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P-Value Calculator

Use this P-value calculator to summarize a data set for P-value. The page returns p-value plus supporting values for z statistic, standard error, and mean difference; check the assumptions before reusing the result.

Primary answer
p-value
Inputs to verify
Sample mean, Hypothesized mean, and Population standard deviation
Use type
Use as an estimate that depends on assumptions.
Keyword intent
P-value calculator

Calculator

P-Value Calculator

Calculates p-value from sample mean, hypothesized mean, population standard deviation. Defaults are filled in so you can review a working example before changing inputs.

Enter the observed sample mean.

Enter the null-hypothesis mean.

Enter the known population standard deviation, not the sample standard deviation.

Enter the number of independent observations.

Choose the alternative before interpreting the p-value.

Result

Result reflects the current submitted inputs.

  • Risk B
  • Reviewed 2026-05-26
  • 4 sources
p-value0.0455
z statistic2
Standard error1
Mean difference2

Breakdown

Alternative
two-sided
Known population standard deviation
10
Sample size
100
  • This MVP calculates p-values for a one-sample mean z-test only.
  • The population standard deviation is known and is not estimated from the sample.
  • The selected alternative hypothesis controls which normal tail area is used.
  • The test statistic follows the standard normal distribution under the null hypothesis.
  • t tests, proportion tests, two-sample tests, paired tests, and exact tests are out of scope.

Accuracy notes

Risk level
B
Reviewed
2026-05-26
Sources
4
Primary result
p-value

Formula logic is kept in a pure calculator module with fixtures, source notes, and page-visible assumptions.

Categories

What the result means

p-value answers the page's main P-value question. Tail probability under the selected one-sample z-test alternative. Read the center or spread metric first, then compare count, minimum, maximum, and sample/population notes. Use z statistic, standard error, and mean difference to explain why p-value moved when an input changed. Review the raw values and decide whether outliers or missing data should be handled before reporting the result.

p-valueTail probability under the selected one-sample z-test alternative.
z statisticStandard normal test statistic.
Standard errorKnown population standard deviation divided by the square root of sample size.
Mean differenceSample mean minus hypothesized mean.

Use the result this way

  1. Start with p-value, then use supporting outputs only to explain the primary answer.
  2. Verify sample mean, hypothesized mean, and population standard deviation before copying the result.
  3. Choose the mode or method first because it can change which formula is applied and stay within the documented minimum and maximum ranges.
  4. Review the raw values and decide whether outliers or missing data should be handled before reporting the result.

User job

How to use this calculator

Use P-Value Calculator when you need p-value, then use z statistic and standard error to check the context for data review, classwork, quality checks, and quick descriptive summaries.

Best for

  • Summarizing a list of values
  • Checking spread, center, or sample assumptions
  • Reviewing a default example before entering your own sample mean and hypothesized mean.

Check before relying

  • Confirm whether the data is a sample or population and whether outliers should stay in the list.
  • This MVP calculates p-values for a one-sample mean z-test only.
  • The population standard deviation is known and is not estimated from the sample.
  • Source context: Penn State Eberly College of Science, reviewed 2026-05-26.

Next useful step

Formula

For one mean with known population standard deviation, z = (sampleMean - hypothesizedMean) / (sigma / sqrt(n)); p-value is calculated from the standard normal CDF and selected alternative. Key assumptions: This MVP calculates p-values for a one-sample mean z-test only. The population standard deviation is known and is not estimated from the sample. The selected alternative hypothesis controls which normal tail area is used.

  • For one mean with known population standard deviation, z = (sampleMean - hypothesizedMean) / (sigma / sqrt(n)); p-value is calculated from the standard normal CDF and selected alternative.
  • This MVP calculates p-values for a one-sample mean z-test only.
  • The population standard deviation is known and is not estimated from the sample.
  • Primary source context: Penn State Eberly College of Science.

Inputs

Enter sample mean, hypothesized mean, population standard deviation, and sample size for data review, summaries, quality checks, and classwork. Before calculating, choose the mode or method first because it can change which formula is applied and stay within the documented minimum and maximum ranges. Sample mean: Enter the observed sample mean. Hypothesized mean: Enter the null-hypothesis mean. Population standard deviation: Enter the known population standard deviation, not the sample standard deviation. Sample size: Enter the number of independent observations.

Sample meanEnter the observed sample mean.
Hypothesized meanEnter the null-hypothesis mean.
Population standard deviationEnter the known population standard deviation, not the sample standard deviation.
Sample sizeEnter the number of independent observations.
Alternative hypothesisChoose the alternative before interpreting the p-value.

Example

Using the default inputs, P-Value Calculator returns p-value of 0.0455. Adjust sample mean, hypothesized mean, population standard deviation, and sample size to match your own scenario.

FAQ

How is p-value calculated here?

For one mean with known population standard deviation, z = (sampleMean - hypothesizedMean) / (sigma / sqrt(n)); p-value is calculated from the standard normal CDF and selected alternative. The first assumption to check is: This MVP calculates p-values for a one-sample mean z-test only.

What does p-value mean for P-value?

Read the center or spread metric first, then compare count, minimum, maximum, and sample/population notes. Secondary values such as z statistic, standard error, and mean difference are there to explain the primary answer, not to replace it.

What should I enter for Sample mean?

Enter the observed sample mean. Choose the mode or method first because it can change which formula is applied and stay within the documented minimum and maximum ranges.

How does Hypothesized mean change p-value?

Enter the null-hypothesis mean. Changing it can alter p-value because the formula uses the submitted inputs together. Also compare sample versus population mode, separators, missing values, outliers, and rounding precision.

Why does the P-value example show 0.0455 for p-value?

The default inputs produce 0.0455 for p-value. Treat that as a format and scale check, then replace every default value with your own inputs.

What should I check before reporting p-value?

Confirm how the values were parsed, whether the data is a sample or population, and whether outliers or missing values should stay in the set.

Sources

Last reviewed: 2026-05-26

  • Reviewed 2026-05-26 · Source n.d.
    STAT 200: One Sample Mean z TestPenn State Eberly College of Science. z test statistic formula, known-sigma assumption, and representative one-sample z-test example.
    Scope
    Introductory statistics lesson for one-sample mean z tests with known population standard deviation.
    Supports
    z test statistic formula, known-sigma assumption, and representative one-sample z-test example.
  • Reviewed 2026-05-26 · Source n.d.
    NIST/SEMATECH e-Handbook: Critical values and p valuesNational Institute of Standards and Technology. Definition of p-value as the probability of a test statistic at least as extreme as the observed value under the null hypothesis.
    Scope
    Engineering Statistics Handbook guidance on p-values and hypothesis-test critical values.
    Supports
    Definition of p-value as the probability of a test statistic at least as extreme as the observed value under the null hypothesis.
  • Reviewed 2026-05-26 · Source n.d.
    NIST/SEMATECH e-Handbook: Normal DistributionNational Institute of Standards and Technology. Standard normal CDF used to convert z statistics to p-values.
    Scope
    Normal distribution PDF, standard normal distribution, and numerical CDF guidance.
    Supports
    Standard normal CDF used to convert z statistics to p-values.
  • Reviewed 2026-05-26 · Source 1964
    Handbook of Mathematical FunctionsU.S. National Bureau of Standards. Error-function approximation used by the local normal CDF implementation.
    Scope
    Numerical approximation for the error function.
    Supports
    Error-function approximation used by the local normal CDF implementation.

Disclaimer

This calculator is an educational estimate based on the inputs and assumptions shown on the page.