Critical Value Calculator
What Is a Critical Value?
A critical value is a cutoff point on the sampling distribution of a test statistic that defines the boundary between the rejection region and the non-rejection region for a hypothesis test. If the observed test statistic falls beyond the critical value, you reject the null hypothesis H₀. If it falls between the critical values, you fail to reject H₀.
The critical value is determined before you look at your data. It depends on four things: (1) the statistical distribution of the test statistic under H₀, (2) the significance level α, (3) whether the test is one-tailed or two-tailed, and (4) the degrees of freedom for distributions that require them. Critical values are the inverse CDF (quantile function) of the relevant distribution evaluated at the appropriate tail probability.
According to the NIST Engineering Statistics Handbook, critical values and rejection regions are the foundational mechanics of the classical frequentist hypothesis test, used uniformly across engineering, medicine, and social science research.
Critical Value Formulas by Distribution
Each distribution uses its own inverse CDF (quantile function). The tail probability fed into that function differs depending on whether the test is left-tailed, right-tailed, or two-tailed.
Z — Standard Normal
Right-tailed:
z* = Φ⁻¹(1 − α)
Left-tailed:
z* = Φ⁻¹(α)
Two-tailed:
z* = ±Φ⁻¹(1 − α/2)
T — Student's t-distribution
Right-tailed:
t* = t⁻¹(1 − α, df)
Left-tailed:
t* = t⁻¹(α, df)
Two-tailed:
t* = ±t⁻¹(1 − α/2, df)
df = n − 1
Chi-square — χ²(df)
Right-tailed (most common):
χ* = χ²⁻¹(1 − α, df)
Left-tailed:
χ* = χ²⁻¹(α, df)
Two-tailed (separate values):
Lower = χ²⁻¹(α/2, df)
Upper = χ²⁻¹(1 − α/2, df)
F — F-distribution
Right-tailed (most common):
F* = F⁻¹(1 − α, df₁, df₂)
Two-tailed (separate values):
Lower = F⁻¹(α/2, df₁, df₂)
Upper = F⁻¹(1 − α/2, df₁, df₂)
df₁ = numerator df
df₂ = denominator df
How Critical Values Are Determined — The Calculation Logic
A critical value is the value of the test statistic that places exactly α probability in the tail(s) of the distribution under the null hypothesis. The three tail rules are:
The critical value satisfies P(X ≥ critical value) = α, which means CDF(critical value) = 1 − α. Reject H₀ if the test statistic exceeds the critical value. Example: for a right-tailed Z test at α = 0.05, z* = 1.645 because exactly 5% of the standard normal distribution lies above 1.645.
The critical value satisfies P(X ≤ critical value) = α, which means CDF(critical value) = α. Reject H₀ if the test statistic falls below the critical value. Example: for a left-tailed Z test at α = 0.05, z* = −1.645 because exactly 5% of the distribution lies below −1.645.
For symmetric distributions (Z and T), α is split equally: α/2 in each tail. The critical values satisfy CDF(−critical value) = α/2 and CDF(critical value) = 1 − α/2. Example: for a two-tailed Z test at α = 0.05, z* = ±1.96 because 2.5% of the distribution lies in each tail. For asymmetric distributions (chi-square, F), the lower and upper critical values must be found from separate quantile lookups.
One-Tailed vs. Two-Tailed Tests — How This Changes the Critical Value
Whether a test is one-tailed or two-tailed changes how α is distributed across the rejection region, which in turn changes the critical value. The tail type must be chosen based on the research hypothesis before examining the data.
| Test Type | Tail Allocation | Z at α = 0.05 | Z at α = 0.01 | Decision Rule |
|---|---|---|---|---|
| Two-tailed | α/2 = 0.025 in each tail | ±1.960 | ±2.576 | Reject if |z| > z* |
| Right-tailed | α = 0.05 in right tail only | +1.645 | +2.326 | Reject if z > z* |
| Left-tailed | α = 0.05 in left tail only | −1.645 | −2.326 | Reject if z < z* |
Notice that the two-tailed critical value at α = 0.05 (z = 1.96) equals the one-tailed critical value at α = 0.025. This is because both allocate 2.5% of the distribution in the upper tail. Choosing the tail direction after seeing the data inflates the Type I error rate; this is why the direction must be specified in advance.
Significance Level (α) and Its Effect on Critical Values
The significance level α is the maximum probability of rejecting a true null hypothesis (Type I error). A smaller α produces a more extreme critical value, making it harder to reach the rejection region.
Two-tailed Z Critical Values at Common α Levels
| Significance Level (α) | Conf. Level (1−α) | Each Tail (α/2) | Z Critical Value (two-tailed) | T Critical Value (df=30) |
|---|---|---|---|---|
| α = 0.10 | 90% | 0.050 | ±1.645 | ±1.697 |
| α = 0.05 | 95% | 0.025 | ±1.960 | ±2.042 |
| α = 0.01 | 99% | 0.005 | ±2.576 | ±2.750 |
| α = 0.001 | 99.9% | 0.0005 | ±3.291 | ±3.385 |
The field norm of using α = 0.05 dates to Ronald Fisher’s early work in experimental statistics. The American Statistical Association’s 2016 statement on p-values, published in The American Statistician, notes that α = 0.05 should not be treated as a universal threshold; the appropriate level depends on field conventions and the consequences of error.
Degrees of Freedom and Their Effect on Critical Values
Degrees of freedom (df) determine the shape of the T, chi-square, and F distributions. For the T-distribution, as df increases, the distribution approaches the standard normal and the critical values converge toward Z critical values.
| Degrees of Freedom (df) | T critical value (two-tailed, α = 0.05) | Difference from Z = 1.96 |
|---|---|---|
| df = 1 | ±12.706 | +10.746 |
| df = 5 | ±2.571 | +0.611 |
| df = 10 | ±2.228 | +0.268 |
| df = 20 | ±2.086 | +0.126 |
| df = 30 | ±2.042 | +0.082 |
| df = 120 | ±1.980 | +0.020 |
| df → ∞ | ±1.960 | 0 |
Small samples have heavier-tailed t-distributions, resulting in larger critical values. This is why using z = 1.96 when σ is unknown and the sample is small understates the true critical value, leading to more Type I errors. For chi-square and F distributions, degrees of freedom change the shape of the entire distribution, so critical values do not converge to Z values the same way.
Critical Value vs. Test Statistic — The Decision Rule
The critical value is determined from the distribution under H₀ before any data are collected. The test statistic is computed from sample data after collection. The decision rule compares them.
Left-tailed test: Reject H₀ if test statistic < critical value.
Two-tailed test: Reject H₀ if |test statistic| > critical value (for Z and T), or if test statistic < lower critical value or > upper critical value (for chi-square and F).
Critical Value vs. P-Value — Two Approaches, One Decision
The critical value approach and the p-value approach are mathematically equivalent: they always produce the same reject/fail-to-reject decision when applied correctly. The choice between them is one of presentation, not of statistical logic.
| Method | What you compare | Decision rule | When useful |
|---|---|---|---|
| Critical Value Approach | Test statistic vs. critical value cutoff | Reject H₀ if test statistic falls in the rejection region | Teaching, manual calculations, checking tables |
| P-Value Approach | P-value vs. α | Reject H₀ if p-value < α | Software output, quantifying strength of evidence |
Critical value approach: z* = ±1.96. Since 2.15 > 1.96, reject H₀.
P-value approach: p-value = 2 × P(Z > 2.15) = 2 × 0.0158 = 0.0316. Since 0.0316 < 0.05, reject H₀.
Both approaches agree: reject H₀.
Critical Values and Confidence Intervals — The Connection
Critical values appear directly in confidence interval formulas as the multiplier applied to the standard error. A 95% confidence interval corresponds to a two-tailed test at α = 0.05, and the critical value that defines the rejection boundary is the same z* or t* used to build the interval.
CI = point estimate ± critical value × standard error
Z interval (95%): x̄ ± 1.960 × (σ/√n)
T interval (95%, df=20): x̄ ± 2.086 × (s/√n)
The critical value is not always 1.96. It depends on the chosen confidence level, the distribution, and the degrees of freedom. See the Confidence Interval Calculator for full examples.
The duality between confidence intervals and hypothesis tests means: if a hypothesized value falls outside a 95% confidence interval, the corresponding two-tailed test would reject H₀ at α = 0.05. Penn State’s STAT 415 covers the formal proof of this duality in the context of interval estimation and hypothesis testing for means and proportions.
The Four Distributions and When to Use Each
Standard Normal Distribution (Z)
The standard normal distribution has mean 0 and standard deviation 1. It is symmetric around zero. Use the Z critical value when the population standard deviation σ is known, when testing proportions (where the normal approximation is valid), or when sample sizes are large enough that the t-distribution converges to Z. The most cited Z critical values are ±1.645 (90% confidence or right-tailed α = 0.05) and ±1.960 (95% confidence).
Student’s T-Distribution
The t-distribution is symmetric around zero and resembles the standard normal, but with heavier tails. The parameter df = n − 1 governs how heavy those tails are. It is used for inference about means when σ is unknown. As df increases the distribution converges to N(0,1). Introduced by William Sealy Gosset writing under the pseudonym “Student,” the t-distribution is described in detail at Penn State STAT 415.
Chi-Square Distribution
The chi-square distribution is right-skewed and defined only for positive values. It is parameterized by degrees of freedom. Chi-square tests are used for goodness-of-fit tests, independence tests in contingency tables, and tests of a single population variance. Because the distribution is not symmetric, the critical values for left-tailed and right-tailed tests are not the negatives of each other. A two-tailed chi-square test requires two separate quantile calculations.
F-Distribution
The F-distribution is defined by two degrees of freedom: numerator df (df₁) and denominator df (df₂). It is right-skewed and takes only positive values. It appears in ANOVA (where the test statistic is a ratio of between-group to within-group variance), in testing equality of two variances, and in overall tests of regression models. Most F tests are right-tailed. The two denominator and numerator degrees of freedom must not be reversed; F(df₁, df₂) ≠ F(df₂, df₁).
| Distribution | Parameters | Symmetric? | Values | Common Tests |
|---|---|---|---|---|
| Z (Standard Normal) | None | Yes | −∞ to +∞ | Large-sample means, proportions |
| T (Student’s t) | df = n − 1 | Yes | −∞ to +∞ | Means with unknown σ |
| Chi-Square | df ≥ 1 | No (right-skewed) | 0 to +∞ | Variance, goodness-of-fit, independence |
| F | df₁, df₂ ≥ 1 | No (right-skewed) | 0 to +∞ | ANOVA, variance ratios, regression F-test |
Worked Examples
Example 1: Z Critical Value, Two-Tailed (α = 0.05)
Tail probability: α/2 = 0.025 in each tail.
Formula: z* = Φ−1(1 − 0.025) = Φ−1(0.975).
Critical value: z* = ±1.960.
Rejection region: Reject H₀ if z < −1.960 or z > 1.960.
Example 2: Z Critical Value, Right-Tailed (α = 0.05)
Formula: z* = Φ−1(1 − 0.05) = Φ−1(0.95).
Critical value: z* = 1.645.
Rejection region: Reject H₀ if z > 1.645.
Example 3: T Critical Value (α = 0.05, df = 20, Two-Tailed)
Tail probability: α/2 = 0.025.
Formula: t* = t−1(0.975, df=20).
Critical value: t* = ±2.086.
Rejection region: Reject H₀ if |t| > 2.086.
Note: Using z* = 1.96 here would understate the critical value. The t-distribution with df = 20 has heavier tails than the standard normal.
Example 4: Chi-Square Critical Value (α = 0.05, df = 5, Right-Tailed)
Formula: χ* = χ²−1(1 − 0.05, df=5) = χ²−1(0.95, 5).
Critical value: χ* = 11.070.
Rejection region: Reject H₀ if χ² > 11.070.
Example 5: F Critical Value (α = 0.05, df₁ = 3, df₂ = 24, Right-Tailed)
Formula: F* = F−1(1 − 0.05, df₁=3, df₂=24) = F−1(0.95, 3, 24).
Critical value: F* = 3.009.
Rejection region: Reject H₀ if F > 3.009.
Example 6: Converting a Confidence Level to a Critical Value
Convert: α = 1 − 0.99 = 0.01. For two-tailed: α/2 = 0.005.
Formula: t* = t−1(1 − 0.005, df=15) = t−1(0.995, 15).
Critical value: t* = ±2.947.
Application: CI = x̄ ± 2.947 × (s/√16).
Example 7: Full Hypothesis Test Using a Critical Value
Critical values: z* = ±1.96.
Sample data: n = 36, x̄ = 104.3, σ = 12.
Test statistic: z = (104.3 − 100) / (12/√36) = 4.3 / 2 = 2.15.
Decision: 2.15 > 1.96, so reject H₀.
Conclusion: At α = 0.05, the sample provides sufficient evidence to conclude μ ≠ 100. This is not the same as proving μ ≠ 100 with certainty; it is a statistical inference conditional on the test assumptions.
Common Mistakes When Finding Critical Values
Finding Critical Values in Excel and Google Sheets
Excel and Google Sheets provide inverse CDF functions for each major distribution. These functions compute critical values directly from α and degrees of freedom.
' ── Z CRITICAL VALUES ──
=NORM.S.INV(1-alpha) ' Right-tailed
=NORM.S.INV(alpha) ' Left-tailed
=NORM.S.INV(1-alpha/2) ' Two-tailed (upper value)
' ── T CRITICAL VALUES ──
=T.INV(1-alpha, df) ' Right-tailed
=T.INV(alpha, df) ' Left-tailed
=T.INV.2T(alpha, df) ' Two-tailed (returns positive value)
' ── CHI-SQUARE CRITICAL VALUES ──
=CHISQ.INV.RT(alpha, df) ' Right-tailed (most common)
=CHISQ.INV(alpha, df) ' Left-tailed
' Two-tailed: use CHISQ.INV(alpha/2, df) and CHISQ.INV.RT(alpha/2, df)
' ── F CRITICAL VALUES ──
=F.INV.RT(alpha, df1, df2) ' Right-tailed (ANOVA, most common)
=F.INV(alpha, df1, df2) ' Left-tailed
' Two-tailed: compute both F.INV(alpha/2, df1, df2) and F.INV.RT(alpha/2, df1, df2)
' ── EXAMPLES ──
=NORM.S.INV(0.975) ' → 1.960 (Z, two-tailed α=0.05)
=T.INV.2T(0.05, 20) ' → 2.086 (T, df=20, two-tailed α=0.05)
=CHISQ.INV.RT(0.05, 5) ' → 11.070 (chi-sq, df=5, right-tailed)
=F.INV.RT(0.05, 3, 24) ' → 3.009 (F, df1=3, df2=24)
These functions work identically in Google Sheets. The argument order matters: degrees of freedom always follow the probability argument. The CHISQ.INV function returns the left-tail quantile; CHISQ.INV.RT returns the right-tail quantile. Verify the direction before reporting results.
Key Statistical Terms Connected to Critical Values
Table: Critical Value Entity Map — 14 Core Concepts
| Term | Symbol / Formula | Definition | Role in Critical Value Calculation |
|---|---|---|---|
| Critical Value | z*, t*, χ*, F* | Cutoff on a distribution separating rejection from non-rejection region | The primary output of this calculator |
| Significance Level | α | Maximum acceptable probability of a Type I error | Sets how extreme the critical value must be |
| Rejection Region | |X| > critical value | Set of test statistic values that lead to rejecting H₀ | Defined by the critical value(s) |
| Tail Probability | α or α/2 | Probability allocated to one or both tails | Input to the inverse CDF |
| Inverse CDF | Φ−1(p) | Function returning the quantile for a given probability | The mathematical operation that computes critical values |
| Degrees of Freedom | df = n − 1 | Parameter controlling the shape of T, chi-square, and F distributions | Required input for T, chi-square, and F critical values |
| Null Hypothesis | H₀ | The default hypothesis tested against the alternative | The distribution under H₀ is used to find critical values |
| Alternative Hypothesis | H₁ | The research hypothesis; determines tail direction | Specifies left-tailed, right-tailed, or two-tailed test |
| Test Statistic | z, t, χ², F | Value computed from sample data and compared to the critical value | Not the critical value itself; compared to it after collection |
| P-Value | P(X ≥ observed | H₀) | Probability of a result at least as extreme as observed under H₀ | Equivalent approach: reject if p-value < α |
| Confidence Level | 1 − α | Proportion of intervals containing the true parameter if repeated | Converts to α via: α = 1 − confidence level |
| Standard Normal | N(0, 1) | Normal distribution with mean 0 and SD 1 | The distribution used for Z critical values |
| T-Distribution | t(df) | Symmetric distribution with heavier tails than Z; parameterized by df | The distribution used for T critical values |
| Quantile | Q(p) | Value below which a given proportion p of a distribution lies | A critical value is the α or (1−α) quantile of the relevant distribution |
Related Topics and Calculators on Statistics Fundamentals
Critical values connect to every hypothesis test and confidence interval in statistics. These resources build out the full picture.
Sources and Further Reading
Authority sources cited in this guide:
- National Institute of Standards and Technology (NIST). Engineering Statistics Handbook. itl.nist.gov
- Penn State STAT 415. Introduction to Mathematical Statistics. online.stat.psu.edu
- Wasserstein, R.L. & Lazar, N.A. (2016). “The ASA Statement on p-Values: Context, Process, and Purpose.” The American Statistician. tandfonline.com
- OpenStax. Introductory Statistics, Chapter 9: Hypothesis Testing with One Sample. openstax.org
- DeGroot, M.H. & Schervish, M.J. Probability and Statistics, 4th ed. Addison-Wesley, 2012.
- R Core Team. R: A Language and Environment for Statistical Computing — qnorm, qt, qchisq, qf functions. stat.ethz.ch
- UCLA Statistical Methods and Data Analytics. Introduction to Power Analysis. stats.oarc.ucla.edu
Frequently Asked Questions
A critical value is a cutoff point on the sampling distribution of a test statistic that defines the boundary between the rejection region and the non-rejection region in a hypothesis test. It is determined by the significance level α, the distribution of the test statistic under H₀, the tail type (one-tailed or two-tailed), and degrees of freedom where applicable. If the observed test statistic falls beyond the critical value, you reject the null hypothesis.
A critical value is calculated as the inverse CDF (quantile function) of the relevant distribution at the appropriate tail probability. For a right-tailed test, CDF(critical value) = 1 − α. For a left-tailed test, CDF(critical value) = α. For a two-tailed test, split α equally across both tails: CDF(lower critical value) = α/2 and CDF(upper critical value) = 1 − α/2. In Excel: NORM.S.INV, T.INV, CHISQ.INV, and F.INV functions perform these calculations.
The z critical value for a 95% confidence interval, which corresponds to a two-tailed test at α = 0.05, is ±1.96. More precisely, it is 1.9600. This value comes from the fact that Φ−1(0.975) = 1.960, meaning 97.5% of the standard normal distribution lies below 1.96 (and 97.5% lies above −1.96), leaving exactly 5% split equally in both tails.
The value 1.96 corresponds to the two-tailed Z critical value at α = 0.05, which is the most widely adopted significance level in many fields of research. It appears so frequently because α = 0.05 became a convention in applied statistics, and the standard normal distribution is used in many large-sample tests. However, 1.96 is not a universal critical value. Different distributions (t, chi-square, F), different α levels, different tail types, and different degrees of freedom all produce different critical values.
Use a t critical value when you are testing a population mean and the population standard deviation σ is unknown. Because σ must be estimated from the sample, the resulting test statistic follows a t-distribution with df = n − 1, not a standard normal. The t-distribution has heavier tails, producing a larger critical value than z (for the same α and tail type). As df increases, t* approaches z*, so for very large samples the practical difference becomes negligible, but the t-distribution is technically correct whenever σ is estimated.
The critical value approach compares the test statistic to a predetermined cutoff based on α, the distribution, and the tail type. The p-value approach computes the probability of observing a test statistic at least as extreme as the one obtained, assuming H₀ is true, and compares this to α. Both methods always reach the same reject/fail-to-reject conclusion: if the test statistic exceeds the critical value, then the p-value will be less than α, and vice versa. The p-value provides more information (how much beyond α you are), but the critical value approach is more direct for checking tables and teaching decision rules.
Yes, for distributions that are symmetric around zero. The standard normal (Z) and Student's t-distribution are symmetric, so left-tailed critical values are negative: for example, the left-tailed z critical value at α = 0.05 is −1.645. For two-tailed Z and T tests, critical values come in pairs: ±z* or ±t*. The chi-square and F distributions are defined only for positive values (they cannot be negative), so their critical values are always positive.
No. The chi-square distribution is right-skewed, so left-tailed and right-tailed critical values are completely different for the same α and df. For example, with df = 10 and α = 0.05: the right-tailed critical value is approximately 18.307 (from CHISQ.INV.RT(0.05, 10)), while the left-tailed critical value is approximately 3.940 (from CHISQ.INV(0.05, 10)). These are not negatives of each other. The distribution’s asymmetry makes it essential to select the correct tail type.
For the T-distribution, larger degrees of freedom produce smaller critical values that converge toward the Z critical value. With df = 1 and α = 0.05 (two-tailed), t* ≈ 12.706. With df = 30, t* ≈ 2.042. With df → ∞, t* → 1.960. This convergence happens because more data reduce the uncertainty about σ, making the t-distribution resemble the standard normal. For the chi-square and F distributions, degrees of freedom change the shape of the entire distribution, not just scale a symmetric curve, so the relationship is more complex.
The rejection region (also called the critical region) is the set of test statistic values for which the null hypothesis H₀ is rejected. It is defined by the critical value(s) and the tail type. For a two-tailed Z test at α = 0.05: the rejection region is z < −1.96 or z > 1.96. For a right-tailed F test at α = 0.05 with df₁ = 3 and df₂ = 24: the rejection region is F > 3.009. The rejection region contains exactly α probability under H₀.
The conversion is α = 1 − confidence level. For a 95% confidence level: α = 1 − 0.95 = 0.05. For a 99% confidence level: α = 1 − 0.99 = 0.01. For two-tailed confidence intervals, each tail receives α/2. So a 95% confidence interval uses α/2 = 0.025 in each tail, which is why z* = 1.96 (corresponding to the 97.5th percentile of the standard normal) is the correct critical value.
The F-statistic is defined as a ratio of two independent chi-square random variables, each divided by its own degrees of freedom: F = (χ²1/df₁) / (χ²2/df₂). The numerator degrees of freedom df₁ comes from the first chi-square (typically between-group variance in ANOVA), and the denominator degrees of freedom df₂ comes from the second (typically within-group or error variance). Because these are separate parameters, the F critical value changes with both values. F(3, 24) and F(24, 3) have different distributions and therefore different critical values for the same α.