How To Find P Value From Test Statistic

Welcome to our p-value calculator! Yous will never again have to wonder how to find the p-value, as here you can decide the ane-sided and two-sided p-values from test statistics, following all the about pop distributions: normal, t-Student, chi-squared, and Snedecor'south F.

P-values appear all over science, notwithstanding many people detect the concept a bit intimidating. Don't worry - in this article nosotros explain not but what the p-value is, but also how to interpret p-values correctly. Take you always been curious near how to summate p-value by hand? We provide you with all the necessary formulae also!

What is p-value?

Formally, the p-value is the probability that the test statistic volition produce values at to the lowest degree as farthermost equally the value it produced for your sample. It is crucial to recall that this probability is calculated under the assumption that the null hypothesis is true!

More intuitively, p-value answers the question:
Assuming that I live in a world where the null hypothesis holds, how probable is it that, for some other sample, the examination I'm performing volition generate a value at to the lowest degree as farthermost equally the ane I observed for the sample I already have?

It is the culling hypothesis which determines what "extreme" really means, and so the p-value depends on the alternative hypothesis that you land: left-tailed, right-tailed, or ii-tailed. In formulas below, S stands for a test statistic, x for the value it produced for a given sample, and Pr(upshot | H₀) is the probability of an effect, calculated nether the assumption that H₀ is true:

Left-tailed test: p-value = Pr(Southward ≤ x | H₀)
Right-tailed test: p-value = Pr(S ≥ x | H₀)
Two-tailed test:
p-value = two * min{Pr(Due south ≤ x | H₀), Pr(Due south ≥ x | H₀)}
(By min{a,b} we denote the smaller number out of a and b.)

If the distribution of the test statistic under H₀ is symmetric near 0, then p-value = 2 * Pr(South ≥ |10| | H₀)

or, equivalently, p-value = 2 * Pr(S ≤ -|x| | H₀)

As a picture is worth a thousand words, let us illustrate these definitions. Hither we use the fact that the probability tin exist neatly depicted as the area under the density curve for a given distribution. We requite ii sets of pictures: ane for a symmetric distribution, and the other for a skewed (non-symmetric) distribution.

Symmetric case: normal distribution

Non-symmetric example: chi-squared distribution

In the last picture (2-tailed p-value for skewed distribution), the area of the left-paw side is equal to the surface area of the correct-hand side.

How to summate p-value from test statistic?

To determine the p-value, you demand to know the distribution of your test statistic under the assumption that the zilch hypothesis is true. Then, with help of the cumulative distribution office (cdf) of this distribution, we tin can limited the probability of the test statistics being at least every bit farthermost every bit its value 10 for the sample:

Left-tailed test: p-value = cdf(ten)
Right-tailed test: p-value = 1 - cdf(x)
Two-tailed test: p-value = 2 * min{cdf(10) , ane - cdf(x)}

If the distribution of the test statistic under H₀ is symmetric well-nigh 0, and then a two-sided p-value can be simplified to p-value = ii * cdf(-|x|), or, equivalently, every bit p-value = ii - 2 * cdf(|ten|)

The probability distributions that are most widespread in hypothesis testing tend to take complicated cdf formulae, and finding the p-value by hand may not be possible. You'll likely need to resort to a computer, or to a statistical tabular array, where people have gathered approximate cdf values.

Well, you lot at present know how to calculate p-value, simply... why practice you need to summate this number in the first place? In hypothesis testing, the p-value approach is an alternative to the disquisitional value approach. Call up that the latter requires researchers to pre-set the significance level, α, which is the probability of rejecting the null hypothesis when it is truthful (so of blazon I error). Once y'all take your p-value, you lot just need to compare it with any given α to speedily decide whether or not to refuse the nix hypothesis at that significance level, α. For details, check the next section, where we explain how to translate p-values.

How to translate p-value?

As nosotros have mentioned above, p-value is the answer to the following question:

Assuming that I live in a world where the null hypothesis holds, how probable is it that, for another sample, the test I'm performing will generate a value at to the lowest degree as farthermost as the one I observed for the sample I already have?

What does that mean for you? Well, yous've got two options:

A high p-value ways that your information is highly compatible with the null hypothesis; and
A small-scale p-value provides evidence against the nada hypothesis, as information technology ways that your event would exist very improbable if the zip hypothesis were truthful.

However, information technology may happen that the zero hypothesis is true, but your sample is highly unusual! For instance, imagine we studied the event of a new drug, and get a p-value of 0.03. This means that, in 3% of similar studies, random chance alone would still be able to produce the value of the test statistic that we obtained, or a value even more farthermost, fifty-fifty if the drug had no outcome at all!

The question "what is p-value" tin likewise be answered equally follows: p-value is the smallest level of significance at which the null hypothesis would exist rejected. And so, if y'all at present want to make a decision near the nothing hypothesis at some significance level α, just compare your p-value with α:

If p-value ≤ α, then yous reject the cipher hypothesis and accept the alternative hypothesis; and
If p-value ≥ α, and so you don't have enough bear witness to decline the null hypothesis.

Patently, the fate of the zip hypothesis depends on α . For case, if the p-value was 0.03, we would pass up the zippo hypothesis at a significance level of 0.05, simply not at a level of 0.01. That'due south why the significance level should be stated in accelerate, and non adjusted conveniently afterward p-value has been established! A significance level of 0.05 is the well-nigh common value, but there'southward nil magical nearly information technology. Here, you tin can encounter what too stiff a faith in the 0.05 threshold can lead to. It's always best to report the p-value, and let the reader to make their own conclusions.

Also, conduct in mind that subject area expertise (and mutual reason) is crucial. Otherwise, mindlessly applying statistical principles, you lot can easily arrive at statistically significant, despite the determination being 100% untrue.

How to utilise the p-value calculator to find p-value from test statistic?

Equally our p-value estimator is here at your service, you no longer need to wonder how to find p-value from all those complicated examination statistics! Here are the steps you demand to follow:

Selection the alternative hypothesis: two-tailed, correct-tailed, or left-tailed.
Tell us the distribution of your test statistic under the nil hypothesis: is it N(0,i), t-Pupil, chi-squared, or Snedecor's F? If you are unsure, check the sections beneath, as they are devoted to these distributions,.
If needed, specify the degrees of freedom of the test statistic's distribution.
Enter the value of test statistic computed for your data sample.
Our calculator determines the p-value from examination statistic, and provides the determination to be fabricated about the nix hypothesis. The standard significance level is 0.05 by default.

Go to the avant-garde mode if you need to increment the precision with which the calculations are performed, or change the significance level.

How to discover p-value from z-score?

In terms of the cumulative distribution office (cdf) of the standard normal distribution, which is traditionally denoted by Φ, the p-value is given by the following formulae:

Left-tailed z-exam:
p-value = Φ(Z==score==)
Right-tailed z-test:
p-value = 1 - Φ(Z==score==)
Two-tailed z-test:

p-value = 2 * Φ(−|Z==score==|) or p-value = 2 - ii * Φ(|Z==score==|)

We use the Z-score if the test statistic approximately follows the standard normal distribution N(0,1). Thanks to the central limit theorem, you can count on the approximation if you have a big sample (say at least 50 information points), and treat your distribution as normal.

A Z-exam most often refers to testing the population mean, or the departure between two population ways, in particular between ii proportions. You tin also find Z-tests in maximum likelihood estimations.

N(0,1) distribution density — Density of the standard normal distribution
StefanPohl / CC0 wikimedia.org

How to find p-value from t?

The p-value from the t-score is given by the post-obit formulae, in which cdf==t,d== stands for the cumulative distribution function of the t-Pupil distribution with d degrees of freedom:

Left-tailed t-exam:
p-value = cdf==t,d==(t==score==)
Correct-tailed t-examination:

p-value = ane - cdf==t,d==(t==score==)

Two-tailed t-test:
p-value = ii * cdf==t,d==(−|t==score==|)

or p-value = ii - two * cdf==t,d==(|t==score==|)

Employ the t-score option if your test statistic follows the t-Educatee distribution. This distribution has a shape similar to N(0,1) (bell-shaped and symmetric), but has heavier tails - the exact shape depends on the parameter chosen the degrees of freedom. If the number of degrees of freedom is large (>30), which generically happens for big samples, the t-Student distribution is practically indistinguishable from normal distribution N(0,1).

t-Student distribution densities — Density of the t-distribution with ν degrees of freedom
Skbkekas / CC BY wikimedia.org

The most common t-tests are those for population means with an unknown population standard deviation, or for the difference betwixt means of 2 populations, with either equal or diff however unknown population standard deviations. In that location'south also a t-test for paired (dependent) samples.

p-value from chi-foursquare score (χ2 score)

Use the χ²-score selection when performing a test in which the examination statistic follows the χ²-distribution.
This distribution arises, if, for case, you take the sum of squared variables, each following the normal distribution Due north(0,ane). Retrieve to bank check the number of degrees of liberty of the χ²-distribution of your exam statistic!

chi-square distribution densities — Density of the χ²-distribution with *chiliad* degrees of freedom
Geek3 / CC Past wikimedia.org

How to discover the p-value from chi-foursquare-score? You tin practice it with help of the following formulae, in which cdf_χ²,d denotes the cumulative distribution part of the χ²-distribution with d degrees of freedom:

Left-tailed χ²-test:
p-value = cdf_χ²,d(χ²_score)
Right-tailed χ²-test:
p-value = 1 - cdf_χ²,d(χ²_score)

Recollect that χ²-tests for goodness-of-fit and independence are right-tailed tests! (see beneath)
Two-tailed χ²-examination: p-value =

two * min{cdf_χ²,d(χ²_score), 1 - cdf_χ²,d(χ²_score)}

(Past min{a,b} we denote the smaller of the numbers a and b.)

The most popular tests which atomic number 82 to a χ²-score are the post-obit:

Testing whether the variance of unremarkably distributed information has some pre-determined value. In this case, the examination statistic has the χ²-distribution with n - 1 degrees of liberty, where n is the sample size. This can be a one-tailed or two-tailed test.
Goodness-of-fit test checks whether the empirical (sample) distribution agrees with some expected probability distribution. In this example, the test statistic follows the χ²-distribution with yard - ane degrees of freedom, where k is the number of classes into which the sample is divided. This is a correct-tailed exam.
Independence test is used to determine if in that location is a statistically significant relationship between two variables. In this case, its examination statistic is based on the contingency table and follows the χ²-distribution with (r - 1)(c - one) degrees of freedom, where r is the number of rows and c the number of columns in this contingency table. This as well is a right-tailed test.

p-value from F-score

Finally, the F-score choice should exist used when you perform a test in which the examination statistic follows the F-distribution, likewise known as the Fisher–Snedecor distribution. The exact shape of an F-distribution depends on ii degrees of freedom.

F-distribution densities — Density of the F-distribution with (d1,d2)-degrees of freedom
IkamusumeFan / CC Past-SA wikimedia.org

To see where those degrees of liberty come from, consider the independent random variables 10 and Y, which both follow the χ²-distributions with d_ane and d₂ degrees of freedom, respectively. In that case, the ratio (10/d_one)/(Y/d₂) follows the F-distribution, with (d₁, d_two)-degrees of freedom. For this reason, the two parameters d₁ and d₂ are likewise called the numerator and denominator degrees of freedom.

The p-value from F-score is given by the following formulae, where we let cdf_F,d₁,d₂ announce the cumulative distribution part of the F-distribution, with (d_one, d₂)-degrees of freedom:

Left-tailed F-test:
p-value = cdf_{F,d_i,d₂}(F_score)
Right-tailed F-test:
p-value = one - cdf_F,d₁,d₂(F_score)
Two-tailed F-test: p-value =

two * min{cdf_F,d₁,d₂(F_score), 1 - cdf_{F,d_one,d_ii}(F_score)}

(By min{a,b} nosotros announce the smaller of the numbers a and b.)

Below we listing the nearly important tests that produce F-scores. All of them are correct-tailed tests.

A examination for the equality of variances in ii unremarkably distributed populations. Its test statistic follows the F-distribution with (northward - 1, m - 1)-degrees of freedom, where n and m are the respective sample sizes.
ANOVA is used to test the equality of means in three or more than groups that come from normally distributed populations with equal variances. We go far at the F-distribution with (k - one, north - k)-degrees of freedom, where chiliad is the number of groups, and n is the total sample size (in all groups together).
A test for overall significance of regression assay. The test statistic has an F-distribution with (thou - 1, northward - m)-degrees of freedom, where northward is the sample size, and k is the number of variables (including the intercept).

With the presence of the linear human relationship having been established in your information sample with the in a higher place test, you tin summate the coefficient of decision, R², which indicates the force of this relationship.
A exam to compare two nested regression models. The test statistic follows the F-distribution with (k₂ - k₁, n - k₂)-degrees of liberty, where grand_i and g_ii are the number of variables in the smaller and bigger models, respectively, and n is the sample size.

You may notice that the F-exam of an overall significance is a particular class of the F-test for comparison 2 nested models: it tests whether our model does significantly better than the model with no predictors (i.eastward., the intercept-only model).

FAQ

Tin can p-value exist negative?

No, the p-value cannot be negative. This is because probabilities cannot be negative and the p-value is the probability of the test statistic satisfying certain weather condition.

What does a high p-value hateful?

A high p-value means that nether the null hypothesis there'south a high probability that for another sample the test statistic volition generate a value at to the lowest degree as farthermost every bit the ane observed in the sample you already have. A high p-value doesn't allow you to pass up the null hypothesis.

What does a low p-value hateful?

A low p-value means that under the null hypothesis there's footling probability that for another sample the test statistic will generate a value at least equally farthermost every bit the i as observed for the sample you already have. A depression p-value is evidence in favor of the alternative hypothesis - it allows you to reject the zip hypothesis.