P Value Calculator

How to Use the P Value Calculator

This p value calculator allows you to find the p value from z statistic, t statistic, F statistic, chi-square statistic, or Pearson’s correlation coefficient (r). Specifically, each of the calculators has its own interface to avoid ambiguity.

To use the calculator:

1. Use the navigation links at the top of the page to jump to the appropriate calculator.
2. Enter the test statistic and any additional required values, such as degrees of freedom or sample size.
3. Enter the significance level, α.
4. Select the test type: two-tailed, left-tailed, or right-tailed.
5. Click Calculate.

Each calculator will return the exact p-value and a clear step-by-step solution showing how the probability is obtained. Where a statistical table does not provide an exact p-value, the step-by-step solution shows you the Excel formula to use to get the correct p value for your test.

Besides providing you with the p-value, the calculator also helps make decision based on the entered significance level using the following rule:

• If p ≤ α, reject the null hypothesis.
• If p > α, fail to reject the null hypothesis.

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 calculator when your hypothesis test follows the standard normal distribution and you already know the calculated z statistic value.

To use the calculator:

1. Go to the P-value from z score section.
2. Enter the z statistic.
3. Enter the significance level, α.
4. Select the test type: two-tailed, left-tailed, or right-tailed.
5. Click Calculate.

The calculator will return the exact p-value, compare it with the significance level you entered, and provide a step-by-step explanation. It will also show how to find the same probability using a standard normal table and Excel.

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

• Left-tailed test, the formula is =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))

Note. You should replace z with the z-statistic you get from a z test. For example, if the computed test statistic is z = 2.10, the right-tailed p value Excel’s formula would be: =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. Thus, p-value = 1 − P(Z < 2.10) = 1-0.9821 = 0.0179

You can verify the result using the p-value from z score calculator and following these simple steps:

1. Go to the P-value from z score section.
2. Enter z = 2.10 and significance level, α = 0.05.
3. Select Right-tailed as the test type
4. Click Calculate.

You can also verify the result using the p-value calculator. To do this:

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

The calculator will instantly return: p-value = 0.017864. You can also use the Excel formula: =1-NORM.S.DIST(2.10,TRUE) to get similar results.

The calculator will also help you make the right decision by comparing the p-value with your significance level. In this case, you would make the decision and conclusion as follows:

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 average test scores.

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. Go to the P-value from t statistic section.
2. Enter the t statistic, degrees of freedom, df, and significance level, α.
3. Select the test type: two-tailed, left-tailed, or right-tailed.
4. Click Calculate.

The calculator will return the exact p-value, compare it with the significance level you entered, and provide a clear step-by-step solution. It will also show the Excel formula used to calculate the p-value from the t distribution.

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 the test statistic is t = 2.35 with df = 18, the Excel formula for a two-tailed p-value is: =T.DIST.2T(ABS(2.35),18)

Note that the degrees of freedom are important because they determine the shape of the t distribution. The same t statistic may 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, the formula becomes: =T.DIST.2T(ABS(2.35),18), which return the two-tailed p-value as 0.030380

You can also verify the result using the p value from t statistic calculator by following these steps:

1. Go to the P-value from t statistic section.
2. Enter t = 2.35, df = 18, and significance level, α = 0.05.
3. Select Two-tailed as the test type.
4. Click Calculate.

The calculator will instantly return the two-tailed p-value as: 0.030380 and show you exactly how to find this value using the correct Excel formula.

The decision and conclusion for this test would be:

Since the p-value (0.030380) is less than the significance level (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 hypothesis tests involving a population variance.

To use the calculator:

1. Go to the P-value from chi-square statistic section.
2. Enter the chi-square statistic, χ², degrees of freedom, df, and the significance level, α.
3. Select the test type: right-tailed, left-tailed, or two-tailed.
4. Click Calculate.

The calculator will return the exact p-value, compare it with the significance level you entered. You’ll also see a clear, step-by-step solution explaining how to find the p-value using Excel.

The Excel formula depends on the direction of the test and include:

• 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)))

Remember. You should replace chi-square with the calculated chi-square statistic and df with the degrees of freedom for your test.

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 a larger chi-square statistic indicates a greater difference between the observed and expected frequencies. Left-tailed and two-tailed chi-square probabilities are mainly used in specific tests, such as some tests involving a population variance.

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. We can use Excel to calculate the exact right-tailed p-value.

For a right-tailed chi-square test, the Excel formula is: =CHISQ.DIST.RT(chi-square,df)

Substituting χ² = 9.84 and df = 3, the Excel formula becomes: =CHISQ.DIST.RT(9.84,3). Therefore, the p-value for the test is: 0.019976

You can verify the result using the p value from chi-square statistic calculator by following these steps:

1. Go to the P-value from chi-square statistic section.
2. Enter χ² = 9.84, df = 3, and significance level, α = 0.05.
3. Select Right-tailed as the test type.
4. Click Calculate.

The calculator will instantly return: p-value = 0.019976. It will also show the Excel formula and explain how to make the statistical decision.

The decision and conclusion for this question will be:

Since the p-value (0.019976) is less than the significance level (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. Go to the P-value from F statistic section.
2. Enter the F statistic, numerator degrees of freedom, df₁, denominator degrees of freedom, df₂, and significance level, α.
3. Select the test type: right-tailed, left-tailed, or two-tailed.
4. Click Calculate.

The calculator will return the exact p-value, compare it with the significance level, and give you the correct decision about your hypothesis. It will also show you exactly how to find this p value using Excel formula.

The Excel formula for finding p value from F statistic 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. This is because larger F statistics provide stronger evidence that the variation explained by the model or differences among the group means are too large to be attributed to random sampling variation alone.

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. Thus, we should use Excel to calculate the exact right-tailed probability. The Excel formula for a right-tailed p-value from F is: =F.DIST.RT(F,df1,df2)

Substituting F = 4.62, df₁ = 2, and df₂ = 27, the formula becomes: =F.DIST.RT(4.62,2,27). Therefore, using excel, the p-value is 0.018809

You can verify the result using the p value from F statistic calculator by following these steps:

1. Go to the P-value from F statistic section.
2. Enter F = 4.62, df₁ = 2, df₂ = 27, and the significance level, α = 0.05.
3. Select Right-tailed as the test type.
4. Click Calculate.

The calculator will instantly return: p-value = 0.018809

It will also show the Excel formula and explain how to make the statistical decision.

The decision and conclusion for this test will be:

Since the p-value (0.018809) is less than the significance level (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. Go to the P-value from correlation coefficient r section.
2. Enter the Pearson correlation coefficient, r, sample size, n, and significance level, α.
3. Select the test type: two-tailed, left-tailed, or right-tailed.
4. Click Calculate.

The calculator will return the exact p-value, compare it with the significance level and give you the right decision. You’ll also get a clear, step-by-step solution explaining how you would get the same p-value using excel.

The first step is to convert the correlation coefficient into a t statistic using:

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

The degrees of freedom are calculated as: df = n-2

After calculating t, use the appropriate Excel formula for the direction of the test:

• 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)

Replace t with the converted t statistic and df with n − 2.

You can also calculate the two-tailed p-value directly from r and n using the Excel formula: =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 formula for a correlation test is: df = n-2, where n is the sample size. Therefore, df = 25-2 = 23

To find the two-tailed p-value for the test, we use the following Excel formula: =T.DIST.2T(ABS(t),df)

Substituting t = 2.484554 and df = 23 gives: =T.DIST.2T(ABS(2.484554),23)

The result is: p-value = 0.020686

You can verify the result using the p value from correlation coefficient calculator by following these steps:

1. Go to the P-value from correlation coefficient r section.
2. Enter r = 0.46, n = 25, and significance level, α = 0.05.
3. Select Two-tailed as the test type.
4. Click Calculate.

The calculator will instantly return: p-value = 0.020686

It will also show the conversion from r to t, the degrees of freedom, the Excel formula, and the statistical decision. The decision is:

Since the p-value (0.020686) is less than the significance level (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 establish that additional study time causes higher scores.

Frequently Asked Questions