SD 1 = 2.7386

SD 2 = 40.0625

F-test = 0.0047

GENERATE WORK

GENERATE WORK

**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:

- 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.
- Press the
**"GENERATE WORK"**button to make the computation. - F-test calculator gives us the p-value of the F-distribution.

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.

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.

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:

- Supporting the alternate hypothesis when the null hypothesis is true;
- Not supporting the alternate hypothesis when the alternate hypothesis is true.

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:

- Insert two sets of data $X$ and $Y$, i.e. a list of real numbers or variables separated by comma;
- Find the sample means $\bar{X}$ and $\bar{Y}$ by using the formulas
$$\bar{X}=\frac{1}{n}\sum_{i=1}^{i=n}x_i=\frac{x_1+\ \ldots+x_n}{n}\quad\mbox{and}\quad \bar{Y}=\frac{1}{n}\sum_{i=1}^{i=n}y_i=\frac{y_1+\ \ldots+y_n}{n}$$
- 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;
- F-statistic (the critical value for F-Test) is determined by the formula $$F=\frac{s_X^2}{s_Y^2}$$

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.

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.