up the upper-tail critical value for 4 degrees of freedom at the 5% significance level.
The critical value is approximately:
[
\chi^2_{\text{critical}}=9.488
]
If your calculated chi-square statistic is greater than 9.488, it falls in the rejection region for this test. If it is less than or equal to 9.488, you do not reject the null hypothesis using this upper-tail decision rule.
Do not assume that every chi-square test uses only an upper-tail cutoff. For example, a two-sided test of a population variance can require both lower and upper critical values.
How to Calculate an F Critical Value
An F critical value comes from the F distribution. It is commonly used in analysis of variance (ANOVA) and other procedures involving ratios of variance estimates.
Finding an F critical value generally requires three inputs:
- The significance level, (\alpha)
- The numerator degrees of freedom, (df_1)
- The denominator degrees of freedom, (df_2)
Example: Find an F critical value
Suppose an F-test has a significance level of 0.05, numerator degrees of freedom of 2, and denominator degrees of freedom of 9. The test is right-tailed.
Look up the F distribution using:
[
\alpha=0.05,\quad df_1=2,\quad df_2=9
]
The upper critical value is approximately:
[
F_{\text{critical}}=4.256
]
If the calculated F-statistic exceeds 4.256, it falls in the rejection region for this right-tailed test. Otherwise, it does not cross the upper cutoff.
F critical values are not interchangeable with t or z critical values. Even when two tests use the same significance level, the distribution and degrees of freedom determine the appropriate cutoff.
How to Find a Critical Value Using a Statistical Table
A statistical table is useful when you need to solve a problem without software or when an assignment specifically requires table lookup.
The process depends on the distribution, but the general approach is similar.
- Choose the correct table. Use a z-table for standard normal values, a t-table for t critical values, a chi-square table for chi-square values, or an F-table for F values.
- Identify the probability convention. Check whether the table uses cumulative probability to the left or probability in the upper tail.
- Locate the significance level. For a two-tailed test, you may need (\alpha/2) or a cumulative probability of (1-\alpha/2).
- Find the degrees of freedom. For t and chi-square tables, locate the relevant row. For an F-table, find both the numerator and denominator degrees of freedom.
- Read the intersection. The value at the appropriate row and column is the critical value.
- Check the sign and tail. A two-tailed t-test has lower and upper critical values, while chi-square and F distributions require attention to their own tail behavior.
One common source of mistakes is reading a table that reports upper-tail probabilities as if it reports cumulative probabilities. For example, a right-tail probability of 0.05 corresponds to a cumulative probability of 0.95, not 0.05.
When a table does not include the exact degrees of freedom or significance level you need, use an appropriate statistical calculator or software rather than silently substituting a different value.
How to Use a Critical Value to Make a Hypothesis-Test Decision
Finding the critical value is only part of the process. To make a decision, compare it with the test statistic calculated from your sample data.
For a two-tailed z-test at (\alpha=0.05), the critical values are −1.96 and +1.96.
| Calculated z-statistic | Decision using the critical-value method |
|---|---|
| −2.20 | Reject the null hypothesis |
| 1.50 | Fail to reject the null hypothesis |
| 2.10 | Reject the null hypothesis |
The first and third statistics fall outside the interval from −1.96 to +1.96. The second falls within it.
For a right-tailed test, reject the null hypothesis when the test statistic reaches or exceeds the upper critical value. For a left-tailed test, reject when the test statistic reaches or falls below the lower critical value.
The direction of the test matters. A large positive statistic may be evidence against the null hypothesis in a right-tailed test but not in a left-tailed test.
Also, failing to reject the null hypothesis does not prove that it is true. It means the available evidence did not meet the rejection criterion at the selected significance level.
How Is a Critical Value Used in a Confidence Interval?
Critical values are also used when estimating a population parameter with a confidence interval. In this setting, the critical value helps determine the margin of error around a sample estimate.
The general formula is:
[
\text{Confidence interval}
=\text{Estimate}\pm
(\text{Critical value}\times\text{Standard error})
]
The appropriate critical value depends on the confidence level and the sampling distribution used by the procedure.
For example, a two-sided 95% confidence interval using the standard normal distribution uses a critical value of approximately 1.96. If the interval estimates a population mean using a t distribution, the critical value depends on the relevant degrees of freedom.
A higher confidence level generally requires a larger critical value and therefore a wider interval, assuming the estimate and standard error stay the same.
If you are calculating an interval rather than conducting a hypothesis test, use the Confidence Interval Calculator to help determine the interval from the relevant inputs.
Common Mistakes When Calculating a Critical Value
Even when the correct table is available, small errors in choosing the probability or distribution can produce the wrong answer.
Using the wrong tail. A one-tailed test at a 5% significance level uses a different cutoff from a two-tailed test at the same alpha. Check the alternative hypothesis before looking up the value.
Confusing alpha with the confidence level. A 95% confidence level corresponds to (\alpha=0.05), not 0.95.
Dividing alpha by two unnecessarily. Divide alpha between two tails only when the procedure is two-tailed.
Using the wrong degrees of freedom. A one-sample t-test commonly uses (n-1), but other tests have different degrees-of-freedom formulas.
Choosing the wrong distribution. A z critical value cannot simply replace a t critical value because both are familiar or because the sample is small or large. Select the distribution based on the statistical procedure and its assumptions.
Misreading the table. Confirm whether the table gives cumulative probabilities or upper-tail probabilities before selecting a column.
Ignoring negative cutoffs. A left-tailed test or the lower side of a two-tailed symmetric test may require a negative critical value.
Treating the critical value as the test statistic. The critical value comes from the chosen statistical procedure and threshold. The test statistic is calculated from the observed data.
Final Takeaway
To calculate a critical value correctly, first identify the statistical distribution, then determine the significance level, test direction, and degrees of freedom where required. Use those inputs to find the appropriate cutoff in a table, calculator, or statistical software.
Always verify the probability convention and the sign of the result before comparing it with your test statistic. Once the critical value is correct, you can use it to determine whether the observed statistic falls in the rejection region or to calculate a confidence interval.