Hypothesis Tests
Chi-Square Critical Value Calculator
A chi-square critical value is the upper-tail cutoff on the chi-square curve that uses exactly the degrees of freedom you type.
Worksheet
Press Calculate to see the result.
A chi-square critical value is the upper-tail cutoff on the chi-square curve that uses exactly the degrees of freedom you type.
What this calculator answers
Chi-square tests of counts are almost always upper-tailed, because a large statistic means the observed counts sit far from the expected counts, and a small statistic means they sit close. The critical value is the point that leaves probability alpha to its right on the chi-square curve with your degrees of freedom. This page returns that upper cutoff and does not search a lower tail. Goodness-of-fit and contingency tests both use this same curve once their own df has been counted.
The headline is the upper critical value at your df and alpha. The page also states the upper-tail rejection rule in words. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Upper-tail chi-square critical value uses this relationship.
Find x so that the probability of a chi-square larger than x equals alpha, on df degrees of freedom.
- df. Degrees of freedom of the chi-square curve.
- alpha. Upper-tail probability.
- critical value. The cutoff with upper area alpha.
- rejection region. Chi-square statistics larger than the cutoff.
The chi-square family is supported on zero and upward, and it is skewed, so there is no plus-or-minus pair the way there is for z. The page searches that curve until the survival probability equals alpha, which is the definition of an upper-tail test. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The loaded lookup is 5 degrees of freedom and an upper-tail alpha of 0.05.
Degrees of freedom are 5 and alpha is 0.05. Chi-square tests of counts are usually upper-tailed, because a large statistic means a poor fit and a small statistic means a close fit. The upper critical value, leaving probability 0.05 to the right, is 11.0705.
Reject when the observed chi-square is greater than 11.0705 on 5 degrees of freedom. There is no negative cutoff. The chi-square curve starts at 0. A goodness-of-fit with k categories usually has df equal to k minus 1, not k. This page used the df you typed, 5.
A contingency table usually has df equal to (rows minus 1) times (columns minus 1). That count is a different worksheet. The result is the cutoff only. A p-value needs an observed chi-square, and none was typed among these 5 degrees of freedom.
| Chi-square critical value | 11.0705 |
|---|---|
| Degrees of freedom | 5 |
| Tail | Upper |
A second situation
One degree of freedom at alpha 0.05, the 2 by 2 table line
Degrees of freedom are 1 and alpha is 0.05. Chi-square tests of counts are usually upper-tailed, because a large statistic means a poor fit and a small statistic means a close fit. The upper critical value, leaving probability 0.05 to the right, is 3.8415.
Reject when the observed chi-square is greater than 3.8415 on 1 degrees of freedom. There is no negative cutoff. The chi-square curve starts at 0. A goodness-of-fit with k categories usually has df equal to k minus 1, not k. This page used the df you typed, 1.
A contingency table usually has df equal to (rows minus 1) times (columns minus 1). That count is a different worksheet. The result is the cutoff only. A p-value needs an observed chi-square, and none was typed among these 1 degrees of freedom.
| Chi-square critical value | 3.8415 |
|---|---|
| Degrees of freedom | 1 |
| Tail | Upper |
Changing df changes the whole curve, so a 1-df cutoff and a 5-df cutoff are not interchangeable. Tightening alpha pushes the cutoff to the right, into a thinner part of the upper tail. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
Alpha moved from 0.05 to 0.01. Degrees of freedom stayed at 5. The headline result went from 11.0705 to 15.0863.
The nudged inputs
Degrees of freedom are 5 and alpha is 0.01. Chi-square tests of counts are usually upper-tailed, because a large statistic means a poor fit and a small statistic means a close fit. The upper critical value, leaving probability 0.01 to the right, is 15.0863.
Reject when the observed chi-square is greater than 15.0863 on 5 degrees of freedom. There is no negative cutoff. The chi-square curve starts at 0. A goodness-of-fit with k categories usually has df equal to k minus 1, not k. This page used the df you typed, 5.
A contingency table usually has df equal to (rows minus 1) times (columns minus 1). That count is a different worksheet. The result is the cutoff only. A p-value needs an observed chi-square, and none was typed among these 5 degrees of freedom.
| Chi-square critical value | 15.0863 |
|---|---|
| Degrees of freedom | 5 |
| Tail | Upper |
How to read the result
Reject when the observed chi-square is at least as large as this cutoff. A small chi-square, near zero, is agreement with the expected counts, not evidence against them. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Pair the critical value with df and alpha, and say the test is upper-tailed. A lone cutoff cannot be matched to a table. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The statistic is referred to a chi-square curve with this df under the null.
- The test is upper-tailed, which is the usual count-data test.
- Df is positive.
- Expected counts are large enough that the chi-square approximation is decent. The cutoff itself does not check that.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Halving alpha as if the test were two-sided z. The upper tail already uses the full alpha.
- Using categories as df in a goodness-of-fit test. The usual df is categories minus 1.
- Using rows times columns as df in a table. The usual df is (rows minus 1) times (columns minus 1).
- Comparing a chi-square statistic with a z cutoff.
Where a different calculator is the honest choice
The chi-square test calculator and the goodness-of-fit calculator compute the statistic from counts and then the p-value. The p-value from chi-square calculator starts from a statistic you already have. This page is only the cutoff. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: P-Value from Chi-Square Calculator, Chi-Square Test Calculator, Chi-Square Goodness of Fit Test Calculator.
Check yourself
What is the upper chi-square critical value at alpha 0.05 with 5 degrees of freedom?
The critical value is 11.0705.
Using the result in a sentence
Goodness-of-fit checks and contingency tables both end at this curve after the df arithmetic. Variance tests that use chi-square directly are the other classic caller. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
A cutoff does not repair tiny expected counts. When several expected cells are under 5, the nominal alpha can be off even if you use the right critical value. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Chi-Square Critical Value Calculator commit to?
This page uses Upper-tail chi-square critical value: Find x so that the probability of a chi-square larger than x equals alpha, on df degrees of freedom.. The chi-square family is supported on zero and upward, and it is skewed, so there is no plus-or-minus pair the way there is for z. The page searches that curve until the survival probability equals alpha, which is the definition of an upper-tail test. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Degrees of freedom, Alpha, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the chi-square critical value is 11.0705. Reject when the observed chi-square is at least as large as this cutoff. A small chi-square, near zero, is agreement with the expected counts, not evidence against them. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Halving alpha as if the test were two-sided z. The upper tail already uses the full alpha. A cutoff does not repair tiny expected counts. When several expected cells are under 5, the nominal alpha can be off even if you use the right critical value. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
The chi-square test calculator and the goodness-of-fit calculator compute the statistic from counts and then the p-value. The p-value from chi-square calculator starts from a statistic you already have. This page is only the cutoff. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The critical value is 11.0705. Pair the critical value with df and alpha, and say the test is upper-tailed. A lone cutoff cannot be matched to a table. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.