ANCOVA (Analysis of Variance) is applied to compare more than two independent sample means, while controlling for a covariate.

Suppose you want to compare mean marks scored in 12 ^th standard, by students in two different schools. The results show that the mean marks for school A are significantly more than that for school B. Here, we tend to infer that school A is much better than school B.

Then comes next issue: selective admission to intelligent students in school A. Now, you want to test whether the difference in the marks is because of quality education or it is because of quality of admitted students to school A. To identify quality of admitted students, you can consider marks obtained in previous years, before the admission to these schools. For example, marks obtained in 10^th (considering students change the school after 10^th standard).

Now you want to compare the mean marks in 12^th AFTER controlling for marks in 10^th. Here the covariate is marks in 10^th standard. So, in this example dependent variable is marks in 12^th standard, independent variable is school of 12^th standard, and covariate is marks previous to the admission to the school. ANCOVA can be applied here to compare two means after controlling for the covariate.

Assumptions
1. Dependent variable and covariate should be continuous and normally distributed in all the categories of independent variable.
2. Samples are drawn using random sampling technique.
3. The observations are independent. There are no observations, which are in more than one group.
4. The residuals should be normally distributed. (homogeneity of variance  / homoscedasticity). It can be tested by Levene's test of homogeneity of variance of errors.
5. There are no outliers.
6. There is one independent variable, which is nominal or ordinal, with more than two possible levels / categories.
7. Covariate should be linearly related to the dependent variable for each category of independent variable. It can be tested by scatter plot for dependent variable and covariate, one each for each category of independent variable.

Null hypothesis and Alternate Hypothesis

Null Hypothesis

A. Sample means for all categories of independent variables are equal (μ_i = μ_j), after controlling for covariate.

 Alternate Hypothesis

A. There are at least two means (at least a pair of means), which are different than each other (μ_i ≠ μ_j), after controlling for covariate.

Where μ_i and μ_j can be any two category means.

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Following is the sample data for marks in 12^th, school and previous marks.

Marks_12_Standard	School	Previous_marks
80	B	64
77	B	61
78	B	60
70	B	54
95	A	90
90	A	92
78	B	52
76	B	57
93	A	85
92	A	90
75	B	57
75	B	65
73	B	55
95	A	88
75	B	51
75	B	69
80	B	60
82	A	83
80	A	98
99	A	92
80	B	56
91	A	94
80	B	68
77	B	65
83	B	67
85	B	70

You can copy and paste above table, and paste in the textbox on the ANCOVA page. (You may need to paste it in excel, to get the data in required format)
Please choose your alpha (usually 5%). Select type III as model type for SSQ. This type III SSQ is more approproiate for most of the situations. For details, please read details about types of SSQ in two-way ANOVA help page.
Click on "Run test". You will get following output.

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  @ Sachin Mumbare

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