Critical Value in Statistics: A Brief Explanation

The critical value is a term used in statistics to describe a boundary value that separates the acceptance and rejection of a null hypothesis. It is a numerical value used to make decisions about the significance of results from a hypothesis test.

In this article, we’ll study the basic definition of the term “critical value” its types, and their uses in real life. In the example section, we’ll cover the methods of calculating the critical values.

What is critical value in statistics?  

The critical value is a value that is compared to a test statistic in hypothesis testing to determine whether the null hypothesis is to be rejected or not. 

Types of Critical values: 

Generally, in basic statistics, two types (T and Z) are discussed but in this section, we’ll cover the basic and advanced types of critical values along with their real-life uses. The types are as follows: 

• T-Critical Value
• Z-Critical Value
• Chi-Square Critical Value
• F-Value

T-Critical Value:

A T-Critical value is used in a t-test to determine whether a sample mean is significantly different from a population mean. It represents the boundary between the acceptance and rejection regions for the t-distribution. The T-Critical value depends on the degrees of freedom and the level of significance (usually represented by alpha). 

In a two-tailed test, the T-Critical value is the value such that the area under the t-distribution to the left of the T-Critical value is equal to alpha/2 and the area to the right of the T-Critical value is also equal to alpha/2.

Real-Life Use of T-critical value:

T-Critical values are commonly used in fields such as psychology, economics, and education to test the significance of differences between groups. For example, in psychology, a t-test may be used to determine whether the average scores of a treatment group are significantly different from those of a control group.

Z-Critical Value:

A Z-Critical value is used in a z-test to determine whether a sample mean is significantly different from a population mean. It is used when the sample size is large (n >= 30) and the population standard deviation is known. 

The Z-Critical value is based on the standard normal distribution (Z-distribution) and represents the boundary between the acceptance and rejection regions for the Z-distribution. Like the T-Critical value, the Z-Critical value depends on the level of significance (usually represented by alpha).

Real Life Use Z-Critical value:

Z-Critical values are commonly used in fields such as biology, medicine, and environmental science to test the significance of differences between groups. For example, in biology, a z-test may be used to determine whether the average height of a species of plant is significantly different from the average height of another species of plant.

Chi-Square Critical Value:

The Chi-Square Critical value is used in a chi-square goodness of fit test to determine whether observed data fits a theoretical distribution. The Chi-Square Critical value represents the boundary between the acceptance and rejection regions for the chi-square distribution. 

The Chi-Square Critical value depends on the degrees of freedom and the level of significance (usually represented by alpha).

Real Life Use Chi-Square Critical Value:

Chi-Square Critical values are commonly used in fields such as sociology, psychology, and marketing to test the significance of differences between observed and expected frequencies. 

For example, in sociology, a chi-square test may be used to determine whether the distribution of family sizes in a population is significantly different from the expected distribution.

F-Value:

The F-Value is used in an F-test to determine whether the variance of two or more populations is equal. It is calculated by dividing the variance of one group by the variance of another group. The F-Value represents the ratio of the variability between two groups to the variability within the groups.

Real-Life Use F-Value:

F-Values are commonly used in fields such as biology. 

Examples of critical value: 

In this section, we’ll describe some mathematical examples to understand the method of finding the critical value of different samples. 

Example 1:

Evaluate the 1-tail and 2-tailed t critical value if α = 0.03 and the degree of freedom is 50 

Solution: 

Step 1: If both α and degree of freedom are given then you just have to check the table to find out the value.  

df	α	df	α	df	α	df	α
1	10.5791	16	2.024	31	1.9522	46	1.9283
2	3.8965	17	2.015	32	1.9499	47	1.9273
3	2.9505	18	2.0071	33	1.9477	48	1.9263
4	2.6008	19	2	34	1.9457	49	1.9253
5	2.4216	20	1.9937	35	1.9438	50	1.9244
6	2.3133	21	1.988	36	1.942
7	2.2409	22	1.9829	37	1.9402
8	2.1892	23	1.9782	38	1.9386
9	2.1504	24	1.974	39	1.9371
10	2.1202	25	1.9701	40	1.9357
11	2.0961	26	1.9665	41	1.9343
12	2.0764	27	1.9632	42	1.933
13	2.0601	28	1.9601	43	1.9317
14	2.0462	29	1.9573	44	1.9305
15	2.0343	30	1.9547	45	1.9294

On 50 degrees the t critical value is 1.9244 for the one-tailed T-test.

df	α	df	α	df	α	df	α
1	21.2052	16	2.3816	31	2.2746	46	2.2395
2	5.6427	17	2.3681	32	2.2712	47	2.238
3	3.8961	18	2.3562	33	2.268	48	2.2365
4	3.2976	19	2.3457	34	2.265	49	2.2351
5	3.0029	20	2.3362	35	2.2622	50	2.2338
6	2.8289	21	2.3278	36	2.2595
7	2.7146	22	2.3201	37	2.257
8	2.6338	23	2.3132	38	2.2546
9	2.5738	24	2.3069	39	2.2524
10	2.5275	25	2.3011	40	2.2503
11	2.4907	26	2.2958	41	2.2482
12	2.4607	27	2.2909	42	2.2463
13	2.4359	28	2.2864	43	2.2445
14	2.4149	29	2.2822	44	2.2427
15	2.397	30	2.2783	45	2.2411

On 50 degrees the t critical value is 2.2338 for the two-tailed T-test.

Sometimes we don’t have the table along with us to overcome this you can check Criticalvaluecalculator to get accurate answers according to the statistical tables of the entered values. 

Example 2: 

Find out the z critical (left-tailed) value if α = 0.4

Solution: 

Step 1: Division

Divide 0.4 by “2”.

= 0.4 / 2 

= 0.20 

Step 2: Subtraction

Subtract the value of α from 1. 

= 1 – 0.20 

= 0.8

Step 3: Check the Z-table.  

The value matches at 0.8 + 0.05. 

So, the approximate value is 0.85.

Summary: 

In this article, we have discussed the basic introduction to the term “critical value”. We have also learned the basic definition of this term, along with its types and their real-life uses. All types i.e., T critical value, Z critical value, F value, Chi square value are explained briefly. 

In the example section, we have tried to cover the method of finding T critical and Z critical values. Hope you have gone through the basic introduction to this statistical term.