# F-Test Calculator

Dataset set x
comma separated input values
Dataset set y
comma separated input values
SD 1  =  2.7386
SD 2  =  40.0625
F-test  =  0.0047
GENERATE WORK

## F-Test - work with steps

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:

1. Enter two samples (observed values) in the box. These values must be real numbers or variables and may be separated by commas. The values can be copied from a text document or a spreadsheet.
2. Press the "GENERATE WORK" button to make the computation.
3. F-test calculator gives us the p-value of the F-distribution.
Input : Two lists of real numbers or variables separated by comma;
Output : A positive real number or a variable.

F-Test Critical Value Formula:
The critical value for F-test is determined by the formula $$F=\frac{s_X^2}{s_Y^2}$$ where $s_X^2$ and $s_Y^2$ are variances for the samples $X$ and $Y$, respectively.

## What is F-Test?

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 is based on the ratio of two variations. It is used to verify whether two independent estimates of variations can be assumed to be estimates of the same variations. If two means or treatments are considerably different, the variation in treatment is greater than the variation due to random dissimilarity among individuals.
No difference would be expected between the variations if estimates of $F$-test were calculated many times for a series of samples drawn from a population of normally distributed variations. Only the positive values are considered in F-distribution, therefore, this is not a symmetrical distribution.
On hypothesis tests, two errors are possible:
1. Supporting the alternate hypothesis when the null hypothesis is true;
2. Not supporting the alternate hypothesis when the alternate hypothesis is true.
An alpha level is the probability of a type I, or we reject the null hypothesis when it is true. The alpha level needs to be halved for two tailed F-Tests.

### How to Calculate F-Test Critical Value

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:

1. Insert two sets of data $X$ and $Y$, i.e. a list of real numbers or variables separated by comma;
2. Find the sample means $\bar{X}$ and $\bar{Y}$ by using the formulas
$$\bar{X}=\frac{1}{n}\sum_{i=1}^{i=n}x_i=\frac{x_1+\ \ldots+x_n}{n}\quad\mbox{and}\quad \bar{Y}=\frac{1}{n}\sum_{i=1}^{i=n}y_i=\frac{y_1+\ \ldots+y_n}{n}$$
3. Find the sample variations $s_X$ and $s_Y$ by using the formulas
$$s_X^2= {\frac{1}{n-1}\sum_{i=1}^{i=n}(x_i-\bar{X})^2}\quad\mbox{and}\quad s_Y^2= {\frac{1}{n-1}\sum_{i=1}^{i=n}(y_i-\bar{Y})^2}$$
where $\bar{X}$ and $\bar{Y}$ are the corresponding sample means;
4. F-statistic (the critical value for F-Test) is determined by the formula $$F=\frac{s_X^2}{s_Y^2}$$
If the variances are equal, then the ratio of the variances is $1$. The larger sample variance should be in the numerator of the F-ratio and the smaller sample variance in the denominator. So, the ratio is always greater than $1$ and simplify the hypothesis test.
The F-test work with steps shows the complete step-by-step calculation for finding the critical value for F-Test of two sets of data $X: 1, 2, 4, 5, 8$ and $Y: 5, 20, 40, 80, 100.$ For other samples, just supply two lists of numbers or variables and click on the "GENERATE WORK" button. The grade school students may use this F-test calculator to generate the work, verify the results derived by hand or do their homework problems efficiently.

### F-Test Practice Problems

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.