# What is a critical value?

In hypothesis testing, a critical value is a point on the test distribution that is compared to the test statistic to determine whether to reject the null hypothesis. If the absolute value of your test statistic is greater than the critical value, you can declare statistical significance and reject the null hypothesis. Critical values correspond to α, so their values become fixed when you choose the test's α.

## Examples of calculating critical values

In hypothesis testing, there are two ways to determine whether there is enough evidence from the sample to reject H0 or to fail to reject H0. The most common way is to compare the p-value with a pre-specified value of α, where α is the probability of rejecting H0 when H0 is true. However, you can also compare the calculated value of the test statistic with the critical value. The following are examples of how to calculate the critical value for a 1-sample t test and a one-way ANOVA.

### Calculating a critical value for a 1-sample t-test

Suppose you are performing a 1-sample t test on ten observations, have a two-sided alternative hypothesis (that is, H1 not equal to), and are using an alpha of 0.10:
1. Choose:
• Mac: Statistics > Probability Distributions > Inverse Cumulative Distribution Function
• PC: STATISTICS > CDF/PDF > Inverse Cumulative Distribution Function
2. In Form of input, select A single value.
3. In Value, enter 0.95 (one minus one-half alpha).
4. In Distribution, select t.
5. In Degrees of freedom, enter 9 (the number of observations minus one).
6. Click OK.

This gives you an inverse cumulative probability, which equals the critical value, of 1.83311. If the absolute value of the t-statistic is greater than this critical value, then you can reject the null hypothesis, H0, at the 0.10 level of significance.

### Calculating a critical value for an analysis of variance (ANOVA)

Suppose you are performing a one-way ANOVA on twelve observations, the factor has three levels, and you are using an alpha of 0.05:
1. Choose:
• Mac: Statistics > Probability Distributions > Inverse Cumulative Distribution Function
• PC: STATISTICS > CDF/PDF > Inverse Cumulative Distribution Function
2. In Form of input, select A single value.
3. In Value, enter 0.95 (one minus one-half alpha).
4. In Distribution, select F.
5. In Numerator degrees of freedom, enter 2 (the number of factor levels minus one).
6. In Denominator degrees of freedom, enter 9 (the degrees of freedom for error).
7. Click OK.

This gives you an inverse cumulative probability (critical value) of 4.25649. If the F-statistic is greater than this critical value, then you can reject the null hypothesis, H0, at the 0.05 level of significance.

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