# Analysis Of Variance (ANOVA)

So today, I will tell you about **ANOVA**(ANalysis Of VAriance ).

**ANOVA:** is a parametric method appropriate for comparing the means for 2 or more independent or dependent groups.

There are 3 types of **ANOVA:**

1). **One Way ANOVA** 2). **Repeated-Measures ANOVA** 3). **Factorial ANOVA**

**One Way ANOVA:** The **one**–**way** **AN**alysis **O**f **VA**riance (**ANOVA**) is used to determine whether there are any significant differences between the means of two or more independent (unrelated) groups (although you tend to only see it used when there are a minimum of three, rather than two groups).

**Repeated-Measures ANOVA:** Repeated measures **ANOVA** is the equivalent of the **one-way ANOVA**, but for **related**, (not **independent**) groups, and **Repeated-Measure** is the extension of the dependent t-test(learn about dependent t-test by clicking on the given link).

**Factorial ANOVA:** More than one **Categorical Independent Variables**. **Factorial ANOVA** measures whether a combination of independent variables predict the value of a dependent variable. The term “way” is often used to describe the number of independent variables measured by an **ANOVA** test.

In this article I will do One Way ANOVA by hand, but in next article I will teach you how to perform ANOVA in R.

For performing ANOVA we need data, I have created a data table just for understanding. Here is our table:

Critical significance value α(alpha) = 0.05.

H(0)[Null Hypothesis] = μ1 = μ2 = μ3.

H(α) = At least one different among the means.

**STEP: 1** Find the degree of freedom between or within repectively **df(bet) or df(within):**

df(bet) = k-1 [k is number groups]

df(within) = N-k [N is total entries in group]

df(bet) = 3-1 = 2

df(within) = 9-3 = 6

df(total) = df(bet)+df(within) = 2+6 = 8

**F(critical)** value for df(bet) and df(within)

F(critical) = 7.20

**STEP 2: Calculate the mean for each condition for group.**

X1 for group one, X2 for group second X3 for group third

X1= 1+2+5=8 mean of X1 = 8/3 = 2.67

X2 = 2+4+2 = 8 mean of X2 = 8/3 = 2.67

X3 = 2+3+4 = 9 mean of X3 = 9/3 = 3

Grand Mean[GM] = G/N= [1+2+5+2+4+2+2+3+4]/9 = 2.78

G= total number of element of groups

N= total number of groups

**STEP 3: Calculate the sum of squares **

SS(total) = Σ(x-x(GM))²

After solving the equations we will get SS(total) = 13.6

Now calculate **SS(within) = Σ(X1- mean(X1))²+Σ(X2- mean(X2))²+Σ(X3- mean(X3))²**

On solving SS(within) equation we will get the value

SS(within)= 13.34

Now, we have SS(total) and SS(within) with the calculated value we will get the SS(bet) value

SS(total) = SS(bet)+SS(within)

SS(bet) = SS(total)-SS(within)

SS(bet) = 13.6-13.34 = 0.26

**STEP 4: Calculating Variance between and within**

Variance or Mean Square MS(bet) = SS(bet)/df(bet) = 0.26/2 = 0.13

Variance or Mean Square MS(within) = SS(within)/df(within) = 13.34/6 = 2.22

**STEP 5:** **Calculate the F value**

F = MS(bet)/MS(within) = 0.13/2.22 = 0.05

Now we have find F and F(critical) value compare these value whether we can reject our null hypothesis or not.

Here what we have

F = 0.05

F(critical) = 7.20

F<<<<F(critical) Based on our calculation we didn’t get to critical region or rejection of Null hypothesis, So we are not able to reject the null hypothesis

H(0)[Null Hypothesis] = μ1 = μ2 = μ3. Their is no significant difference between the 3 groups of our table.

**In my next article I will explain ANOVA using R, Till then stay tuned and enjoy**

If you have any doubts please mention in comment box or reach me @ irrfankhann29@gmail.com.