Testing equality of two variances
Many times it is desirable to compare two variances rather than comparing two means. F test is used to compare two population variances or population standard deviations.
In this tutorial we will discuss six steps approach used in hypothesis testing to test whether two population variances are same or not.
FTest for equality of two variances
Let $X_1, X_2, \cdots, X_{n_1}$
be a random sample of size $n_1$ from a population with variance $\sigma^2_1$ and $Y_1,Y_2, \cdots, Y_{n_2}$
be a random samples of sizes $n_2$ from a population with variance $\sigma^2_2$.
Let $\overline{x} = \frac{1}{n_1} \sum X_i$ and $s_1^2 =\frac{1}{n_11}\sum (X_i \overline{x})^2$ be the sample mean and sample variance of first sample respectively.
Let $\overline{y} = \frac{1}{n_2} \sum Y_i$ and $s_2^2 =\frac{1}{n_21}\sum (Y_i \overline{y})^2$ be the sample mean and sample variance of second sample respectively.
Assumptions
a. The two populations are independent.
b. The two samples are simple random samples.
c. The two populations are normally distributed.
Step by Step Procedure
Step by step procedure of $F$ test for equality of two variances is as follows:
Step 1 State the hypothesis testing problem
The hypothesis testing problem can be structured in any one of the three situations as follows:
Situation  Hypothesis Testing Problem 

Situation A :  $H_0: \sigma^2_1=\sigma^2_2$ against $H_a : \sigma^2_1 < \sigma^2_2$ (Lefttailed) 
Situation B :  $H_0: \sigma^2_1=\sigma^2_2$ against $H_a : \sigma^2_1 > \sigma^2_2$ (Righttailed) 
Situation C :  $H_0: \sigma^2_1=\sigma^2_2$ against $H_a : \sigma^2_1 \neq \sigma^2_2$ (Twotailed) 
Step 2 Define the test statistic
The test statistic for testing above hypothesis is
$$ F =\frac{s_1^2}{s_2^2} $$
The test statistic $F$ follows $F$ distribution with $n_11$ and $n_21$ degrees of freedom.
Step 3 Specify the level of significance
Specify the value of level of significance $\alpha$.
Step 4 Determine the critical values
For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.
 For lefttailed alternative hypothesis: Find the $t$critical value using
$$ \begin{aligned} P(F<F_{1\alpha, n_11, n_21}) = \alpha. \end{aligned} $$
 For righttailed alternative hypothesis:
$$ \begin{aligned} P(F > F_{\alpha,n_11,n_21}) = \alpha. \end{aligned} $$
 For twotailed alternative hypothesis:
$$ P(F < F_{ 1  \alpha/2,n_11,n_21} \text{ or } F> F_{\alpha/2,n_11,n_21}) = \alpha. $$
Step 5 Computation
Compute the test statistic under the null hypothesis $H_0$ using equation
$$ F_{obs} =\dfrac{s_1^2}{s_2^2} $$
Step 6 Decision (Traditional Approach)
It is based on the critical values.

For lefttailed alternative hypothesis: Reject $H_0$ if
$F_{obs}\leq F_{1\alpha,n_11,n_21}$
. 
For righttailed alternative hypothesis: Reject $H_0$ if
$F_{obs}\geq F_{\alpha,n_11,n_21}$
. 
For twotailed alternative hypothesis: Reject $H_0$ if
$F_{obs} < F_{1\alpha/2,n_11,n_21}$
or$F_{obs} > F_{\alpha/2,n_11,n_21}$
.
OR
Step 6 Decision (pvalue Approach)
It is based on the $p$value.
Alternative Hypothesis  Type of Hypothesis  $p$value 

$H_a: \sigma^2_1<\sigma^2_2$  Lefttailed  $p$value $= P(F\leq F_{obs})$ 
$H_a: \sigma^2_1>\sigma^2_2$  Righttailed  $p$value $= P(F\geq F_{obs})$ 
$H_a: \sigma^2_1\neq \sigma^2_2$  Twotailed  $p$value $= 2P(F\geq F_{obs})$ 
If $p$value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.
The above six step approach helps you to understand how to test the hypothesis about the equality of two variances or equality of two standard deviations.