P Value Calculator

How to Use the P Value Calculator

This calculator helps you find p-value from z, p-value from t, p-value from F, p-value from chi-square, or p-value from r. To use the calculator:

1. Choose the test statistic you want to use (Z, T, F, Chi-Square, or Pearson r).
2. Enter the required values. Depending on the statistic, you may also need to enter degrees of freedom or sample size.
3. Select the appropriate tail (two-tailed, left-tailed, or right-tailed).
4. Click Calculate.

The calculator returns the exact p-value and displays the corresponding probability area on the distribution graph. You can also expand “Solution” to see how the p-value was calculated and the equivalent Excel formula.

Tip. For Pearson correlation, the calculator first converts r to a t statistic using (df = n – 2), then uses the t distribution to find the p-value.

What Is a P-Value?

A p-value is a statistical measure that tells you how likely it is to get a test statistic as extreme as the one observed, assuming the null hypothesis is true. Smaller p-values tend to provide sufficient evidence against the null hypothesis (H₀), whereas larger p-values do not provide sufficient evidence to reject the null hypothesis.

How to Interpret a P-Value

The most common way to interpret a p-value is to compare it with a significance level. The decision rule is:

• Reject the null hypothesis (H0) if p-value ≤ α
• Fail to reject the null hypothesis (H0) if p-value > α

Let’s assume that the significance level, α = 0.05; the table below shows the right decision for some chosen p-values.

Note. A p-value does not prove that the null hypothesis is false. It only tells you whether there is sufficient evidence against the null hypothesis.

One-Tailed vs Two-Tailed P-Values

Before calculating a p-value, you need to know the test type. In hypothesis testing, the most common test types are:

• Left-tailed test
• Right-tailed test
• Two-tailed test

Still wondering how to know the right test? It all depends on the alternative hypothesis. Here’s a quick guide to help you determine the test type for your hypothesis problem:

• Use a left-tailed test when the alternative hypothesis says the parameter is less than a certain value. In this case, the p-value is the area to the right of the test statistic.
• Use a right-tailed test when the alternative hypothesis says the parameter is greater than a certain value. The p-value for such tests is the area to the right of the test statistic.
• Use a two-tailed test when the alternative hypothesis says the parameter is different from a certain value. For a two-tailed test, you can find the p-value by multiplying the smaller one-tailed p-value by 2.

Which P-Value Calculator Should I Use?

Choose the calculator that matches the test statistic you already have. Each calculator has its own inputs, p-value result, and step-by-step solution.

The table below provides a quick summary of each of the p-value calculators and when to use them.

Does your test statistic require you to use the degrees of freedom, and you don’t know how to compute them? Use the degrees of freedom calculator.

P Value from Z Calculator

The p value from z score calculator finds the probability associated with a given z test statistic. Use this option when your hypothesis test follows the standard normal distribution and you already know the calculated z statistic.

To use the calculator:

1. Select Z Score under Find P-Value From.
2. Enter the z statistic.
3. Select the test type: two-tailed, left-tailed, or right-tailed.
4. Click Calculate.

The calculator will return the exact p-value and shade the corresponding probability area on the standard normal distribution. You can also expand “Solution” to see the probability used and the equivalent Excel formula.

The Excel formula depends on the direction of the test. Use the following formula for each test:

• Left-tailed test: =NORM.S.DIST(z,TRUE)
• Right-tailed test: =1-NORM.S.DIST(z,TRUE)
• Two-tailed test: =2*(1-NORM.S.DIST(ABS(z),TRUE))

Replace z with the z statistic obtained from your hypothesis test.

For example, if the test statistic is z = 2.10, the Excel formula for a right-tailed p-value is:

=1-NORM.S.DIST(2.10,TRUE)

Example 1: P Value from Z

A researcher wants to determine whether a new teaching method improves the population mean examination score. The hypothesis test produces a z statistic of 2.10. Find the p-value and make a decision at the 0.05 significance level.

Solution

Since the researcher is testing for an improvement, this is a right-tailed test.

To find the right-tailed p-value using the z-score table, follow these steps:

1. Look up z = 2.10 in the z table. Most z tables give the left-tailed probability. Therefore:P(Z < 2.10) = 0.9821
2. Subtract the left-tailed probability from 1 to get the right-tailed p-value.p-value = 1 − 0.9821= 0.0179

You can also find the exact p-value using the calculator:

1. Select Z Score.
2. Enter 2.10 as the z statistic.
3. Select Right-tailed.
4. Click Calculate.

The calculator will instantly return: p-value = 0.017864

It will also shade the area to the right of z = 2.10 on the standard normal distribution.

You can get the same result in Excel using:

=1-NORM.S.DIST(2.10,TRUE)

To make the decision, compare the p-value with the significance level, α = 0.05.

Since 0.017864 < 0.05, the result is statistically significant at the 5% level. Therefore, the researcher would reject the null hypothesis and conclude that the new teaching method improves the population mean examination score.

P Value from T Calculator

Use the p value from t calculator when you have a t statistic and the corresponding degrees of freedom. This is common when conducting a one-sample t test, paired t test, two-sample t test, Welch’s t test, or a significance test for a regression coefficient.

To use the calculator:

1. Select T Statistic under Find P-Value From.
2. Enter the t statistic and degrees of freedom, df.
3. Select the test type: two-tailed, left-tailed, or right-tailed.
4. Click Calculate.

The calculator will return the exact p-value and shade the corresponding probability area on the t distribution. You can also expand “Solution” to see the probability used and the equivalent Excel formula.

The Excel formula depends on the direction of the test. Use the following formulas:

• Left-tailed test: =T.DIST(t,df,TRUE)
• Right-tailed test: =T.DIST.RT(t,df)
• Two-tailed test: =T.DIST.2T(ABS(t),df)

Replace t with the calculated t statistic and df with the degrees of freedom for your test.

For example, if t = 2.35 and df = 18, the Excel formula for a two-tailed p-value is:

=T.DIST.2T(ABS(2.35),18)

Note that degrees of freedom are important because they determine the shape of the t distribution. The same t statistic can produce different p-values when the degrees of freedom are different.

Example 2: P Value from T

A researcher conducts a t test to determine whether the population mean sleep duration of students differs from 8 hours. The test produces a t statistic of 2.35 with 18 degrees of freedom. Find the p-value and make a decision at the 0.05 significance level.

Solution

Since the researcher wants to determine whether the mean sleep duration differs from 8 hours in either direction, this is a two-tailed test.

From the question, we know that:

• T statistic, t = 2.35
• Degrees of freedom, df = 18
• Significance level, α = 0.05
• Test type: Two-tailed

Unlike most printed t tables, which usually provide critical values or a range containing the p-value, Excel can calculate the exact p-value directly.

For a two-tailed test, the Excel formula is:

=T.DIST.2T(ABS(t),df)

Substituting t = 2.35 and df = 18 gives:

=T.DIST.2T(ABS(2.35),18)

Therefore:

p-value = 0.030380

You can also verify the result using the calculator:

1. Select T Statistic.
2. Enter 2.35 as the t statistic.
3. Enter 18 as the degrees of freedom.
4. Select Two-tailed.
5. Click Calculate.

The calculator will instantly return: p-value = 0.030380

It will also shade both tails of the t distribution and show the Excel formula under “Solution”

To make the decision, compare the p-value with α = 0.05.

Since 0.030380 < 0.05, the result is statistically significant at the 5% level. Therefore, the researcher would reject the null hypothesis and conclude that the population mean sleep duration of students differs from 8 hours.

P Value from Chi-Square Calculator

Use the p value from chi-square calculator when your test statistic follows a chi-square distribution. This is common for chi-square goodness-of-fit tests, chi-square tests of independence, and some hypothesis tests involving a population variance.

To use the calculator:

1. Select Chi-Square under Find P-Value From.
2. Enter the chi-square statistic, χ², and degrees of freedom, df.
3. Select the test type: right-tailed, left-tailed, or two-tailed.
4. Click Calculate.

The calculator will return the exact p-value and shade the corresponding probability area on the chi-square distribution. You can also expand the “Solution” section to see the probability used and the Excel formula.

The Excel formula depends on the direction of the test:

• Left-tailed test: =CHISQ.DIST(chi-square,df,TRUE)
• Right-tailed test: =CHISQ.DIST.RT(chi-square,df)
• Two-tailed test: =MIN(1,2*MIN(CHISQ.DIST(chi-square,df,TRUE),CHISQ.DIST.RT(chi-square,df)))

Replace chi-square with the calculated chi-square statistic and df with the degrees of freedom.

For example, if χ² = 9.84 and df = 3, the Excel formula for the right-tailed p-value is:

=CHISQ.DIST.RT(9.84,3)

Most chi-square goodness-of-fit tests and tests of independence use a right-tailed p-value. This is because larger chi-square statistics indicate a greater difference between the observed and expected frequencies.

Example 3: P Value from Chi-Square

A researcher wants to determine whether gender and preferred learning method are associated. A chi-square test of independence produces the following results:

• Chi-square statistic, χ² = 9.84
• Degrees of freedom, df = 3

Find the p-value and make a decision at the 0.05 significance level.

Solution

Since this is a chi-square test of independence, we use a right-tailed test.

From the question, we know that:

• Chi-square statistic, χ² = 9.84
• Degrees of freedom, df = 3
• Significance level, α = 0.05
• Test type: Right-tailed

A printed chi-square table usually provides critical values or a range containing the p-value rather than the exact probability.

For a right-tailed chi-square test, the Excel formula is:

=CHISQ.DIST.RT(chi-square,df)

Substituting χ² = 9.84 and df = 3 gives:

=CHISQ.DIST.RT(9.84,3)

Therefore:

p-value = 0.019976

You can also calculate the p-value using the calculator:

1. Select Chi-Square.
2. Enter 9.84 as the chi-square statistic.
3. Enter 3 as the degrees of freedom.
4. Select Right-tailed.
5. Click Calculate.

The calculator will instantly return:

p-value = 0.019976

It will also shade the area to the right of χ² = 9.84.

To make the decision, compare the p-value with α = 0.05.

Since 0.019976 < 0.05, the result is statistically significant at the 5% level. Therefore, the researcher would reject the null hypothesis and conclude that there is sufficient evidence of an association between gender and preferred learning method.

P Value from F Statistic Calculator

Use the p value from F statistic calculator when your test statistic follows an F distribution. This is common in analysis of variance (ANOVA), overall regression model tests, and tests that compare variances.

To use the calculator:

1. Select F Statistic under Find P-Value From.
2. Enter the F statistic.
3. Enter the numerator degrees of freedom, df₁, and denominator degrees of freedom, df₂.
4. Select the test type: right-tailed, left-tailed, or two-tailed.
5. Click Calculate.

The calculator will return the exact p-value and shade the corresponding probability area on the F distribution. You can also expand the “Solution” section to see the probability used and the equivalent Excel formula.

The Excel formulas are:

• Left-tailed test: =F.DIST(F,df1,df2,TRUE)
• Right-tailed test: =F.DIST.RT(F,df1,df2)
• Two-tailed test: =MIN(1,2*MIN(F.DIST(F,df1,df2,TRUE),F.DIST.RT(F,df1,df2)))

Replace F with the calculated F statistic, df1 with the numerator degrees of freedom, and df2 with the denominator degrees of freedom.

For example, if F = 4.62, df₁ = 2, and df₂ = 27, the Excel formula for the right-tailed p-value is:

=F.DIST.RT(4.62,2,27)

Most ANOVA and overall regression F tests use a right-tailed p-value because larger F statistics provide stronger evidence against the null hypothesis.

Example 4: P Value from F Statistic

A researcher conducts a one-way ANOVA to compare the population mean examination scores of students taught using three different teaching methods. The ANOVA produces the following results:

• F statistic, F = 4.62
• Numerator degrees of freedom, df₁ = 2
• Denominator degrees of freedom, df₂ = 27

Find the p-value and make a decision at the 0.05 significance level.

Solution

Since this is a one-way ANOVA, we use a right-tailed F test.

From the question, we know that:

• F statistic, F = 4.62
• Numerator degrees of freedom, df₁ = 2
• Denominator degrees of freedom, df₂ = 27
• Significance level, α = 0.05
• Test type: Right-tailed

A printed F table usually provides critical values rather than the exact p-value.

For a right-tailed F test, the Excel formula is:

=F.DIST.RT(F,df1,df2)

Substituting F = 4.62, df₁ = 2, and df₂ = 27 gives:

=F.DIST.RT(4.62,2,27)

Therefore:

p-value = 0.018809

You can also calculate the p-value using the calculator:

1. Select F Statistic.
2. Enter 4.62 as the F statistic.
3. Enter 2 as df₁.
4. Enter 27 as df₂.
5. Select Right-tailed.
6. Click Calculate.

The calculator will instantly return:

p-value = 0.018809

It will also shade the area to the right of F = 4.62.

To make the decision, compare the p-value with α = 0.05.

Since 0.018809 < 0.05, the ANOVA result is statistically significant at the 5% level. Therefore, the researcher would reject the null hypothesis and conclude that the three population means are not all equal. In other words, there is sufficient evidence that at least one teaching method has a different population mean examination score.

P Value from Correlation Coefficient (r) Calculator

Use the p value from r calculator when you want to test whether a Pearson correlation coefficient is statistically significant. The calculator converts the correlation coefficient, r, into a t statistic and then uses the t distribution to find the p-value.

To use the calculator:

1. Select Pearson r under Find P-Value From.
2. Enter the Pearson correlation coefficient, r.
3. Enter the sample size, n.
4. Select the test type: two-tailed, left-tailed, or right-tailed.
5. Click Calculate.

The calculator will return the exact p-value and shade the corresponding probability area on the t distribution. You can also expand the “Solution” to see the conversion from r to t, the degrees of freedom, and the Excel formula.

The first step is to convert r into a t statistic using:

t = r*SQRT((n-2)/(1-r^2))

The degrees of freedom are:

df = n-2

After calculating t, use the appropriate Excel formula:

• Left-tailed test: =T.DIST(t,df,TRUE)
• Right-tailed test: =1-T.DIST(t,df,TRUE)
• Two-tailed test: =T.DIST.2T(ABS(t),df)

You can also calculate a two-tailed p-value directly from r and n using:

=T.DIST.2T(ABS(r*SQRT((n-2)/(1-r^2))),n-2)

Use a two-tailed test when you want to determine whether the population correlation differs from zero in either direction. Use a right-tailed test when the alternative hypothesis specifies a positive correlation and a left-tailed test when it specifies a negative correlation.

Example 5: P Value from r

A researcher wants to determine whether study time is related to examination score. A sample of 25 students produces a Pearson correlation coefficient of 0.46. No direction for the relationship was specified before the data were analyzed. Find the p-value and make a decision at the 0.05 significance level.

Solution

Since the researcher wants to determine whether study time and examination score are related without specifying a positive or negative direction, this is a two-tailed test.

From the question, we know that:

• Correlation coefficient, r = 0.46
• Sample size, n = 25
• Significance level, α = 0.05
• Test type: Two-tailed

To calculate the p-value, we first convert the correlation coefficient into a t statistic.

The formula is:

t = r*SQRT((n-2)/(1-r^2))

Substituting r = 0.46 and n = 25 gives:

t = 0.46*SQRT((25-2)/(1-0.46^2))

= 0.46*SQRT(23/0.7884)

= 2.484554

The degrees of freedom are:

df = n-2

= 25-2

= 23

To find the two-tailed p-value, use:

=T.DIST.2T(ABS(t),df)

Substituting t = 2.484554 and df = 23 gives:

=T.DIST.2T(ABS(2.484554),23)

Therefore:

p-value = 0.020686

You can also calculate the p-value using the calculator:

1. Select Pearson r.
2. Enter 0.46 as the correlation coefficient.
3. Enter 25 as the sample size.
4. Select Two-tailed.
5. Click Calculate.

The calculator will instantly return: p-value = 0.020686

It will also show the corresponding t distribution with both tails shaded. Under “Solution”, you can see the conversion from r to t and the Excel formula used.

To make the decision, compare the p-value with α = 0.05.

Since 0.020686 < 0.05, the correlation is statistically significant at the 5% level. Therefore, the researcher would reject the null hypothesis and conclude that there is sufficient evidence of a relationship between study time and examination score.

Because r = 0.46 is positive, the sample results indicate that students who spend more time studying tend to obtain higher examination scores. However, a statistically significant correlation does not by itself show that additional study time causes higher examination scores.

Frequently Asked Questions