What is critical value in statistics?
In statistics, the critical value is a numerical value that is used to calculate the level of significance, the degree of freedom, and the distribution of the test statistic. The critical value is denoted by ‘c’.
Types of critical value:
Generally, there are five types of the term "Critical value"
- T critical value
- Z critical value
- Chi-square value
- F value
- R-value
Formulas for critical values
Formulas vary depending on the type of critical value being calculated.
Formula for T-Value
T = (α/2, df)
Df = n – 1 (n is the sample size)
where:
- Alpha (α) is the significance level
- df represents the degree of freedom
Formula for Z-Value
Z = (α / 2)
where:
- α / 2 gives us the Z-Value known as the normal standard distribution
Formula for Chi-Square value
X2 = ∑ (O – E)2 / E
Where:
- χ2 is the chi-square test statistic
- ∑ is the sum of values
- O is the observed frequency for each category in the contingency table
- E is the expected frequency for each category in the contingency table, calculated as (row total * column total) / grand total
Formula for F Value
F-Value = (σ1)2 / (σ2)2
Where:
- (σ1)2 is the larger sample variance
- (σ2)2 is the smaller sample variance
Applications of Critical Value:
The critical value helps us in various fields, some real-life applications of this term are as follows:
Quality control:
In manufacturing and production, critical values are used to determine if a product meets certain quality standards.
Medical testing:
Critical values are used in medical testing to determine if a result is abnormal or requires immediate attention.
Finance and investments:
Critical values are used in finance and investments to determine if a particular investment or financial decision is sound.
Environmental monitoring:
Critical values are used in environmental monitoring to determine if certain levels of pollutants or other substances are hazardous to human health or the environment.
Sports:
In sports, critical values can be used to determine if a particular athlete's performance meets a certain standard or record.
Calibration of instruments:
In scientific research, critical values are used to calibrate instruments and equipment.
Example section:
The calculation of critical value specifically depends on the specific statistical test being performed. In this section, we’ll cover the method of finding the Critical value.
Example 1:
Find the critical value of a sample size of 5 with a level of significance of 0.1.
Solution:
Step 1:
α = 0.1
n = 5
df = n – 1 = 4
Step 2:
Calculate the t-value of the left critical probability using the t-critical value calculator or t-table:
t-value = 1.5332 (right-tailed probability)
t-value = -1.5332 (left-tailed probability)
t-value = ±2.1318 (two-tailed probability)
