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Critical Value — Definition, Formula & Examples

A critical value is the boundary score (such as z∗z^* or t∗t^*) on a sampling distribution that separates the rejection region from the non-rejection region in a hypothesis test, or that defines the margin of error in a confidence interval. It is determined by the chosen confidence level or significance level α\alpha.

Given a significance level α\alpha and a reference distribution (standard normal or tt-distribution with dfdf degrees of freedom), the critical value c∗c^* is the value such that the probability of the test statistic falling in the tail(s) beyond ± c∗\pm\,c^* equals α\alpha. For a two-tailed test, P(∣Z∣>z∗)=αP(|Z| > z^*) = \alpha; for a one-tailed test, P(Z>z∗)=αP(Z > z^*) = \alpha (right tail) or P(Z<−z∗)=αP(Z < -z^*) = \alpha (left tail). This concept is distinct from the calculus critical number, which is a point where a function's derivative is zero or undefined.

Key Formula

Margin of Error=z∗⋅σnort∗⋅sn\text{Margin of Error} = z^* \cdot \frac{\sigma}{\sqrt{n}} \quad \text{or} \quad t^* \cdot \frac{s}{\sqrt{n}}
Where:
  • z∗z^* = Critical value from the standard normal distribution
  • t∗t^* = Critical value from the t-distribution with df = n − 1
  • σ\sigma = Population standard deviation (known)
  • ss = Sample standard deviation
  • nn = Sample size

How It Works

You choose a confidence level (say 95%) or a significance level (α=0.05\alpha = 0.05). Then you look up the value on the appropriate distribution — standard normal for large samples or known σ\sigma, or tt-distribution when using a sample standard deviation with smaller samples. For a 95% two-tailed interval, you need the zz-score that leaves 2.5% in each tail, giving z∗=1.96z^* = 1.96. In hypothesis testing, you compare your calculated test statistic to the critical value: if the test statistic is more extreme, you reject the null hypothesis. In confidence intervals, the critical value scales the standard error to create the margin of error.

Worked Example

Problem: Find the critical value z* for a 95% confidence interval, then compute the margin of error if σ = 10 and n = 64.
Step 1: Determine the tail area. A 95% confidence level leaves 5% total in two tails, so each tail has 2.5%.
α2=0.052=0.025\frac{\alpha}{2} = \frac{0.05}{2} = 0.025
Step 2: Look up the z-score that has 0.025 in the upper tail of the standard normal distribution.
z∗=1.96z^* = 1.96
Step 3: Compute the standard error.
σn=1064=108=1.25\frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{64}} = \frac{10}{8} = 1.25
Step 4: Multiply the critical value by the standard error to get the margin of error.
ME=1.96×1.25=2.45\text{ME} = 1.96 \times 1.25 = 2.45
Answer: The critical value is z∗=1.96z^* = 1.96, and the margin of error is 2.452.45.

Another Example

Problem: A researcher runs a two-tailed t-test at the α = 0.05 level with a sample of n = 20. The test statistic is t = 2.30. Should the researcher reject the null hypothesis?
Step 1: Find the degrees of freedom.
df=n−1=20−1=19df = n - 1 = 20 - 1 = 19
Step 2: Look up the two-tailed critical value for α = 0.05 with 19 degrees of freedom in a t-table.
t∗≈2.093t^* \approx 2.093
Step 3: Compare the test statistic to the critical value. Since |2.30| > 2.093, the test statistic falls in the rejection region.
∣t∣=2.30>2.093=t∗|t| = 2.30 > 2.093 = t^*
Answer: Yes, reject the null hypothesis because the test statistic exceeds the critical value.

Visualization

Why It Matters

Critical values appear on every AP Statistics exam in both the confidence-interval and hypothesis-testing sections. Beyond the classroom, any field that uses significance testing — medical trials, quality control, social-science research — relies on critical values to decide whether observed results are statistically significant. Mastering this concept lets you interpret published studies and make data-driven decisions with a clear standard of evidence.

Common Mistakes

Mistake: Using the full α instead of α/2 for a two-tailed test or confidence interval.
Correction: For two-tailed procedures, split α evenly between the two tails. A 95% confidence interval uses 0.025 per tail, giving z* = 1.96, not the z-value for 0.05 in one tail (1.645).
Mistake: Using a z* critical value when the population standard deviation is unknown and the sample is small.
Correction: When σ is unknown and you estimate it with s, use the t-distribution with df = n − 1. The t* value is larger than the corresponding z*, producing a wider interval that accounts for extra uncertainty.