How To Calculate Critical Value in Statistics

You can use the T.INV() function to find the critical value of t for one-tailed tests in Excel, and you can use the T.INV.2T() function for two-tailed tests. Example: Calculating the critical value of t in ExcelTo calculate the critical value of t for a two-tailed test with df = 29 and α = .05, click any blank cell and type:

In statistics, critical value is the measurement statisticians use to calculate the margin of error within a set of data and is expressed as: Critical probability (p*) = 1 – (Alpha / 2), where Alpha is equal to 1 – (the confidence level / 100).

Find Critical Value in Standard Normal Z Distribution

What’s the importance of critical value?

The critical value is extremely important in terms of evaluating validity, accuracy and the range at which errors or discrepancies within the sample set can occur. This value is an essential factor in calculating the margin of error. Likewise, the critical value can give you insight into the characteristics of the sample size youre evaluating.

For instance, expressing the critical value as a t statistic is important for accurately measuring small sample sizes or data sets where the standard deviation is unknown. Expressing the critical value as the cumulative probability, or the Z-score, allows for a more accurate evaluation of a larger data set, typically with 40 or more samples in the set. The critical value becomes extremely important for assessing validity and accuracy, along with discrepancies within different sizes of populations you study.

What is critical value?

In statistics, critical value is the measurement statisticians use to calculate the margin of error within a set of data and is expressed as:

Critical probability (p*) = 1 – (Alpha / 2), where Alpha is equal to 1 – (the confidence level / 100).

You can express the critical value in two ways: as a Z-score related to cumulative probability and as a critical t statistic, which is equal to the critical probability. Additionally, the critical value describes several characteristics about the margin of error that statisticians can then use to determine the validity of the data they study.

For instance, suppose a statistician is analyzing a population study about the effects of sunlight on mood disorders. Within a sample size of the population there will be a margin of error that describes the rate at which any discrepancies will occur within the data set, such as any outliers.

How to calculate critical value

Calculating the critical value of a data set is fairly straightforward. You can also express the critical value in one of two ways, depending on your sample size. The following steps provide a guide for how to do this:

1. Compute the alpha value

Find the alpha value before calculating the critical probability using the formula alpha value (α) = 1 – (the confidence level / 100). The confidence level represents the probability of a statistical parameter also being true for the population you measure. This value is typically represented with a percent value. For instance, a confidence level of 95% within a sample set indicates that the specific criteria has a 95% probability of being true for the entire population. Using a confidence level of 95%, you would complete the formula to find the alpha value:

Alpha value = 1 – (95/100) = 1 – (0.95) = 0.05. In this case, the alpha value is 0.05.

2. Calculate the critical probability

Using the alpha value from the first formula, calculate the critical probability. This will be the critical value, which you can then express as a t statistic or a Z-score. Using the previous example alpha value of 0.05, complete the formula to find the critical probability:

Critical probability (p*) = 1 – (0.05 / 2) = 1 – (0.025) = 0.975. The critical probability in this example is then 0.975, or 97.5%.

3. Use the critical t statistic for small sample sets

If you are measuring a small sample size, the critical t statistic is the appropriate expression for the critical probability. Express the critical probability of 97.5% as the t statistic like this:

Degree of freedom (df) = the sample size – 1. This means that the number of samples you have in your study subtracted by one will equal the degree of freedom. So if you have a sample size of 25, subtract one from this value to get the degree of freedom. In this case, it would be 24.

4. Express critical value as a Z-score for large data sets

For population sizes larger than 40 samples in a set, you can express the critical value as a Z-score. The Z-score should have a cumulative probability that is equal to the critical probability. The cumulative probability refers to the probability that a random variable will be less than or equal to a specific value. This probability must be equal to the critical probability, or the critical value.

Types of critical value systems

You can use different types of critical value testing systems to evaluate the statistical significance of a given population or sample youre studying. The statistical significance will tell you if the results you obtain from your tests are valid. Here are the types of critical value systems that statisticians use when calculating significance:


Chi-squares come from two types of chi-square tests: the goodness of fit and the independence chi-square tests. The goodness of fit chi-square test helps determine if a small set of sample data matches the whole population. In the independence chi-square test, youll compare two variables in order to determine the relationship between them.


T-scores result from standardized tests. For instance, the SATs are one example of a standardized test that can result in t-scores. The t-score in statistics allows you to convert an individual test score into a standardized form, which you can then use to compare other test scores to.


Z-scores are the standard scores you derive from a data set. The Z-score will tell you how far a given data point is from the mean of your sample. This type of critical value will tell you the measure of how many standard deviations above or below the raw score your population mean is.


The following example shows how you can calculate the critical value (critical probability) of a sample set using the p-value (or the critical probability) approach:

Suppose you want to compare the likelihood of the test statistic being more or less than the significance level, or the alpha value, of your sample set. You can calculate the critical value by using the p-value, or the critical probability. This means the p-value will correspond to the probability of gaining sample data thats as extreme as the initial test statistic.

If the p-value of your hypothesis test equals 0.01, for instance, you can then reject a null hypothesis with any significance level greater than or equal to 0.01. If your significance level is less than or equal to 0.01, you would not reject the null hypothesis. The p-value of 0.01 in this case will equal the critical value. Additionally, this value will be beneficial for evaluating the strength and validity of the evidence against a null hypothesis without specific reference to your significance level.


What is a critical value in statistics?

Determine the critical value for a 95% level of confidence (p<0.05). The critical value for a 95% two-tailed test is ± 1.96.

What is critical formula?

Critical values are essentially cut-off values that define regions where the test statistic is unlikely to lie; for example, a region where the critical value is exceeded with probability alpha if the null hypothesis is true.

How do you find the critical value in a hypothesis test?

Critical Angle Formula = the inverse function of the sine (refraction index / incident index).

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