A test statistic is judged against the distribution it would follow if the null hypothesis were true, and that distribution supplies the critical value and the p-value. Four distributions cover the tests of a first and second econometrics course, and seeing how the t, the chi-square and the F are built from the normal, by squaring, adding and dividing, shows which test uses which. (Unit-root tests, met later, have their own tables.)
1. The four distributions side by side
Take a regression of log wages on education, experience and tenure for 526 workers (the one read line by line in how to read regression output). Its output reports a t-statistic of 2.39 for experience and an F-statistic of 80.39 for the regression as a whole. Whether 2.39 or 80.39 is large depends on how such a statistic behaves when the null hypothesis is true: its reference distribution. The table sets out the four that econometrics uses.
| Standard normal | t | Chi-square | F | |
|---|---|---|---|---|
| Shape | Bell, symmetric about 0 | Bell, symmetric about 0, heavier tails | Skewed to the right | Skewed to the right |
| Values it can take | Any | Any | Zero or more | Zero or more |
| Degrees of freedom | None | One number, k | One number, k | Two numbers, m and d |
| Built from standard normals | Z itself | Z divided by the root of (an independent chi-square ÷ k) | Sum of k squared independent Z's | (chi-square ÷ m) divided by (an independent chi-square ÷ d) |
| Used to test | One mean or coefficient, large sample | One mean or coefficient, variance estimated (exact for normal errors) | Several restrictions, large sample | Several restrictions in a regression (exact for normal errors) |
| A 5% critical value | 1.96 (two-sided) | 2.04 with 30 degrees of freedom (two-sided) | 7.81 with 3 | 2.62 with 3 and 522 |
Two rows decide where the rejection region goes. The normal and the t are symmetric and can be negative, so their tests can be one-sided or two-sided. The chi-square and the F are built from squares, so they are never negative, and the tests on this page that use them reject for large values.
The sections below build each distribution from standard normal variables, which we write \(Z\): normal, with mean 0 and variance 1. Capital letters stand for random variables: \(Z\), \(T_k\), \(F_{m,d}\). Books use other letters; the choice is cosmetic.
What students say
“Nicky is an excellent econometrics tutor. He explains complex concepts clearly and makes them much easier to understand.”
“I was completely lost in my economics class, and then I started working Nicky and he totally saved me. He was excellent at explaining the material and extremely accommodating to meet with me many times despite us being in different time zones! Highly recommend working with Nicky!”
“Nicky is a very good and thorough teacher. He has excellent knowledge of econometrics and takes the time to work through your questions. Highly recommended!”
“Really great tutor! Encouraging, knowledgeable, friendly”
“Very good lesson, Nicky is very helpful.”
“Very helpful and clear.”
“I gather it was very useful, thank you!”
“Fantastic!”
2. What does "degrees of freedom" mean?
Degrees of freedom count the independent pieces of information that go into a sum of squares.
Five workers are paid £10, 12, 13, 16 and 19 an hour. Their mean is 14, and the deviations from the mean are -4, -2, -1, 2 and 5. These five numbers are not five independent pieces of information, because deviations from a mean always add to zero. Given the first four (which add to -5), the fifth has to be 5. Only four are free. The same count gives the divisor of the sample variance. Deviations from the sample mean are on average smaller than deviations from the population mean, because the sample mean is the value that makes their sum of squares as small as possible; under random sampling, with \(n\) observations from a population with variance \(\sigma^2\), their sum of squares averages \((n - 1)\sigma^2\) across samples, not \(n\sigma^2\). Hence the sample variance divides it by \(n - 1\), the number of free deviations, and not by \(n\): \(50/4 = 12.5\) (lesson 3 works through this).
The same count applies in a regression. Each coefficient estimated puts one condition on the residuals, so the degrees of freedom left are the number of observations minus the number of coefficients, including the intercept: 526 - 4 = 522 in the wage regression. More degrees of freedom mean the variance is estimated from more information, and the t distribution has thinner tails.
3. The normal distribution
The central limit theorem says that an average of many independent observations from the same population is approximately normal whatever the distribution of the observations themselves, provided they have a finite variance. Under random sampling a sample mean is such an average, and a regression coefficient is the population coefficient plus a weighted sum of the errors, to which the same result applies. Both are centred on the population value (for a coefficient, provided nothing left in the error moves with the regressors), and in a large sample the standard error is close to the standard deviation it estimates. Hence in large samples the following ratio is approximately standard normal, which is why the normal distribution is the starting point:
$$ \frac{\text{estimate} - \text{population value}}{\text{standard error}} . $$A standard normal variable lies within 1 of zero with probability 68.3%, within 1.96 with probability 95.0% and within 3 with probability 99.7%. The second of these is the source of the 1.96 that appears in every large-sample 95% confidence interval and two-sided 5% test.
| Significance level | 10% | 5% | 1% |
|---|---|---|---|
| One-sided critical value | 1.282 | 1.645 | 2.326 |
| Two-sided critical value | 1.645 | 1.960 | 2.576 |
The table gives the point beyond which the stated share of the distribution lies: in one tail for a one-sided test, split between the two tails for a two-sided one.
4. The chi-square distribution: squaring and adding
Some hypotheses involve several quantities at once, for example that three coefficients are all zero. Each estimate, standardised, is approximately a \(Z\) in large samples. Positive and negative departures from zero both count as evidence, so a natural summary is to square each one and add them up. If the standardised estimates were independent, the distribution of that sum would be the chi-square. Estimates from one regression are in general correlated, so the statistic used in practice adjusts for that, and in large samples the result behaves like a sum of squares of independent standard normals, with a chi-square distribution.
Because it is a sum of squares, a chi-square variable cannot be negative. A large value says that the squared quantities are too big, taken together, to be the squares of standard normals, so a chi-square test of this kind rejects in the upper tail only. The 5% critical values are 3.84 for 1 degree of freedom, 5.99 for 2, 7.81 for 3, 11.07 for 5 and 18.31 for 10.
With one degree of freedom the chi-square is \(Z^2\) itself, and \(Z^2\) is above its 5% critical value, 3.841, exactly when \(Z\) is beyond 1.960 in either direction: 3.841 is the square of 1.960, up to rounding in the last digit. A chi-square test with one degree of freedom is a two-sided normal test in another form.
5. The t distribution: a normal divided by an estimated standard deviation
The normal approximation of section 3 treats the standard error as if it were known, but it is computed from the same sample as the estimate. In a large sample that makes no practical difference. In a small sample the standard error can come out well below or well above its true value. Suppose the errors are normally distributed. Then with 4 degrees of freedom the standard error is below half its true value in about 9% of samples, and the ratio of section 3 is then more than twice what it would be with the true standard deviation in the denominator. These cases fatten the tails more than the opposite cases thin them, so the ratio has heavier tails than the normal. The t distribution is the exact distribution of that ratio when the errors are normal and the denominator is estimated. Without normality it is an approximation, and the approximation improves as the sample grows.
The t distribution is symmetric and bell-shaped like the normal, with more of its probability in the tails. How much more depends on the degrees of freedom:
| Degrees of freedom | 3 | 10 | 30 | 120 | 522 | Normal |
|---|---|---|---|---|---|---|
| Two-sided 5% critical value | 3.182 | 2.228 | 2.042 | 1.980 | 1.965 | 1.960 |
| Probability of a value beyond ±1.96 | 14.5% | 7.8% | 5.9% | 5.2% | 5.1% | 5.0% |
The first row is the multiple needed to leave 5% in the two tails. The second row shows what happens if 1.96 is used regardless: with 3 degrees of freedom a "5%" test would in fact reject a true null in 14.5% of samples. As the degrees of freedom grow, the estimated standard deviation settles on the true one (the law of large numbers), so the denominator settles down to 1 and the t distribution approaches the standard normal.
So the same t-statistic is stronger or weaker evidence depending on the sample behind it. The coefficient on experience in the wage regression has \(t = 2.39\). With that regression's 522 degrees of freedom the two-sided p-value is 0.017. The same 2.39 from a regression with 10 degrees of freedom would have a p-value of 0.038, and with 3 degrees of freedom 0.097, which is not significant at the 5% level.
6. The F distribution: a ratio of two chi-squares
To test several restrictions on a regression at once we need the counterpart of the t-statistic: a sum of squared quantities in the numerator, as in section 4, and an estimated variance in the denominator, as in section 5. If the errors are normally distributed, both are chi-squares, independent of each other, each divided by its degrees of freedom, and the distribution of their ratio is the F.
Like the chi-square, an F variable is never negative, and an F-test of restrictions rejects only for large values. If the restrictions are true, the numerator and the denominator are both estimates of the same variance, so the ratio varies around 1: with 3 and 522 degrees of freedom it exceeds 2.62 in only 5% of samples. A ratio far above 1 says the restrictions cost the regression much more fit than chance alone would account for.
In the wage regression the F-statistic for "all three slope coefficients are zero" is 80.39, with 3 and 522 degrees of freedom. The 5% critical value of that F distribution is 2.62. The statistic is thirty times larger, and the null is rejected.
7. How the four connect, in numbers
The definitions above can be checked directly. We drew standard normal numbers and built each distribution from them exactly as defined, 100,000 times each.
| Built from standard normals | Distribution | 5% critical value | Share of 100,000 simulated values beyond it |
|---|---|---|---|
| Z | Standard normal | ±1.96 | 4.96% |
| Sum of 3 squared Z's | Chi-square, 3 | 7.815 | 4.97% |
| Z over root of (chi-square with 5 ÷ 5) | t, 5 | ±2.571 | 5.01% |
| (chi-square with 3 ÷ 3) over (chi-square with 20 ÷ 20) | F, 3 and 20 | 3.098 | 5.06% |
Each row builds one variable from standard normals and counts how many of the 100,000 simulated values fall beyond that distribution's 5% critical value. A share of 5% counted over 100,000 independent draws has a standard error of \(\sqrt{0.05 \times 0.95/100{,}000} = 0.0007\), or 0.07 percentage points, and all four counts are within one standard error of 5%.
The definitions also tie the distributions to each other:
- A t squared is an F. Squaring \(T_k\) gives \(Z^2\) over \(\chi^2_k/k\), which is an F with 1 and \(k\) degrees of freedom. The critical values match: with 30 degrees of freedom, \(2.042^2 = 4.17\), the 5% critical value of F with 1 and 30. The same holds for a single coefficient in a regression. In the wage regression the experience coefficient has \(t = 2.391\). Its square, 5.72, is the F-statistic for the null that this one coefficient is zero. The 5% critical value of F with 1 and 522 degrees of freedom is 3.86, which is 1.965 squared, and the p-value is 0.017 for both.
- A chi-square with one degree of freedom is a squared normal. Its 5% critical value, 3.841, is the square of 1.960 (up to rounding in the last digit).
- With many degrees of freedom, t becomes normal and F becomes a scaled chi-square. The denominator of each settles down to 1. For the t, 1.965 at 522 degrees of freedom against 1.960. For the F with 3 restrictions, \(3 \times 2.622 = 7.87\) at 522 denominator degrees of freedom, against the chi-square value of 7.81.
8. Which test uses which distribution?
| Test | What it checks | Reference distribution | Example on this site |
|---|---|---|---|
| z-test | One mean or coefficient, standard deviation known or sample large | Standard normal | Mean grade against 65, z = 1 |
| t-test | One mean or coefficient, standard deviation estimated | t with degrees of freedom: observations minus 1 for a mean; observations minus the number of coefficients for a regression | Coefficient on experience, t = 2.39 with 522 |
| F-test | Several restrictions on a regression at once | F with (number of restrictions, residual degrees of freedom) | All slopes zero, F(3, 522) = 80.39 |
| Chi-square tests (Wald, likelihood ratio, LM, Q) | Several restrictions at once, in large samples | Chi-square with degrees of freedom equal to the number of restrictions | Autocorrelations jointly zero, in reading the correlogram |
The rule of thumb: one restriction, normal or t; several restrictions, chi-square or F. Within each pair, the t and the F are the versions for regression with an estimated error variance, and the normal and the chi-square are their large-sample counterparts. The likelihood-ratio test in maximum likelihood estimation is a chi-square test.
9. Critical values and p-values in Stata, R and Python
Each program has functions that return a critical value for a given tail area, or a tail area for a given statistic. These are the values used on this page:
| Quantity | Value |
|---|---|
| Normal, two-sided 5% critical value | 1.960 |
| t, two-sided 5% critical value, with 3, 8, 24, 30 and 522 degrees of freedom | 3.182, 2.306, 2.064, 2.042, 1.965 |
| Chi-square with 3 degrees of freedom, 5% critical value | 7.815 |
| F with 3 and 522 degrees of freedom, 5% critical value | 2.622 |
| Two-sided p-value of t = 2.39 with 522 degrees of freedom | 0.0172 |
| Probability that a standard normal exceeds 1 | 0.1587 |
Stata: the code
display invnormal(0.975) display invttail(3, 0.025) display invttail(8, 0.025) display invttail(24, 0.025) display invttail(30, 0.025) display invttail(522, 0.025) display invchi2tail(3, 0.05) display invFtail(3, 522, 0.05) display 2*ttail(522, 2.39) display 1 - normal(1)
Stata: what it prints
. display invnormal(0.975) 1.959964 . display invttail(3, 0.025) 3.1824463 . display invttail(8, 0.025) 2.3060041 . display invttail(24, 0.025) 2.0638986 . display invttail(30, 0.025) 2.0422725 . display invttail(522, 0.025) 1.9645189 . display invchi2tail(3, 0.05) 7.8147279 . display invFtail(3, 522, 0.05) 2.6219805 . display 2*ttail(522, 2.39) .01720207 . display 1 - normal(1) .15865525
R: the code
# critical values and tail probabilities qnorm(0.975) qt(0.975, df = c(3, 8, 24, 30, 522)) qchisq(0.95, df = 3) qf(0.95, df1 = 3, df2 = 522) 2 * pt(-2.39, df = 522) pnorm(1, lower.tail = FALSE)
R: what it prints
> # critical values and tail probabilities > qnorm(0.975) [1] 1.959964 > qt(0.975, df = c(3, 8, 24, 30, 522)) [1] 3.182446 2.306004 2.063899 2.042272 1.964519 > qchisq(0.95, df = 3) [1] 7.814728 > qf(0.95, df1 = 3, df2 = 522) [1] 2.621981 > 2 * pt(-2.39, df = 522) [1] 0.01720207 > pnorm(1, lower.tail = FALSE) [1] 0.1586553
Python: the code
from scipy import stats # critical values and tail probabilities print(stats.norm.ppf(0.975)) print(stats.t.ppf(0.975, [3, 8, 24, 30, 522])) print(stats.chi2.ppf(0.95, 3)) print(stats.f.ppf(0.95, 3, 522)) print(2 * stats.t.cdf(-2.39, 522)) print(stats.norm.sf(1))
Python: what it prints
1.959963984540054 [3.18244631 2.30600414 2.06389856 2.04227246 1.96451894] 7.814727903251179 2.6219805333662087 0.01720207232318519 0.15865525393145707
In R the functions beginning with q return critical values and those beginning with p return probabilities. Stata's functions ending in tail work with upper-tail areas. In Python, ppf is the inverse of the cumulative distribution and sf is the upper-tail area.
10. Common mistakes
- Using 1.96 whatever the sample size. With 3 degrees of freedom the two-sided 5% critical value of t is 3.182, and a test that uses 1.96 rejects a true null in 14.5% of samples.
- Looking for a lower-tail rejection region in a chi-square or F-test of restrictions. Both statistics are built from squares, and evidence against the restrictions makes them large.
- Swapping the two degrees of freedom of an F. The number of restrictions comes first and the residual degrees of freedom second: F(3, 522), not F(522, 3).
- Testing several coefficients with several t-tests. Three separate t-tests are not a test of the joint hypothesis that all three coefficients are zero. That hypothesis needs one F-test.
- Counting degrees of freedom as n. For a mean they are n - 1, and for a regression n minus the number of coefficients, including the intercept.
11. Check your understanding
Five questions, marked on the page; every option has an explanation, including the wrong ones. (Without JavaScript the explanations are shown beneath each question.)
1. Which two of the four distributions are symmetric about zero?
Correct. Both are bell-shaped and centred on zero. The chi-square and the F are built from squares, so they start at zero and are skewed to the right.
A chi-square variable is a sum of squared normals, so it cannot be negative and cannot be symmetric about zero.
The F is a ratio of two sums of squares, so it is never negative. The square of a t variable is an F, but the t itself is symmetric and the F is not.
2. A regression with one explanatory variable reports a slope t-statistic of 9.53 with 3 degrees of freedom. What is the F-statistic for the regression?
3.09 is the square root of 9.53. The F-statistic is the square of t, not its root.
F equals t only for t = 0 or t = 1. With a single restriction F is the square of t.
19.06 doubles t. The link between the two is a square: F = t².
Correct. With one restriction, F = t² = 9.53² = 90.8. (The software squares t before rounding it: 9.5263² = 90.75, the F it prints.)
3. What happens to the two-sided 5% critical value of the t distribution as the degrees of freedom rise from 3 to 30 to 522?
It moves the other way. More degrees of freedom mean a better estimate of the standard deviation, thinner tails and a smaller critical value.
Correct. With few degrees of freedom the t distribution has heavy tails and needs a large multiple. As the degrees of freedom grow it becomes the standard normal.
1.96 is the value for the standard normal. For the t distribution the critical value depends on the degrees of freedom as well as the level.
4. A regression has three explanatory variables and normally distributed errors. You test that all three slope coefficients are zero at once. Which distribution does the test statistic follow if the null is true?
The t distribution handles one restriction at a time. Three restrictions tested together need a statistic that combines them, and that statistic follows an F distribution.
A single standardised coefficient is approximately normal in large samples. A joint test of three restrictions is not a single standardised quantity.
Correct. Three restrictions give 3 numerator degrees of freedom; the denominator has n minus the four coefficients estimated. For the wage regression this is F(3, 522).
5. Why can a chi-square statistic or an F statistic never be negative?
Correct. A chi-square is a sum of squared standard normals, and an F is a ratio of two such sums. Evidence against the restrictions makes these statistics large, so the tests on this page reject in the upper tail.
Nothing is discarded or made positive after the event. The statistics are sums of squares, so they are non-negative by construction.
There are no negative values to drop. Large positive values are the ones that count against the null.
12. Questions people ask
The t distribution is symmetric about zero and tests one restriction, such as a single coefficient being zero. The F distribution takes only positive values, is skewed to the right, and tests several restrictions at once. They are linked: the square of a t variable with k degrees of freedom is an F with 1 and k degrees of freedom.
All four are built from standard normal variables. The sum of k squared independent standard normals is a chi-square with k degrees of freedom. A standard normal divided by the square root of an independent chi-square over its degrees of freedom is a t. The ratio of two independent chi-squares, each divided by its degrees of freedom, is an F.
They count the independent pieces of information left for estimating the variance. For a test on one mean they are n - 1, because the deviations from the sample mean must add to zero. In a regression they are n minus the number of coefficients estimated. Fewer degrees of freedom mean heavier tails and larger critical values.
The standard normal and the t are symmetric about zero, so their tests can be one-sided or two-sided. The chi-square and the F are built from squares: they cannot be negative, they are skewed to the right, and the tests of restrictions built on them reject for large values. As the degrees of freedom grow, the t approaches the normal.
Both test several restrictions together. The F-test is the exact version for a regression with normal errors and is what regression software prints. The chi-square test is its large-sample counterpart, used for likelihood-ratio tests and for tests on autocorrelations. In large samples they agree: the number of restrictions times F is approximately chi-square.
13. Where next
- Hypothesis testing explained: how a reference distribution turns a statistic into a p-value and a decision.
- Confidence intervals and t-tests: the normal and t multiples at work.
- How to read regression output: the t-statistics and the F-test in a real regression table.
- The course lessons on inference: lesson 6, lesson 7 and lesson 8.
- The central limit theorem (lesson 4 of the course), the reason the normal distribution is the starting point, and the binomial and normal distributions at A-Level standard.
- Maximum likelihood estimation and reading the correlogram, for chi-square tests in use.
Send me your question: module, topic and deadline
I tutor statistics and econometrics one-to-one, from distribution theory and a first hypothesis test through to regression, panel data and time series. Bring the question you are stuck on and we will work through it in the free 30-minute consultation.
Book a free 30-minute consultation Statistics tuition: how it works
Numbers and simulation. Critical values and probabilities computed with scipy and checked in R. Figure 4: one draw of 100,000 rows of 33 independent standard normal numbers, numpy default_rng(2026); rng.standard_normal((100000, 33)). Counting columns from 0, each row gives Z (column 0), a chi-square with 3 degrees of freedom (columns 1 to 3), a t with 5 (column 4 over columns 5 to 9) and an F with 3 and 20 (columns 10 to 12 over columns 13 to 32). Wage regression: Wooldridge, Introductory Econometrics: A Modern Approach, data set WAGE1 (526 workers, 1976 Current Population Survey); results quoted, data not republished. Five-worker example: illustrative data.