Input Data :
Data set x = 1, 2, 4, 5, 8
Data set y = 5, 20, 40, 80, 100
Total number of elements = 5
Objective :
Find the test of significance by using F-test.
Solution :
mean1 = (1 + 2 + 4 + 5 + 8)/5
= 20/5
mean1 = 4
mean2 = (5 + 20 + 40 + 80 + 100)/5
= 245/5
mean2 = 49
SD1 = √(1/5 - 1) x ((1 - 4)2 + ( 2 - 4)2 + ( 4 - 4)2 + ( 5 - 4)2 + ( 8 - 4)2)
= √(1/4) x ((-3)2 + (-2)2 + (0)2 + (1)2 + (4)2)
= √(0.25) x ((9) + (4) + (0) + (1) + (16))
= √(0.25) x 30
= √7.5
SD1 = 2.7386
SD2 = √(1/5 - 1) x ((5 - 49)2 + ( 20 - 49)2 + ( 40 - 49)2 + ( 80 - 49)2 + ( 100 - 49)2)
= √(1/4) x ((-44)2 + (-29)2 + (-9)2 + (31)2 + (51)2)
= √(0.25) x ((1936) + (841) + (81) + (961) + (2601))
= √(0.25) x 6420
= √1605
SD2 = 40.0625
variance x = (SD1)2
= (2.7386)2
variance x = 7.5
variance y = (SD2)2
= (40.0625)2
variance y = 1605
F = Varinance of dataset xVarinance of dataset y
F = 7.51605
F = 0.0047
F-Test calculator calculates the F-Test statistic (Fisher Value) for the given two sets of data $X$ and $Y$. It's an online statistics and probability tool requires two sets of real numbers or valuables.
It is necessary to follow the next steps:
Test that uses F-distribution, named by Sir Ronald Fisher, is called an F-Test. F-distribution or the Fischer-Snedecor distribution, is a continuous statistical distribution used to test whether two observed samples have the same variance.
A variable has an F-distribution if its distribution has the shape of a special type of curve, called an F-curve. Some examples of F-curves are shown in the picture below.
The F-distribution for testing two population variances has two numbers of degrees of freedom, the number of independent pieces of information for each of the populations. The first number of degrees of freedom for an F-curve is called the degrees of
freedom for the numerator, and the second is called the degrees of freedom for the
denominator. The degrees of freedom corresponding to the variance in the numerator, d.f.N, and the degrees of freedom corresponding to the variance in the denominator, d.f.D.
Degrees of freedom is sample size minus 1. F-distribution is not symmetrical and spans only non-negative numbers.
F-Test compares two variances, $s_X$ and $s_Y$, by dividing them. Since variances are positive, the result is always a positive number. The critical value for F-Test is determined by the equation $$F=\frac{s_X^2}{s_Y^2}$$ In the following we will give a stepwise guide for calculation the critical value for F-test for two sets of data with help of this calculator:
Analysis of variance (ANOVA) uses F-tests to statistically test the equality of means. F-Test is also used in regression analysis to compare the fits of different linear models.
It should be pointed out that the F-test function is categorized under Excel's Statistical functions. It gives the result of an F-Test for two given arrays or ranges. The function returns the two-tailed probability that the variances in the two arrays are not significantly different. This function is mostly used in financial analysis, especially it is useful in risk management.
Practice Problem 1:
A random sample of $13$ members has a standard deviation of $27.50$ and a random sample of $16$ members has a standard deviation of $29.75$. Find the p-value for F-Test.
Practice Problem 2:
Conduct a two tailed F-Test. The first sample has a variance of $118$ and size of $41$. The second sample has a variance of $65$ and size of $21$.
The F-test calculator, work with steps, formula and practice problems would be very useful for grade school students (K-12 education) to learn what is F-test in statistics and probability, how to find it. It's applications is of great significance in the hypothesis testing of variances.