A critical value is the cut-off your test statistic has to beat. You choose a significance level, look up the value that marks the edge of the rejection region, and if your statistic is more extreme than it, you reject the null hypothesis.

1.1 The idea

Every hypothesis test produces a single number, the test statistic, that measures how far your data sit from what the null hypothesis predicts. The question is how far is far enough to be worth taking seriously.

The critical value answers that. You decide in advance how often you are willing to be wrong when the null is actually true, usually 5%, and the critical value is the point beyond which only 5% of statistics would fall if the null were true. Anything past it is unusual enough that you stop believing the null.

1.2 The rejection region

The critical value marks the boundary of the rejection region, the shaded area in the tails where you would reject. Its size is exactly alpha, your significance level. A two-tailed test at 5% has two rejection regions of 2.5% each; a one-tailed test has a single region of 5%.

This is why the one-tailed critical value is smaller. All of your alpha is concentrated in one tail rather than split between two, so the boundary sits closer to the centre and the test is easier to pass in that direction. It is also why choosing the tail after seeing which way the data went is indefensible: it silently doubles your false positive rate.

1.3 Which distribution do you need?

1.4 Why t is not z

If you knew the population standard deviation you would use z. You almost never do, so you estimate it from the same sample you are testing, and that estimate carries its own uncertainty. The t-distribution has heavier tails to account for it, which pushes the critical value further out.

The gap is dramatic at small samples and negligible at large ones. At df = 1 the t critical value is 12.706 against z at 1.960. At df = 10 it is 2.228. At df = 30 it is 2.042. By df = 1000 it is 1.962, essentially the normal value. There is never a penalty for using t, so use it whenever the standard deviation was estimated.

1.5 Why chi-square and F only have one tail

Both statistics are built from squared quantities, so they cannot be negative and their distributions are skewed rather than symmetric. More importantly, the evidence against the null always pushes them upward: a poor fit produces a large chi-square, and a real group difference produces a large F. A small value means the data agree with the null unusually well, which is not evidence against it.

So both are conventionally right-tailed, with one critical value rather than two. The exception is a test about a variance, where you may genuinely care about both directions and need the lower critical value too. This calculator reports both bounds for chi-square when you ask for a two-tailed test.

1.6 Critical values versus p-values

They are two ways of making the same comparison. The critical value approach asks "is my statistic beyond the cut-off?". The p-value approach asks "how much area lies beyond my statistic, and is it less than alpha?". The answers always agree.

Before computers, tables of critical values were the only practical option, because computing an exact tail area by hand was infeasible. Software changed that, and journals now expect exact p-values. Critical values survive because they make the logic visible: you can see the boundary, see where your statistic landed, and see the rejection region shaded. That is why this calculator draws it rather than just printing a number.

4.1 What the number actually represents

The critical value is the point on the distribution beyond which only alpha of the probability lies, assuming the null hypothesis is true. At alpha = 0.05 two-tailed, z = 1.960 means that if the null were true, only 5% of samples would produce a statistic further from zero than 1.960.

So when your statistic clears it, you are saying: either something unusual happened, or the null hypothesis is wrong. You choose to believe the second. The 5% is the rate at which you have agreed to be wrong when the null is actually true.

4.2 The rejection region, and why its size equals alpha

The shaded area in chart 1 is the rejection region. Its total area is exactly alpha by construction, because that is how the critical value was defined: find the point that leaves alpha in the tail.

This is worth sitting with, because it explains everything else on the page. A smaller alpha means a smaller rejection region, which means the boundary moves further out, which means a larger critical value and a harder test. A two-tailed test splits the same alpha into two regions of alpha over two, so each boundary sits closer in than a one-tailed boundary would, but you now have to clear it on whichever side you land.

4.3 Why the one-tailed value is smaller, and why that matters

At alpha = 0.05, the two-tailed z critical value is 1.960 and the one-tailed is 1.645. All 5% is concentrated in a single tail rather than split into two lots of 2.5%, so the boundary sits closer to the centre.

That makes a one-tailed test more powerful in the direction you chose, and completely blind in the other. It is legitimate only if you fixed the direction before collecting data and would treat a result in the opposite direction exactly as you would treat no result at all. If a surprising reversal would interest you, you need two tails.

Choosing after seeing the data is not a grey area. It converts a nominal 5% test into a 10% test while reporting it as 5%, which is why journals increasingly ask you to pre-register the direction.

4.4 Degrees of freedom, and why t starts so high

The t critical value at df = 1 is 12.706. At df = 10 it is 2.228, at df = 30 it is 2.042, and by df = 1000 it is 1.962, essentially the normal 1.960.

The reason is that the t-distribution accounts for uncertainty in your estimate of the standard deviation. With one degree of freedom that estimate is almost worthless, so the distribution has extremely heavy tails and the bar is set punishingly high. As the sample grows, the estimate stabilises, the tails thin, and t converges on z.

This is honest rather than harsh. A tiny sample genuinely cannot establish much, and the critical value is simply refusing to pretend otherwise.

4.5 Chi-square: why it is right-tailed and skewed

A chi-square statistic is a sum of squared standardised deviations, so it cannot be negative. Its distribution is right-skewed, with a mean equal to its degrees of freedom, and it becomes more symmetric as df grows.

The evidence against the null always pushes it upward: a poor fit between observed and expected counts produces a large chi-square. A small chi-square means your data fit the null better than expected, which is not evidence against it. That is why goodness-of-fit and independence tests use only the right tail.

The exception is a test about a variance, where a variance that is unusually small can be as interesting as one that is unusually large. There you need both bounds, and this calculator reports them when you select a two-tailed test.

4.6 F: two degrees of freedom, and the order matters

An F statistic is a ratio of two variances, so it needs a degrees of freedom for each. Critically, F(3, 20) is not the same as F(20, 3): the first has a 5% critical value of 3.098 and the second of 8.660. Swapping them is a silent error that will not throw a warning anywhere.

In ANOVA the numerator df is groups minus one and the denominator is total observations minus the number of groups. In a variance ratio test both are n minus one for their respective samples, and the convention is to put the larger variance on top, which forces the statistic above 1 and makes the test right-tailed.

4.7 Critical value or p-value: they are the same comparison

Comparing your statistic against a critical value and comparing your p-value against alpha are mathematically identical operations. If the statistic is beyond the boundary then the tail area beyond it is smaller than alpha, and vice versa. They cannot disagree.

Tables of critical values exist because computing an exact tail area by hand was infeasible before computers. Software removed that constraint, journals now expect exact p-values, and the tables have become mostly a teaching device. They survive for a good reason though: they make the logic visible. You can see the boundary, see the shaded region, and see where your statistic landed, which a bare "p = 0.032" does not convey.

4.8 What clearing the bar does and does not tell you

Rejecting the null means the effect is detectable at your sample size. It does not mean the effect is large, important, or real in any deeper sense. With a large enough sample any non-zero difference clears any critical value, which is why a significant result from n = 50,000 may describe a difference nobody would act on.

Equally, failing to clear it is not evidence that the null is true. It means this sample could not distinguish the null from the alternative, which is as much a statement about your power as about reality. Report a confidence interval, which shows the range of effects still compatible with your data, and an effect size, which is independent of sample size.

4.9 Reading the four charts

Chart 1 draws the distribution with the rejection region shaded and both the critical value and your statistic marked, which is the clearest possible picture of the decision. Chart 2 shows how the critical value rises as alpha falls, quantifying the cost of a stricter test. Chart 3 plots the critical value against degrees of freedom, with the normal limit dashed behind it for t, so the convergence is visible. Chart 4 compares one-tailed and two-tailed values at several alphas.

4.10 The most common mistakes

Comparing a statistic against the wrong distribution's critical value. Looking up 1.96 for a t-test with n = 8 will make you reject things you should not. Nothing in your output will warn you.

Halving alpha instead of the tail area, or vice versa. For a two-tailed test at alpha = 0.05 you look up the 0.975 quantile, not the 0.95. Getting this backwards gives 1.645 where 1.960 belongs.

Using a two-tailed critical value with a one-tailed alternative hypothesis, which is conservative but throws away power you were entitled to, if you genuinely committed to the direction in advance.

Reversing the F degrees of freedom. Always numerator first.

Treating the boundary as a cliff. A statistic at 1.95 and one at 1.97 are not meaningfully different pieces of evidence, but the decision rule treats them as opposites. This is the strongest argument for reporting the exact p-value and an interval rather than a bare verdict.

▶ Run the analysis above to auto-fill all five examples with your results.

1. Paste your numbers into the data column. From my data opens first, with one column ready to go. Choose the test you are running, enter your values comma-separated, and the calculator works out the test statistic, finds the matching critical value and tells you whether to reject.
2. Or switch tab if you already know your statistic. Z, T, Chi-square and F give you the critical value from alpha and degrees of freedom alone. Compare my statistic takes a value you already have and returns the decision and the p-value.
3. Pick the right distribution. Use z only when the population standard deviation is genuinely known. Use t whenever you estimated it from your sample, which is almost always. Use chi-square for counts and categorical data, and F for ANOVA or comparing two variances.
4. Enter degrees of freedom carefully. This is where most errors happen. It is n − 1 for a one-sample or paired t-test, n₁ + n₂ − 2 for a pooled two-sample test, categories − 1 for goodness of fit, and (rows − 1) × (columns − 1) for a test of independence.
5. For F, get the order right. Numerator first, denominator second. F(3, 20) and F(20, 3) are different distributions with critical values of 3.098 and 8.660, and nothing will warn you if you swap them.
6. Choose alpha and the tail before you look at your data. This is not a formality. Switching to one tail after seeing which direction the result went converts a 5% test into a 10% test while you report it as 5%.
7. Leave chi-square and F right-tailed. The tab switches automatically because that is the convention: only a large statistic is evidence against the null. Two tails make sense only for a test about a variance, and both bounds are reported if you choose it.
8. Add your test statistic for a decision. Every distribution tab has an optional field for it. Fill it in and you get the verdict, the exact p-value, and your statistic marked on the chart alongside the rejection region.
9. Press Find Critical Value. Nothing is computed until you do, and changing any input clears the results so you never read stale numbers.
10. Read the rejection region, then the caveat. The two pills show exactly when to reject and when not to. But clearing the bar tells you the effect is detectable, not that it matters, so pair every test with a confidence interval and an effect size.

Excel has a function for every distribution, but they are frustratingly inconsistent about what they expect: some take alpha, some take the cumulative probability, some are right-tailed and some are left-tailed. Below is the whole workflow in ten steps, each with a picture of what your sheet should look like.

Step 1. Put alpha and the tail count in their own cells so every formula below can point at them. This is what makes the sheet reusable.

Step 2. NORM.S.INV(1-alpha/2) returns 1.959964 for a two-tailed test. Dividing alpha by the tail count makes the same formula work for one tail as well.

Step 3. T.INV.2T takes alpha directly, NOT alpha/2, and returns the two-tailed value 2.228139 at df = 10. This inconsistency between the z and t functions catches almost everyone.

Step 4. T.INV is the one-tailed version and takes the CUMULATIVE probability, so you pass 1-alpha. It returns 1.812461. T.INV and T.INV.2T are not the same function with different names.

Step 5. Passing alpha rather than 1-alpha gives the left-tailed value, -1.812461. Same function, mirrored result.

Step 6. CHISQ.INV.RT is the right-tail version and is what you want for goodness of fit and independence tests. At alpha 0.05 with df 5 it gives 11.070498.

Step 7. For a two-sided variance test you need both bounds. CHISQ.INV takes the LEFT tail, so alpha/2 gives the lower bound and 1-alpha/2 the upper.

Step 8. F.INV.RT gives the right-tailed F critical value. Numerator df first, denominator second, and the order is not interchangeable.

Step 9. Swapping the two degrees of freedom gives 8.660190 instead of 3.098391. Nothing will warn you if you do this by accident, which is why it is worth checking twice.

Step 10. The decision in one formula. Note that =IF(B5<0.05,...) on the p-value gives the identical answer, because the two rules are the same comparison.

The complete function reference

Which functions take alpha and which take the cumulative probability

The rule of thumb: anything ending in .RT or .2T takes alpha. Everything else takes the cumulative probability. It is not a satisfying rule, but it is a reliable one.

Six mistakes that catch people out

• Passing alpha to NORM.S.INV. =NORM.S.INV(0.05) returns −1.645, not 1.96. You need 1-alpha/2 for a two-tailed value.
• Passing alpha/2 to T.INV.2T. That function already halves it internally. =T.INV.2T(0.025,10) gives you the 2.5% two-tailed value of 2.634, not the 5% value of 2.228.
• Using CHISQ.INV when you meant CHISQ.INV.RT. At alpha 0.05 with df 5 they give 1.145 and 11.070 respectively. One is nearly ten times the other.
• Swapping the F degrees of freedom. F(3,20) is 3.098 and F(20,3) is 8.660.
• Comparing a t statistic against a z critical value. Nothing errors. You just get the wrong answer, and at small n you get it badly.
• Hard-coding 1.96 everywhere. It is only correct for z at alpha 0.05 two-tailed. Point at a cell holding alpha instead, and the whole sheet updates when you change it.

R is the cleanest of the three. Every distribution follows the same naming pattern, so once you know one you know them all: q for the quantile (critical value), p for the cumulative probability, d for the density and r for random draws.

# Critical Value Calculator in R (base R, no packages)

alpha <- 0.05

# ---- 1. Z critical values ---------------------------------------------
z_two <- qnorm(1 - alpha/2)              # 1.959964
z_one <- qnorm(1 - alpha)                # 1.644854
# identical, using the right tail directly:
z_one_alt <- qnorm(alpha, lower.tail = FALSE)

# ---- 2. T critical values ---------------------------------------------
df    <- 10
t_two <- qt(1 - alpha/2, df)             # 2.228139
t_one <- qt(1 - alpha, df)               # 1.812461

# ---- 3. Chi-square (right-tailed by convention) -----------------------
chi_df   <- 5
chi_crit <- qchisq(1 - alpha, chi_df)    # 11.070498
# both bounds, for a two-sided test about a variance:
chi_lo <- qchisq(alpha/2, chi_df)
chi_hi <- qchisq(1 - alpha/2, chi_df)

# ---- 4. F critical value (ORDER MATTERS) ------------------------------
d1 <- 3; d2 <- 20
f_crit <- qf(1 - alpha, d1, d2)          # 3.098391
f_swap <- qf(1 - alpha, d2, d1)          # 8.660190, completely different

# ---- 5. Compute a statistic and make the decision ---------------------
x      <- c(52,48,55,61,47,58,50,63,45,56,54,49)
mu0    <- 50
n      <- length(x)
t_stat <- (mean(x) - mu0) / (sd(x) / sqrt(n))
t_c    <- qt(1 - alpha/2, n - 1)
p_val  <- 2 * pt(-abs(t_stat), n - 1)
reject <- abs(t_stat) > t_c              # identical to p_val < alpha

cat(sprintf("Z   alpha=%.2f: two-tailed %.6f, one-tailed %.6f\n",
            alpha, z_two, z_one))
cat(sprintf("T   df=%d:       two-tailed %.6f, one-tailed %.6f\n",
            df, t_two, t_one))
cat(sprintf("CHI df=%d:        right-tailed %.6f, two-sided [%.6f, %.6f]\n",
            chi_df, chi_crit, chi_lo, chi_hi))
cat(sprintf("F   (%d,%d):       right-tailed %.6f   F(%d,%d) = %.6f\n",
            d1, d2, f_crit, d2, d1, f_swap))
cat(sprintf("DECISION  t(%d) = %.6f, critical +/-%.6f, p = %.6f\n",
            n-1, t_stat, t_c, p_val))
cat(sprintf("          reject = %s   (p < alpha is %s: they agree)\n",
            reject, p_val < alpha))

# ---- 6. One figure ----------------------------------------------------
grid <- seq(-5, 5, length.out = 700)
pdf  <- dt(grid, n - 1)
plot(grid, pdf, type = "l", lwd = 2.4, col = "#4338ca", bty = "n",
     xlab = "t value", ylab = "density",
     main = sprintf("Rejection Region at alpha = %.2f, two-tailed, df = %d",
                    alpha, n - 1))

# shade both rejection tails
tail_r <- grid[grid >=  t_c]
tail_l <- grid[grid <= -t_c]
polygon(c(tail_r, rev(tail_r)), c(dt(tail_r, n-1), rep(0, length(tail_r))),
        col = adjustcolor("#f59e0b", 0.40), border = NA)
polygon(c(tail_l, rev(tail_l)), c(dt(tail_l, n-1), rep(0, length(tail_l))),
        col = adjustcolor("#f59e0b", 0.40), border = NA)

abline(v = c(-t_c, t_c), col = "#b45309", lty = 2, lwd = 1.6)
text(t_c, dt(0, n-1) * 0.72, sprintf(" critical %.3f", t_c),
     pos = 4, cex = 0.8, col = "#b45309")
points(t_stat, dt(t_stat, n-1), pch = 17, cex = 1.6, col = "#be185d")

legend("topright", bty = "n", cex = 0.85,
       legend = c(sprintf("t-distribution, df = %d", n-1),
                  sprintf("rejection region, area = %.2f", alpha),
                  sprintf("your t = %.3f", t_stat)),
       col = c("#4338ca", "#f59e0b", "#be185d"),
       lwd = c(2.4, 6, NA), pch = c(NA, NA, 17))

What the script prints

Z   alpha=0.05: two-tailed 1.959964, one-tailed 1.644854
T   df=10:       two-tailed 2.228139, one-tailed 1.812461
CHI df=5:        right-tailed 11.070498, two-sided [0.831212, 12.832502]
F   (3,20):       right-tailed 3.098391   F(20,3) = 8.660190
DECISION  t(11) = 1.934605, critical +/-2.200985, p = 0.079165
          reject = FALSE   (p < alpha is FALSE: they agree)

These are the same numbers the Python script produces and the same numbers the calculator at the top of this page produces.

Line-by-line explanation

• Block 1 shows two equivalent ways to get a one-tailed value: pass 1 - alpha, or pass alpha with lower.tail = FALSE. The second is clearer about intent and avoids arithmetic errors.
• Block 2 is identical in structure, with df added. That consistency is R's main advantage over Excel here.
• Block 3 gives the right-tailed chi-square value you want for goodness of fit, plus both bounds for the variance case. Note that qchisq(1 - alpha, df) and qchisq(alpha, df, lower.tail = FALSE) are the same thing.
• Block 4 demonstrates the F ordering trap explicitly: 3.098 becomes 8.660 when you swap the arguments, and nothing errors.
• Block 5 computes a statistic and makes the decision both ways. The two reject checks must agree, and if they ever do not you have made an arithmetic error somewhere.
• Block 6 draws the rejection region. This is the figure worth showing a class, because the shaded area is alpha, made visible.

Useful one-liners

SciPy follows one consistent pattern across every distribution: .ppf() for the quantile (the critical value), .cdf() for the cumulative probability, .sf() for the survival function (the right tail), and .pdf() for the density. The script below was run before being published, so the output shown underneath is real.

# Critical Value Calculator in Python
# z, t, chi-square and F critical values, with rejection regions and decisions.
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from scipy import stats

alpha = 0.05

# ---- 1. Z critical values ---------------------------------------------
z_two   = stats.norm.ppf(1 - alpha/2)      # 1.959964
z_one   = stats.norm.ppf(1 - alpha)        # 1.644854

# ---- 2. T critical values ---------------------------------------------
df = 10
t_two   = stats.t.ppf(1 - alpha/2, df)     # 2.228139
t_one   = stats.t.ppf(1 - alpha, df)       # 1.812461

# ---- 3. Chi-square critical value (right-tailed by convention) --------
chi_df   = 5
chi_crit = stats.chi2.ppf(1 - alpha, chi_df)          # 11.070498
# both bounds, for a two-sided test about a variance:
chi_lo   = stats.chi2.ppf(alpha/2, chi_df)
chi_hi   = stats.chi2.ppf(1 - alpha/2, chi_df)

# ---- 4. F critical value (right-tailed; ORDER MATTERS) ----------------
d1, d2  = 3, 20
f_crit  = stats.f.ppf(1 - alpha, d1, d2)              # 3.098391
f_swap  = stats.f.ppf(1 - alpha, d2, d1)              # 8.660168, different!

# ---- 5. Compute a statistic and make the decision ---------------------
x   = np.array([52,48,55,61,47,58,50,63,45,56,54,49], dtype=float)
mu0 = 50.0
n   = x.size
t_stat = (x.mean() - mu0) / (x.std(ddof=1) / np.sqrt(n))
t_c    = stats.t.ppf(1 - alpha/2, n - 1)
p_val  = 2 * stats.t.sf(abs(t_stat), n - 1)
reject = abs(t_stat) > t_c            # identical to: p_val < alpha

print(f"Z   alpha={alpha}: two-tailed {z_two:.6f}, one-tailed {z_one:.6f}")
print(f"T   df={df}:       two-tailed {t_two:.6f}, one-tailed {t_one:.6f}")
print(f"CHI df={chi_df}:      right-tailed {chi_crit:.6f}, "
      f"two-sided [{chi_lo:.6f}, {chi_hi:.6f}]")
print(f"F   ({d1},{d2}):      right-tailed {f_crit:.6f}   "
      f"F({d2},{d1}) = {f_swap:.6f}  <- order matters")
print(f"DECISION  t({n-1}) = {t_stat:.6f}, critical +/-{t_c:.6f}, "
      f"p = {p_val:.6f}")
print(f"          reject = {reject}   (p < alpha is {p_val < alpha}: they agree)")

# ---- 6. One figure ----------------------------------------------------
fig, ax = plt.subplots(figsize=(9, 5))
grid = np.linspace(-5, 5, 700)
pdf  = stats.t.pdf(grid, n - 1)
ax.plot(grid, pdf, color="#4338ca", lw=2.4, label=f"t-distribution, df = {n-1}")
ax.fill_between(grid, pdf, where=np.abs(grid) >= t_c, color="#f59e0b",
                alpha=.40, label=f"rejection region, area = {alpha}")
for c in (-t_c, t_c):
    ax.axvline(c, color="#b45309", ls="--", lw=1.6)
ax.text(t_c, stats.t.pdf(0, n-1)*0.72, f" critical {t_c:.3f}",
        fontsize=9, color="#b45309")
ax.plot([t_stat], [stats.t.pdf(t_stat, n-1)], marker="^", ms=13,
        color="#be185d", ls="none", label=f"your t = {t_stat:.3f}")
ax.set_xlabel("t value")
ax.set_ylabel("density")
ax.set_title(f"Rejection Region at alpha = {alpha}, two-tailed, df = {n-1}")
ax.legend(frameon=False, fontsize=9)
ax.spines[["top", "right"]].set_visible(False)
fig.tight_layout()
fig.savefig("critical_value.png", dpi=150)
print("saved critical_value.png")

Actual output from running the script

Z   alpha=0.05: two-tailed 1.959964, one-tailed 1.644854
T   df=10:       two-tailed 2.228139, one-tailed 1.812461
CHI df=5:      right-tailed 11.070498, two-sided [0.831212, 12.832502]
F   (3,20):      right-tailed 3.098391   F(20,3) = 8.660190  <- order matters
DECISION  t(11) = 1.934605, critical +/-2.200985, p = 0.079165
          reject = False   (p < alpha is False: they agree)

Line-by-line explanation

• Block 1 uses stats.norm.ppf, which takes the cumulative probability. There is no alpha-taking variant to confuse it with, unlike Excel.
• Block 2 is structurally identical with df added as the second argument. Every SciPy distribution follows this shape.
• Block 3 gives the right-tailed chi-square value for goodness of fit, plus both bounds for a two-sided variance test. stats.chi2.isf(alpha, df) is an equivalent shortcut for the right tail.
• Block 4 makes the F ordering trap explicit: F(3,20) is 3.098 and F(20,3) is 8.660. Both are valid calls, so nothing will error.
• Block 5 computes a statistic and makes the decision two ways. The printed line confirms that the critical value rule and the p-value rule agree, which they must.
• Block 6 shades the rejection region. fill_between with a where mask handles both tails in a single call.

Useful one-liners

11.1 Z critical values

The only table that fits on one screen, because the normal distribution has no degrees of freedom. These seven pairs cover essentially every z test you will ever run.

11.2 T critical values, two-tailed

Find your degrees of freedom down the left and your alpha across the top. The bottom row is the normal distribution, which is where t converges as df grows.

11.3 T critical values, one-tailed

The same table for a directional hypothesis. Note that the one-tailed value at 0.025 equals the two-tailed value at 0.05, which is the arithmetic behind the warning about switching tails after seeing your data.

11.4 Chi-square critical values

Column headings are the area in the right tail. For a standard goodness-of-fit or independence test at α = 0.05, use the 0.05 column. The first five columns are for the lower tail of a two-sided variance test.

Useful sanity check: the mean of a chi-square distribution equals its degrees of freedom. If your statistic is close to df, the fit is about as good as chance would predict. The critical value always sits well above df.

11.5 F critical values at α = 0.05

Numerator degrees of freedom across the top, denominator down the side. This is the table used for ANOVA at the conventional 5% level.

11.6 F critical values at α = 0.01

11.7 The same alpha, four distributions

Notice the relationship in the df = 1 rows: 3.8415 is exactly 1.9600 squared, and 6.6349 is 2.5758 squared. A chi-square with one degree of freedom is the square of a standard normal, which is not a coincidence but a definition.

11.8 Degrees of freedom, by test

11.9 Excel, R and Python side by side

11.10 How to read any of these tables

Every number below was computed with the calculator on this page and cross-checked against SciPy. Each example has its own colour and its own figure showing the distribution with the rejection region shaded. Examples 5 and 7 are the two mistakes worth studying: both flip a decision and neither produces any error message.

A critical value is the last thing you look up and the first thing that can be gamed. Everything that makes it meaningful is decided before you see a single number.

1. State the null and alternative hypotheses in writing. The alternative determines the tail, and writing it down before collecting data is what stops it from drifting to match your results.
2. Fix alpha in advance. 0.05 is a convention Fisher suggested for convenience, not a law. Choose it deliberately: a stricter alpha protects against false positives at the cost of missing real effects.
3. Choose the tail from the hypothesis, not the data. One tail is legitimate only if you would treat a result in the opposite direction exactly as you would treat no result. If a reversal would interest you, use two tails.
4. Pick the distribution from the design. Means with an estimated SD go to t, counts go to chi-square, variance ratios and ANOVA go to F, and z only when the population standard deviation is genuinely known.
5. Work out the degrees of freedom before you run anything. Section 11.8 lists them by test. This is where most silent errors originate.
6. Check the assumptions of the test itself, not just the arithmetic. The critical value is correct regardless; the question is whether the statistic follows that distribution at all.
7. Do not peek and continue collecting. Checking whether you have cleared the bar and then adding more data until you do inflates the false positive rate well above your stated alpha.
8. Run one primary test. Twenty tests at 5% give you a 64% chance of at least one false positive. Pre-register the primary comparison, or correct for multiplicity and say that you did.
9. Report the exact p-value, not just the verdict. Journals expect it, and a bare "significant at 0.05" hides the difference between p = 0.049 and p = 0.0001.
10. Report a confidence interval and an effect size alongside. The critical value comparison answers a yes-or-no question that is rarely the one anyone actually cares about.
11. State alpha, the tail and the degrees of freedom in the write-up. A critical value without them cannot be checked by a reader.
12. Keep a record of the decisions made before data collection. Pre-registration exists precisely because these choices are unverifiable after the fact.

Use z when

• The population standard deviation is genuinely known. Rare outside textbooks, quality control with a long-established process, and standardised tests with published norms.
• You are working with proportions, where the standard error is determined by p and n rather than estimated separately.
• The sample is very large, where t has converged on z anyway. Above about n = 100 the difference is under 1%.
• You are building a confidence interval for a percentage, which is where 1.96 earns its fame.

Use t when

• You estimated the standard deviation from your sample. This covers almost every real analysis of means.
• The sample is small. Below about n = 30 the difference between t and z is large enough to change conclusions, as Example 7 shows.
• You are testing a regression coefficient, where the residual standard error is estimated from the data.
• You are unsure. There is never a penalty for using t. It is slightly conservative at large n and correct at small n.

Use chi-square when

• You have counts in categories and want to test whether they match an expected pattern.
• You are testing independence in a contingency table.
• You are testing a hypothesis about a single variance, which is the one case where two tails make sense.
• You are assessing model fit via a likelihood ratio, which follows a chi-square distribution asymptotically.

Use F when

• You are comparing three or more group means via ANOVA.
• You are comparing two variances directly.
• You are testing overall regression significance, or comparing nested models.

Do not use a critical value at all when

What a critical value assumes

The critical value itself is exact arithmetic: it is the quantile of a named distribution and cannot be wrong. What can be wrong is the assumption that your statistic follows that distribution. Everything below is about that.

Limitations of the approach itself

• It answers a yes-or-no question. Statistics at 1.95 and 1.97 receive opposite verdicts despite being nearly identical evidence. The threshold is a convenience, not a natural boundary.
• It says nothing about effect size. With a large enough sample, any non-zero difference clears any critical value. Significance and importance are different questions.
• Failing to clear it is not evidence of no effect. It means this sample could not distinguish the hypotheses, which is partly a statement about your power.
• Alpha is a convention. There is nothing special about 5%. Fisher proposed it as a convenient default and it hardened into a rule he did not intend.
• It does not give the probability that the hypothesis is true. That is a Bayesian quantity requiring a prior. The p-value is the probability of the data given the null, not the reverse.
• Printed tables force interpolation. Real tables jump from df 30 to 40 to 60, and interpolating between rows is an approximation that software makes unnecessary.
• Multiple testing is not handled. Each critical value controls one test's error rate, not the family's.
• The distribution may not apply at all. The critical value is exact for the assumed distribution and irrelevant if your statistic does not follow it.

A critical value is the cut-off your test statistic has to beat. You fix a significance level, look up the point beyond which only that proportion of the distribution lies, and reject the null hypothesis if your statistic is more extreme. The shaded region in chart 1 has an area of exactly alpha, and that single fact explains everything else on this page.

Four distributions cover almost all of applied work. Z is used when the population standard deviation is genuinely known, which is rare, and its value depends only on alpha and the tail: 1.9600 two-tailed at 5%, 1.6449 one-tailed. T is used whenever you estimated the standard deviation from your own data, which is almost always, and it needs degrees of freedom. Chi-square handles counts and categorical data. F handles ANOVA and variance ratios and needs two degrees of freedom in a fixed order.

Three things about those values are worth internalising. First, the tail choice matters enormously: at alpha 0.05 the two-tailed z is 1.9600 and the one-tailed is 1.6449, because all the alpha sits in one tail rather than being split. Choosing the tail after seeing which direction your data went converts a nominal 5% test into a 10% test while you report it as 5%, and nothing in your output reveals it. Second, degrees of freedom change everything except z: the t critical value falls from 12.7062 at df = 1 through 2.2281 at df = 10 to 1.9840 at df = 100, converging on the normal value. Small samples are held to a much higher bar, which is honest rather than harsh. Third, chi-square and F are right-tailed by convention, because only a large statistic is evidence against the null; the exception is a test about a variance, where an unusually small value can matter too.

Two errors on this page flip decisions and produce no warning whatsoever. Using the normal critical value of 1.9600 where a t(5) value of 2.5706 belongs turns a correct retain into a false reject, and the write-up looks entirely normal. Swapping the F degrees of freedom turns 3.0984 into 8.6602, which does the opposite: a real effect declared non-significant. Both are silent, both are common, and both are avoidable by checking the distribution and the argument order before you look anything up.

Critical values and p-values are the same comparison read from opposite ends of the distribution, so they cannot disagree. Tables existed because computing an exact tail area by hand was infeasible; software removed that constraint and journals now expect exact p-values. What tables still do better is make the reasoning visible, which is why every result here is drawn as well as printed.

The last point is the one most worth keeping. A critical value answers a yes-or-no question, and it is rarely the question anyone actually has. Clearing the bar means the effect is detectable at your sample size, not that it is large or important, and with enough data any non-zero difference clears any threshold. Failing to clear it is not evidence that nothing is there. Report the exact p-value, a confidence interval and an effect size, and treat the threshold as one input to a judgement rather than the judgement itself.

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