How to Find Critical Value in Two Steps (With Examples)

By Indeed Editorial Team

Published November 5, 2021

The Indeed Editorial Team comprises a diverse and talented team of writers, researchers and subject matter experts equipped with Indeed's data and insights to deliver useful tips to help guide your career journey.

You can use critical value in hypothesis analysis to calculate the margin of error within a collection of data. Establishing critical value is important because it can allow researchers to determine if they can prove their hypotheses. Understanding how to find critical value can help you identify statistical miscalculations and determine statistical significance, which is vital if you are considering a career in statistics.

In this article, we provide steps to calculate critical value, explore the definition of T statistics and Z-scores, learn how to express critical value as both T statistics and Z-scores, review frequently asked questions about statistical concepts, and learn about the types of critical value systems.

How to find critical value

Here's a list of steps that you can use if you're interested in learning how to find critical value:

1. Find the alpha value

Here's the formula you can use to determine the alpha value (a):

a = 1 - (confidence level/100)

The alpha value determines whether the calculation is statistically significant, and the confidence level signifies the odds of the statistical factor also being true for the population you're measuring. For example, if the confidence level is 85%, here is the equation to determine the alpha value:

a = 1 - (85/100) = 0.15

2. Calculate critical probability

The next step is finding the critical probability, or critical value, using the alpha value that was calculated in the first equation. In this equation, "p*" represents the critical probability, which is equal to subtracting one from half the alpha value:

Critical probability = (p*): p* = 1 - a/2

Once you calculate the critical probability, you can format into a T statistic or a Z-score. T statistics and Z-scores are slightly different statistical measures that you can use to test two points in a hypothesis. Here's an example of finding the critical probability using the previously calculated alpha value of 0.15:

Critical probability = 1 - (0.15/2) = 0.925 = 92.5%

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What is a T statistic?

You can express critical value as a T statistic. Both T statistics and Z-scores can be used to find and compare two points of a hypothesis. You may use T statistics when working with small sample sizes or if you don't know the population's standard deviation because there isn't enough information. The Central Limit Theorem states that as sample sizes grow to infinity, the sample mean becomes normally distributed. This means that larger data sets typically have more accurate information.

It's important to put T statistics in context by adding background information to explain their real value. For example, saying that "the weight is 75" may not be very helpful, but clarifying that "the sum of three packages creates a total weight of 75 lbs" can add much more information about the relative significance of that weight.

What is a Z-score?

Sometimes called a "standard score," a Z-score's purpose is to provide an estimate of how different a mean may be from a specific data point. You can use Z-scores to compare your results to a standard population. For example, if you're comparing height data, a Z-score may tell you how one person's height compares to the average population's mean height. When your sample size is too small, you can use a T statistic instead.

How to express critical value as a T statistic

To learn how to express critical value as a statistic, here's a list of steps you can follow:

1. Find the degree of freedom (df)

The degree of freedom (df) is the maximum number of values that can vary based on your data sample. The degree of freedom is equal to the sample size minus one. Here's the formula to calculate the degree of freedom:

Degree of freedom (df) = sample size - 1

2. Convert to a T statistic

The critical T statistic (t*) is the T statistic that has a degree of freedom (df) and a cumulative probability that is equal to the critical probability (p*). There isn't a single formula for finding the T statistic because its value depends on the type of test you're performing. A T statistic test with only one sample may have different results from a paired T statistic test because they can include such diverse variables.

Expressing critical value as a Z-score

Expressing critical value as a Z-score means using your Z-score as a cumulative probability that's equal to the critical probability (p*). Cumulative probability is the likelihood that the value of a random variable occurs within a certain range, meaning the variable is less than or equal to a specific value.

Here's the formula for a basic Z-score equation where "x" represents the raw score, "μ" represents the population mean, and "σ" signifies the population's standard deviation:

Z-score = (z): z = (x - μ)/σ

Example of Z-score calculation

Here's an example of a Z-score equation using a test score of 200, a mean of 50, and a standard deviation of 20:

z = (200-50) / 20 = 7.5

In this scenario, the Z-score is 7.5 standard deviations above the statistical test average.

Frequently asked questions about critical value

Critical value is a complicated concept, so it's common to have questions about how it works or about the meaning of the involved terms. Here are some frequently asked questions and answers about statistical concepts:

What's the difference between standard error and standard deviation?

Standard error is the extent of difference between a sample mean and the population mean, while standard deviation refers to the variance in measurement within a single sample and mean. For example, if the average income from a random sampling of 100 people is $10,000, and the standard deviation, or "σ," of that sample is $1000. The standard deviation reveals how much a single income differs from the average of $10,000.

The standard error of the sample, where $10,000 is the average and the standard deviation is $1,000, is 100. The standard of error for the sample is 1%, meaning there's only a slight chance of random errors occurring.

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What are null and alternative hypotheses?

A null hypothesis signifies that there are zero differences between the specified populations and no errors with the sampling or experiments. An alternative hypothesis is an expected disparity between two or more variables of your research. Each type of hypothesis relates to the conclusion of a statistical test or experiment being accepted or rejected.

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What is statistical significance?

Statistical significance is a term used by researchers to signify the unlikelihood of their studies occurring under the null hypothesis of a statistical test. If a set of data has statistical significance, you can attribute your conclusions about it to a specific cause, rather than random chance. The term is completely subjective and relies on the variables of a specific study or hypothesis.

What does P-value and variability mean?

A P-value, also known as a probability value, is a number that states the odds of your data occurring under the null hypothesis of your test. A small P-value shows that there's an adequate amount of evidence to support an alternative hypothesis. A large P-value means that the evidence against the null hypothesis is not strong enough to reject it.

In statistical analysis, variability is the measure of statistical dispersion. Variability, in relation to statistics, can tell you how far apart data points are from one another. This value can also show you how far a data point is from the centre of distribution.

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What are the types of critical value systems?

Here is a list of the three main kinds of critical value systems, known as chi-squares, T-scores, and Z-scores:

  • Chi-squares: Chi-squares come from two forms of chi-square tests, known as the independence test and the goodness of fit test. The independence test compares two variables in order to see their relations, while the goodness of fit test helps determine if the data from the sample matches the population in reference.

  • T-scores: A T-score, also referred to as a T-value, associates with R-tests and regression tests. When data follows a T-distribution, a T-score can show the distance between an observation from the mean.

  • Z-scores: A Z-score, also known as a standard score, is a measurement that shows the amount a value differs from the mean in terms of standard deviation.

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