Calculate T, Z, χ2, F and r (2024)

Welcome to our critical value calculator. This tool is used to find the left, right and two-tailed critical value for z, t-student, chi-square, r and F-distribution. Whether you're a student, researcher, or professional, this tool is designed to meet your statistical needs.

The critical value is the most important component in hypothesis testing as well as in the confidence interval. If the hypothesis test is one-tailed then it gives one critical value and if the hypothesis test is two-tailed then it gives two critical values i.e. one is positive and the other is negative.

Table of Contents

  • How to Use the Calculator
  • Related Calculators
  • What are Critical Values?
  • Different Types of Critical Values
  • How to find value from table ?
  • Where does Critical Values Are Used ?
  • Practical Examples:

How to Use the Critical Values Calculator

The critical value calculator is an easy-to-use tool for determining critical value whether it is a one-tailed test or two-tailed test.

  1. Test type : in this fields enter the type of test from z-score, t-score, chi-square \( \chi^2 \), F-score or correlation coefficient r.
  2. Tail type : In this fields enter the type of test like one tailed ( left tailed or right tailed ) or two tailed according to hypothesis testing.
    • For T Critical Value: Enter the degrees of freedom and the significance level \(\alpha\).
    • For Z Critical Value: Enter the significance level \(\alpha\).
    • For Chi-Square Critical Value: Enter the degrees of freedom and the significance level \(\alpha\).
    • For F Critical Value: Enter the degrees of freedom for both the numerator and the denominator, and the significance level \(\alpha\).
    • For r Critical Value: Enter sample size, and the significance level \(\alpha\).
  3. Calculate : Click on calculate button to get the desired critical value.

Related Calculators :

Below are more calculators which use the critical value to perform statistical analysis.


What are Critical Values?

The critical value also known as critical point is used to decide whether to reject or fail to reject the null hypothesis while hypothesis testing. It is also used to find the lower and upper limits of the confidence interval. The critical value has left-tailed, right-tailed, and two-tailed critical values.

  • Left-Tailed Test : As this is a left-tailed test, if the test statistic falls to the left of the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
  • Right-Tailed Test : As this is a right-tailed test, if the test statistic falls to the right of the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
  • Two-Tailed Test : As this is a two-tailed test, if the test statistic falls in the left or right tail beyond the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
  • Calculate T, Z, χ2, F and r (1)

    The above figure shows left, right and two tailed critical value along with there rejection region for z distribution.

Different types of critical values

The different types of critical values are explained below.

t critical value :

In t distribution, the significance level \( ( \alpha ) \) and degrees of freedom are required to find the critical values. This distribution is used when the population standard deviation is unknown.

Z Critical Value :

In z distribution, the only significance level is required to find the critical values. This distribution is used when the population standard deviation is known.

Chi-Square Critical Value :

In Chi-Square tests, the degrees of freedom and the significance level \(\alpha \) are used to find the critical values

F Critical Value :

In f test the significance level\( ( \alpha ) \)and the degrees of freedom for the denominator and numerator are used to find the critical value. Note in the f test there are two degrees of freedom i.e \( df_1 \) and \( df_2 \)

Correlation Coefficient (r) critical Value

The sample size and significance level \( \alpha \) are used to find the critical value. The critical values for the correlation coefficient are used to determine the significance of the correlation between two variables.

How to find value from table ?

There are numbers in critical value tables, but the procedure is the same to find the critical value. Let's take one example of how to find the critical value on the z table.

Step 1 : Determine the Significance Level \( (\alpha) \)

  • If the hypothesis test is a two tailed test, divide \( \alpha \) by 2.
  • If the hypothesis test is a one tailed test, use \( \alpha \) directly.

Step 2 : Calculate the Critical Probability:

  • For a left-tail test: Use the significance level.
  • For a right-tail test: Subtract the significance level from 1.
  • For two-tailed tests: Use both the above probabilities.

Step 3 : Locate the Critical Value:

  • Use a Z table to find the Z-score corresponding to the calculated probability.
  • The Z table provides the area to the left of a Z-score in a standard normal distribution.

Critical value table :

Left tailed critical value table for z test

Calculate T, Z, χ2, F and r (2)

Right tailed critical value table for z test

Calculate T, Z, χ2, F and r (3)

Two tailed critical value table for z test

Calculate T, Z, χ2, F and r (4)

Where does Critical Values Are Used ?

The critical values are used in various fields like statistical hypothesis testing and research. Here’s a summary of where they are applied:

Decision Making:

Based on the calculated test statistic ( T, Z, r, Chi-Square), critical values determine whether to reject or fail to reject the null hypothesis.

Significance Levels :

Critical values, which are usually set at \(0.05\) or \(0.01\), aid in assessing the degree of confidence in the statistical findings.

Two-Tailed vs. One-Tailed Tests:

Depending on whether the test is two-tailed (non-directional hypothesis) or one-tailed (directional hypothesis), different critical values apply.

Practical Examples:

The critical values are used in medical research for testing the effectiveness of a new drug compared to a placebo.

It can be used in market research foranalyzing survey data to determine if there is a preference between two products.

It can also be used in quality control to assess if a manufacturing process meets specified standards.

Why Use Our Critical Values Calculator ?

  • Accuracy: Our calculator provides precise critical values for your statistical tests.
  • User-Friendly: The interface is intuitive, making it easy to input data and obtain results quickly.
  • Versatile : Calculate T, Z, Chi-Square,r, and F critical values in one place.
  • Free and Accessible: No cost and accessible online anytime, anywhere.
Calculate T, Z, χ2, F and r (2024)

FAQs

How do you calculate Z for Z test? ›

The formula for the z test statistic is given as follows: z = ¯¯¯x−μσ√n x ¯ − μ σ n . ¯¯¯x x ¯ is the sample mean, μ μ is the population mean, σ σ is the population standard deviation and n is the sample size.

How to calculate one sample t test? ›

The four steps are listed below:
  1. Calculate the sample mean. ¯y = y1 + y2 + ⋯ + ynn.
  2. Calculate the sample standard deviation. ˆσ = √(y1 − ¯y)2 + (y2 − ¯y)2 + ⋯ + (yn − ¯y)2n − 1.
  3. Calculate the test statistic. t = ¯y − m0ˆσ/√n.
  4. Calculate the probability of observing the test statistic under the null hypothesis.

What is the full equation to calculate the t statistic? ›

t test formula (1 sample) t = M – µ Sx Sample mean (M) minus population mean you are comparing your sample to (µ), divided by the standard error (Sx).

How do you calculate %Z? ›

The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.

How do you calculate z-score easily? ›

There are three variables to consider when calculating a z-score: the raw score (x), the population mean (μ), and the population standard deviation (σ). To get the z-score, subtract the population mean from the raw score and divide the result by the population standard deviation.

What is the formula for the F test? ›

The F-test is a type of hypothesis testing that uses the F-statistic to analyze data variance in two samples or populations. The F-statistic, or F-value, is calculated as follows: F = σ 1 σ 2 , or Variance 1/Variance 2. Hypothesis testing of variance relies directly upon the F-distribution data for its comparisons.

How do you calculate the t-test? ›

The t-score formula for an independent t-test is: t equals the mean of population 1 minus the mean of population 2 divided by the product of the pooled standard deviation and the square root of one over the sample size of sample 1 plus one over the sample size of sample 2.

What is the difference between the z-test and the t-test? ›

Purpose: The T-test is employed to compare means of small samples (usually when the sample size is less than 30), while the Z-test is used to compare means of large samples (typically when the sample size is equal to or greater than 30).

How do you calculate t from z? ›

To eliminate thesecharacteristics, z scores often are converted to T scores. This isaccomplished using the simple formula: T score = 10(z score) + 50. Forexample, a z score of -2.5 becomes a T score of 25.

When should you use the z test? ›

z -tests are a statistical way of testing a hypothesis, when we know the population variance σ2 . We use them when we wish to compare the sample mean μ to the population mean μ0 . However, if your sample size is large, n≥30 n ≥ 30 , then you can still use z -tests without knowing the population variance.

When to use t score vs z-score? ›

Z score is the standardization from the population raw data or more than 30 sample data to a standard score, while the T score is the standardization from the sample data of less than 30 data to a standard score.

How do you calculate the value of Z? ›

If you know the mean and standard deviation, you can find the z-score using the formula z = (x - μ) / σ where x is your data point, μ is the mean, and σ is the standard deviation.

How do we calculate the z-score of a one proportion z test? ›

Statistics - One Proportion Z Test

The test statistic is a z-score (z) defined by the following equation. z=(p−P)σ where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and σ is the standard deviation of the sampling distribution.

How do you find the Z value of a sample? ›

Calculating Z Scores

Use the following format to find a z-score: z = X - μ / σ. This formula allows you to calculate a z-score for any data point in your sample. Remember, a z-score is a measure of how many standard deviations a data point is away from the mean.

What is the formula for the z-score t test? ›

As evidenced above, zscores are often negative and may contain decimal places. To eliminate thesecharacteristics, z scores often are converted to T scores. This isaccomplished using the simple formula: T score = 10(z score) + 50.

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