Critical Value Calculator

Critical Value Calculator

Critical value calculator is a useful tool to measure chi-square value, t value, z value, and many more. This calculator can be used to calculate the above mentioned critical values as well as f value and r value. Those days are gone when you needed to look in the tables and scan hundreds of values to find a corresponding value for your data. You can use eCalculator critical t value calculator to find any type of critical value. With a simple and compelling interface, our t critical value calculator makes it easy to perform calculations for calculating critical values for any kind of testing.

How to use a critical value calculator?

To calculate a critical value using this critical values calculator, identify the type of critical value you want to find. There is a separate tab for t value, z value, chi-square value, f value, and r value each.

Click on one of the tabs for which you need to calculate the critical value. Each critical value requires different variables for the calculation. To find t value, chi-square value, or r value, it will ask for the degrees of freedom and significance level, and for z value, only significance level is required. Whereas, to calculate f value, it will ask for the significance level and two degrees of freedom. After entering the relevant values in the given input boxes, click the “Calculate” button to get critical value. This rejection region calculator will show you the result right away after clicking the button.

This Tool, which is also known as t table calculator, produces a detailed result against each calculation. It will give you the critical value, upper probability, and right-tailed t-value. It will also provide the one-tailed and two-tailed probabilities for ±z.

What is a critical value?

A critical value specifies the margin of error in a distribution graph when analyzing statistics. If a research statistic's absolute value is greater than the critical value, then a statistical sense denies an accepted hypothesis.

A distribution graph is divided by critical value into sections that represent "rejection regions." Essentially, if a value of a study slips into a rejected region, a recognized hypothesis, which is also known as the null hypothesis, must be denied. The null hypothesis cannot be overruled if the test value falls within the allowed range.

The testing of sample data includes surveys and research such as SAT scores, voting habits, population figures, the percentage of body fat, and blood pressure.

Distribution curve and Hypothesis testing

Hypothesis tests check whether your data was collected from a population that adheres to a hypothesized distribution of probabilities. It is categorized by an alternative hypothesis and null hypothesis. Critical value importance for hypotheses study is a point in the distribution graph that is evaluated with a check to decide if a null hypothesis, which studies seek to refute, should be dismissed.

In a set of observations, the importance of a null hypothesis implies that there is no statistical significance, and it is believed to be so until statistical proof invalidates it from a hypothesis that is considered to be an alternate.

A normal distribution curve, the bell-like curve, reflects logically how frequently an experiment delivers a specific result. Normal Distribution has a mean (average of a set of values), median (middle value in a data set), or mode (most frequent value in a data set). 50% of the values of the normal distribution are less than the mean, and 50% of the values of the normal distribution are greater than the mean.

Most of the data points are comparatively similar in a normal distribution. The symmetry of a normal distribution characterizes it perfectly, which makes half of the data observations found on both sides of the chart from the center. This means that they appear with smaller labels on the high and low points of the graph within a range of values.

Critical values are not part of the range of common data points due to these effects. That is why a null hypothesis is forfeited when research figures surpass the critical value.

A two-tailed test or a one-tailed which are also called right-tailed or left-tailed tests can be applied for critical values. Statisticians decide what test to implement first based on the data.

How to find critical value?

The critical value formula is the standard equation for the probability:

p = 1 – α/2

Where alpha (α) represents the significance or confidence level, and p is the probability.

In this equation, the alternative hypothesis is considered as alpha. If this hypothesis is true, the null hypothesis is not accepted. Suppose, if a researcher wants to achieve a significance level of 0.06, it implies that there is a 6% probability that a discrepancy occurs.

The critical value can be calculated in z scores or t score if the distribution of a sampling range is usual or near to normal. You can use our f critical value calculator to find a critical value.

What should you calculate: T score or Z score?

It doesn’t matter if you calculate z score or t score if the size of the sample is more than 40. Although both approaches quantify similar results, most statistical beginning textbooks use the z score.

If the size of the sample is minimal, and there is no established normal population standard deviation, the t score will be used. The t score is a measure of probability whereby statistics will evaluate specific data sets with a normal distribution. Note that small samples of non-normal populations should not use the t score.

Standard deviation tests the distribution of numbers into a set of values and indicates the amount of variation. Low standard deviation implies that numbers are within the mean distribution, and a high standard deviation indicates that numbers are spread in a broader range.

How to calculate the z score?

As shown in the following calculations, the critical value of the z score can be used to calculate the margin of error:

Margin of error = Standard deviation of the statistic × Critical value

Margin of error = Standard error of the statistic × Critical value

The z-score (also known as the standard normal probability score) shows the number of standard deviations from a statistical item. In hypothesis testing, a z score table is used to check the difference and proportions of two mean values. Z table determines the percentage of data at any stage of the graph.

The z value formula is:

z = (X – μ) / σ

Where μ is the mean of population data, X is the element value, σ is the standard deviation.

Let’s calculate the z score by using an example. Let's assume, for starters, that you have 80 marks in a test. We will calculate the z score if the mean value of the test score is 40, and the standard deviation is 25. Substitute the values in the above equation to get the z score.

X = 80, μ = 40, σ = 25

z = (80 – 40) / 25
= 40 / 25

z = 1.6

For a test score of 80, with a mean value of 40 and a standard deviation value of 25, the z score will be 1.6.

In this case, your score is 1.6, which deviates above the average value. You can use our z critical value calculator to find the z value.

Interpreting the z score

If the value is greater than 0, the number is higher than the mean. The sample is lower than the mean if the score is lower than 0. The standard deviation will be greater than mean by one if the score is equal to 1. The standard deviation will be less than mean by one if the score is equal to -1. For elements in a large set, about 68% fall from-1 to 1, around 95% fall from-2 to 2, about 99 percent decrease from-3 to 3.

How to find t critical value?

To find the critical value of t use this standard formula for t score:

t = [ x – μ ] / [ s / √ ( n ) ]

Where x is the mean of the sample, μ is the population mean, n is the size of the sample, and s is the standard deviation of the sample.

We also need to calculate degrees of freedom (df), by subtracting one from the size of the sample. Let’s say the size of the sample is 8. df = n – 1 = 8 – 1 = 7

T distribution is a constant probability distribution member occurring in cases where the sample standard deviation is uncertain, and the mean of a widely distributed population is minimal. It implies that a mean of a sample can be identified based on a random sample with n as the size of the sample if mean is equal to or less than x. Cumulative probability relates to the possibility of a suspected statistic dropping within a certain range. Statisticians use tα to represent cumulative probability 1 – α along with t statistics.

The t value is partially calculated by using the t distribution table to evaluate the degrees of freedom (df). Let's assume you have a sample size of five for demonstration purposes and want to run a right-tailed test.

To calculate the degrees of freedom, subtract one from the size of the sample. Here, the sample size is 5, which means degrees of freedom will be 4. We will assume that the alpha level is 0.025 for this example. In the t distribution table given below, search for the degrees of freedom and its parallel alpha level. The critical value will be found at the intersection of a row and column.


α = 0.1





ta= 1.2816 



































For the alpha level 0.025, sample size 5, and degrees of freedom 4, the critical value of t will be 2.776.

Let’s calculate the t score by using the t score formula.

An electronics manufacturer produces flash drives. The manager of the company says that the flash drives from their factory last for 300 days. Quality specialists arbitrarily select 20 flash drives for testing, which lasted for an average of 380 days, with a standard deviation of 90 days. If the flash drives last for 380 days, then what would be the probability of 20 flash drives having an average life that’s less than 300 days?

t = [ x – μ ] / [ s / √ ( n ) ]

x = 300, μ = 380, s = 90, n = 20

Substitute the values in the t score formula.

t = [ 300 – 380 ] / [ 90 / √20 ]
= -80 / [90 / 4.472136]
= -80 / 20.1246
t = -3.9752

The degrees of freedom will be calculated by the same method by subtracting one from the size of the sample. df = 20 -1 = 19. The critical value is 0.2491 in this case. You can use our t value calculator to find the t value at one click.

There is a 24.91% probability that the average flash drive for 20 randomly chosen drives would be less than or equal to 300 days if the life of a flash drive is 380 days.

If the research results are higher than this critical value, the “flash drives have a lifespan of 380 days” argument should be dismissed. This is the case if testing is conducted, over 24.91% of sample flash drives, which will last for 300 days or less.

What is Chi-Square Value?

The chi-square statistic compares two different variables to determine whether they are related. The numbers are determined by looking at the chart in the table. The chi-square table is identical to other distribution tables; to look up the information, you need some details. For Chi-square, you will have to study the probability and degrees of freedom that are usually provided in the question.

One way to show a relationship between two categorical variables is a Chi-square statistic. There are two kinds of variables in statistics: numeric variables, which are countable and non-numeric, which are categorical in nature. The chi-squared value is a single number that represents the disparity between the measured counts and the numbers you predict if the population has no connection whatsoever. The chi-square values vary, and the use of these values depends on the hypothesis being tested and the method of collection of data. Expected values are compared with the gathered values, and the same process is used by all types of variations. Use the above chi square critical value calculator to find the chi-square value instantly.

Chi-Square value interpretation

If the chi-square value is low, it implies that your two data sets have a high correlation. In definition, if your values measured and predicted are equal, chi-square would be zero, an occurrence that in real life is unlikely to occur. It doesn't seem as easy to decide the statistically significant difference by a chi-square test. It is difficult to access the range of chi-square tests in relation to the significant difference. If we could assume that the test numbers in the chi-square are more than 10, that would be fantastic, but that's sadly not the case.

You can compare your chi-square value calculated from a Chi-square distribution table to a critical value to check the difference between them. The significance difference exists if the critical value is lower than the chi-square value.

P-value can also be used here. Write the alternative and null hypothesis to use a p-value. Build a chi-square curve with a p-value for your findings. Small p-values, which are less than 5%, typically show a significant difference.

Only numbers can be used with Chi-square statistics. The proportions, ratios, percentages, mean or equivalent mathematical/statistical measures cannot be used. If you have a percentage, you have to convert that percentage to a number to use it in a chi-square test.

What is the importance of critical value?

Statisticians and researchers also operate with a small percentage of a survey community as data are gathered. Research with population samples does not mean that the actual results of the population are represented. Analysts conduct hypothesis checks using these critical values to check whether the statistics are indicative of the actual population.

Critical value determination is critical for statistical data testing. It is one of the main factors to support or contradict commonly accepted facts in hypothesis testing. Proper statistical analysis and assessments help the public to correct information that is inaccurate or obsolete. Testing of a hypothesis is valuable in a variety of fields, including psychology, economics, political science, and business quality management.

Application of critical value in real-life

Critical values have several uses in real-life. When studying a large variety of fields, validating statistical knowledge is essential. This wide variety of fields includes social research such as finance, psychology, sociology, political, and anthropological sciences.

First of all, it retains quality control. It involves sample evaluations of businesses and review of test results in schools. Additionally, testing of theories is important for the scientific and medical communities because it is necessary to spread hypothesis and concepts.

When you consider work that examines behavior, the analysis probably uses sample population testing the hypotheses that are established before. Experts perform research from the voting behavior of the population to which houses citizens tend to buy. Research such as the risk of violence in male teens or obese parents or children in certain states is other indicators that use critical values to assess distribution.

In the health care sector, the issues of how often measles, diphtheria, small poxes, or polio are found in an environment that are important to public safety. Monitoring will allow people to learn if certain levels of safety increase alarmingly. In the age of anti-vaccine activists, this is especially relevant now. Add values in z critical value calculator for the calculation of any critical value for any of these purposes.

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