Understanding the Critical Value: Its Significance and Types

Critical value is used to determine the find the statistical result of significance. It helps decide whether to accept or reject a null hypothesis based on a chosen significance level (α). It serves as a reference point for making informed decisions in hypothesis testing.

For decisions making critical values play an important role. Researchers use critical value to assess if a statistical test’s results are statistically significant or if they may have happened by chance. 

In this article, we will examine the definition of critical value, category, and working steps of critical value. Also, in this article with the help of examples topic will be explained.

Critical Value

A critical value is a concept used in statistics, particularly in hypothesis testing and confidence intervals. It’s a point (or points) on the scale of the observed test statistic that defines regions where the distribution of the test statistic is unlikely, given that a null hypothesis is true. If the observed test statistic falls within these regions, one might reject the null hypothesis.

How do critical values work?

Here are the working criteria of critical value.

• Selecting a Significance Level (α): The most common values are 0.05 and 0.01, but other levels can be chosen depending on the specific study and its requirements. Researchers or statisticians choose a significance level.
• Selecting a Statistical Test: Depending on the research question and the data, an appropriate statistical test (e.g., t-test, chi-squared test, z-test) is chosen.
• Calculating the Test Statistic: The test statistic is calculated based on the sample data and the chosen statistical test.
• Determining Critical Values: Critical values are pre-determined values from a reference distribution (e.g., t-distribution, chi-squared distribution, normal distribution) that correspond to the chosen significance level and the degrees of freedom associated with the test.

Types of Critical Value

In this section, we will discuss the types of critical value.

• Z-Critical Value (Z-Score): The Z-critical value is used in hypothesis testing when dealing with a normally distributed population or when the sample size is large (typically n > 30). The Z-critical value corresponds to a specific significance level (α) and is often found in standard normal distribution tables or can be calculated using software.
• T-Critical Value (T-Score): The t critical value is used when dealing with smaller sample sizes and the population follows a normal distribution. The sample size and the degrees of freedom (df), which affect the dispersion of the t-distribution, define the T-critical value. Commonly used T-critical values are found in t-distribution tables or calculated using software.
• Chi-Squared Critical Value: The chi-squared critical value is used in chi-squared tests, which are often applied when analyzing categorical data or testing for independence in contingency tables. The critical values are determined by the degrees of freedom and the chosen significance level. Chi-squared critical values can be found in chi-squared distribution tables or computed using software.
• F-Critical Value: The F-critical value is used in the analysis of variance (ANOVA) and regression analysis. The choice of significance level determines the specific F-critical value to be used.

Critical value is a crucial component of hypothesis testing, helping researchers make informed decisions about their data and hypotheses.

How to Find the Critical Value?

Here are a few solved examples to learn how to find critical value.

Example number 1:

If the value of α = 0.05 and the number 21 determines the critical value t₀.

Solution:

Given data in the given question

α = 0.05 and n= 21.

Critical value =?

Step 1:

df = n-1

df= 21 -1

df = 20

Step 2:

When α = 0.05 in one-tailed column

value t₀ ≈ -1.725

Step 3:

Because the value of the test is -1.725 is left tailed, so, the critical value is negative.

Example number 2:

If α = 0.5% and 90 degrees of freedom, determine the crucial t value (one tail and two tails).  

Solution

Find the values

Level of significance = 0.5

df = 90

A tabular representation of this value answer is given below 

df	α	df	α	df	α	df	α	df	α	df	α	df	α	df	α	df	α
1	0.0005	11	0.0005	21	0.0005	31	0.0005	41	0.0005	51	0.0005	61	0.0005	71	0.0005	81	0.0005
2	0.0005	12	0.0005	22	0.0005	32	0.0005	42	0.0005	52	0.0005	62	0.0005	72	0.0005	82	0.0005
3	0.0005	13	0.0005	23	0.0005	33	0.0005	43	0.0005	53	0.0005	63	0.0005	73	0.0005	83	0.0005
4	0.0005	14	0.0005	24	0.0005	34	0.0005	44	0.0005	54	0.0005	64	0.0005	74	0.0005	84	0.0005
5	0.0005	15	0.0005	25	0.0005	35	0.0005	45	0.0005	55	0.0005	65	0.0005	75	0.0005	85	0.0005
6	0.0005	16	0.0005	26	0.0005	36	0.0005	46	0.0005	56	0.0005	66	0.0005	76	0.0005	86	0.0005
7	0.0005	17	0.0005	27	0.0005	37	0.0005	47	0.0005	57	0.0005	67	0.0005	77	0.0005	87	0.0005
8	0.0005	18	0.0005	28	0.0005	38	0.0005	48	0.0005	58	0.0005	68	0.0005	78	0.0005	88	0.0005
9	0.0005	19	0.0005	29	0.0005	39	0.0005	49	0.0005	59	0.0005	69	0.0005	79	0.0005	89	0.0005
10	0.0005	20	0.0005	30	0.0005	40	0.0005	50	0.0005	60	0.0005	70	0.0005	80	0.0005	90

FAQs

Q. Number # 1:

How do I find critical values for a specific test?

Answer:

Critical values are typically found in statistical tables corresponding to the chosen significance level and degrees of freedom. Alternatively, you can use statistical software or calculators to compute critical values.

Q. Number # 2:

Can critical values be negative?

Answer:

Yes, critical values can be negative or positive, depending on the direction of the hypothesis test (left-tailed, right-tailed, or two-tailed).

Q. Number # 3:

When are critical values used in hypothesis testing?

Answer:

Critical values are used in hypothesis testing when comparing the test statistic to determine whether an observed effect or difference in data is statistically significant or if it could have occurred by random chance.

Conclusion

In this article, we have discussed the definition of critical value, category, and working steps of critical value. Also, in this article with the help of examples topic will be explained. Anyone may easily defend this article after reading this article.