What is ANOVA
An ANOVA test is a way to find out if experiment results are significant or not. They help you to figure out if you need to accept the alternate hypothesis or reject the null hypothesis. Mainly, you test groups to see if there is a difference between them or not.
ANOVA: Single factor
It is also known as oneway ANOVA between groups. This test two groups to see if there’s a difference between them.
ANOVA: Two – Factor without Replication
This type is used when we have one group and we are doubletesting that same group.
ANOVA: Two – Factor with Replication
Oneway ANOVA
Oneway ANOVA is used to compare two means from two independent groups using the F distribution. The Alternate hypothesis for the test is that the two means are not equal.
Two Way ANOVA
Two Way ANOVA it’s an extension of OneWay ANOVA. A TwoWay ANOVA, there are two independents. Use a Twoway ANOVA when you have one measurement variable and two nominal variables.
How to calculate ANOVA
Calculate ANOVA by hand
For example, we have 3 group named Group 1, Group 2, Group 3. The Results of the study are in the following table:
Group 1 
Group 2 
Group 3 

n 
70 
70 
70 
M 
4.0 
3.7 
3.4 
s^2 
4.4 
5.2 
6.1 
Where,
n = Sample Size
M = Mean
S^2 = Sample Variance
Step 1:
Create null and alternative hypothesis H_{0} and H_{A}
H_{0}: mean group 1 = mean group 2 = mean group 3
H_{A}: mean group 1 ≠ mean group 2 ≠ mean group 3
Step 2:
Determine the “degrees of freedom” also called df
df between (Numerator df) = 31 = 2
df within (denominator df) = 69 + 69 + 69 = 207
(df 1 = 701 = 69, df 2 = 701 = 69, df 3 = 701 = 69)
Step 3: Use the FTable to get the values for this Ftest ANOVA.
F for df 2, 207 is 3.0718
Note: As the closest value to 207 is 102 i.e. why I took df 2, 102
Step 4: Run the FTest to determine the f value
Step 5: Calculate the Grand mean which is (4.0 + 3.7 + 3.4) / 3 = 3.7
Step 6: Calculate the variance of the means
= [(4.0 – 3.7)^{2} + (3.7 – 3.7 )^{2} + (3.4 – 3.7)^{2}] / 2
Where,
4.0, 3.7, and 3.4 are individual group means
And 2 is the df between.
= (0.09 + 0 + 0.09) / 2
= 0.09
Step 7: Calculate the variance between the groups
= variance of the means * N (N is sample size)
= 0.09 * 70 = 6.3
Step 8: Calculate the within variance
= (4.4 + 5.2 + 6.1)/3
See that Group 1 has a variance 4.4 and group 2 has a variance or 5.2, and group 3 has a variance of 6.1. To get the average of the three group variances, we add them together and divide by 3.
= 5.233
Step 9: Now Calculate F
F = variance between the groups/ within variance
= 6.3/5.233
= 1.20
F test is the between variance divided by the within variance.
Our Ftest result here is 1.20
Step 10: Determine the final result and conclude our ANOVA Ftest
Ftest is 1.20.
Cutoff value is 3.07
As Ftest is less than cutoff value we will not reject H_{0} i.e. mean group 1 = mean group 2 = mean group 3.
Calculate ANOVA in excel
Here we have taken a data of age among the three types of people i.e. people who can take care of own personal needs, people who can take care of own personal needs with some difficulty, people who cannot take care of own personal needs without help.
Below is the data
Where,
Age (1) shows the age of people who can take care of own personal needs
Age (2) shows the age of people who can take care of own personal needs with some difficulty
Age (3) shows the age of people who cannot take care of own personal needs without help
Step 1:
Create null and alternative hypothesis H_{0} and H_{A}
H_{0}; Mean of Age (1) = Mean of Age (2) = Mean of Age (3)
H_{A}; Mean of Age (1) ≠ Mean of Age (2) ≠ Mean of Age (3)
Step 2: To Calculate ANOVA for this go to Data Analysis in Data Tab
Step 3: Then in Data Analysis select ANOVA: Single Factor.
Note: You may select any 3 of ANOVA depends on you data type here my data is of single factor
Step 4: After Selecting ANOVA select your Data in Input Range i.e. the data of which you want to check mean difference and write your alpha value in my case i.e. 0.05 below is the screen shot
Step 5: After selecting all the things and click ok then you will get the whole table of ANOVA through which you can interpret your result below is the screen shot
Step 6: Using above table we can interpret the result i.e.
F value = 18.95993069
P value = 3.58066E07
F critical value = 3.142808517
P value is not significant as its value is less than our α and calculated F value is greater than the F critical value which conclude that we will not accept our H_{0} and reject it which means the Mean age among them is not equal.