What is the Critical Value? Defined and explained with calculations

The critical value is a statistical measurement that is frequently used to calculate the significance level according to the one and two-tailed tables. In this blog post, we’ll explain everything you need to know about critical values, including the difference between t and z critical values, how to find the critical value, and when to use a t or z critical value.

What is a critical value?

A critical value is the value of a statistic that determines statistical significance. It is also known as a t-value or z-value. There are two types of critical values – t critical values and z critical values.

T Critical Values

The t critical value is the smallest difference between two sampling distributions which would still allow us to conclude that the difference between the two populations could be attributed to chance alone.

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This means that if the difference between these two populations were less than or equal to the t critical value, then there was no reason for us to believe that there was an actual difference between these populations and it could have been due to chance alone.

For example, if we had a sample size of 20 people and we wanted to know whether they all ate at Mcdonalds’ on their last day alive, then we would use a t critical value of 3.5 as our cutoff point in order to determine whether or not there was an actual difference between these populations.

If the difference in the number of Mcdonalds’ visits among our 20 subjects was greater than 3.5, then it could be said with 95% confidence that this discrepancy was simply due to chance and not due to any real differences between these groups of people (i.e., they may have all eaten at different McDonalds on their last day alive).

Z Critical Values

The z-value is similar to the t-value, but it takes into account how extreme the differences in data are from one sample instance to another sample instance.

In other words, z-values take into account how much variation exists within each group rather than just looking at how much variation exists across groups (as with t-values).

So, if we had a sample size of 20 people and we wanted to know whether or not they all ate at Mcdonalds’ on their last day alive, then we would use a z critical value of 2 as our cutoff point in order to determine whether or not there was an actual difference between these populations.

If the difference in the number of McDonalds visits among our 20 subjects was greater than 2, then it could be said with 95% confidence that this discrepancy was simply due to the difference between these groups of people (i.e., they may have all eaten at different McDonalds on the last day alive).

Difference Between t & z Critical Values

The t and z critical values are used when conducting statistical tests.

How to find t critical value?

Here we are going to learn how to calculate the t and z critical values. First of all, you must have a number of samples and the significance level and then you have to check the values according to the one or two-tailed table.

Both z and t tables are available you just have to take the table and search for your required result. It is a pretty difficult task to check the values in the table according to the given inputs as there were hundreds of values present in the table.

To tackle this difficulty, a critical value calculator can be used as it will calculate the t and z values according to the tables within seconds with a screenshot of the table to let you know where the values lie.

Here are a few examples of t and z critical values.

Examples of t-critical value

Example 1: For one-tailed

Calculate the t critical value if the significance level is 0.03 and the number of samples is 12.

Solution

Step 1: First of all, write the given information.

Significance level = 0.03

Number of samples = 12

Step 2: Now subtract 1 from the given sample (12) to get the degree of freedom.

Degree of freedom = Df = 12 – 1

Degree of freedom = Df = 11

Step 3: Select the one-tailed or two-tailed t distribution table and search the degree of freedom in the first column of the t table.

Step 4: After that search the significance level in the first row of the t table.

Step 5: Now go forward from the degree of freedom and go down from the significance level, and choose the value where both terms intersect. That would be the t critical value.

t critical value = 2.0961

Example 2: For two-tailed

Calculate the t critical value if the significance level is 0.5 and the number of samples is 102.

Solution

Step 1: First of all, write the given information.

Significance level = 0.5

Number of samples = 102

Step 2: Now subtract 1 from the given sample (102) to get the degree of freedom.

Degree of freedom = Df = 102 – 1

Degree of freedom = Df = 101

Step 3: Select the two-tailed t distribution table and search the degree of freedom in the first column of the t table.

Step 4: After that search the significance level in the first row of the t table.

Step 5: Now go forward from the degree of freedom and go down from the significance level, and choose the value where both terms intersect. That would be the t critical value.

t critical value = ± 0.6769

Example of z critical value

Example

If the significance level is 0.05 then calculate the z critical value.

Solution

Step 1: First of all, take the significance level and then divide it by 2.

Significance level = 0.05

Significance level / 2 = 0.05 / 2 = 0.025

Step 2: Now subtract half of the significance level from 1.

1 – Significance level / 2 = 1 – 0.025 = 0.975

Step 3: Now search 0.975 in the z critical value table.

Step 4: After finding the value select the significance level and degree of freedom and add them.

z critical value = 1.9 + 0.06

z critical value = 1.96

Wrap Up

Now you can grab all the basics of the t and z critical values as we have discussed both the terms along with solved examples. You can either use t and z distribution tables or the above-mentioned calculator to solve the problems of critical value.