Target: To check if the diffrence between the average (mean) of two or more groups (populations) is significant, using sample data
ANOVA is usually used when there are at least three groups, since for two groups you can use t tests.
Example1: A farmer use four differnt fertilizers in four fields. The farmer compare sample average of apples' weight from each field to decide if the difference between the fertilizers is significant.
Independent samples | |
Normal distribution of the analyzed population | |
Equal standard deviation, σ_{1}=σ_{2}=...=σ_{k} |
Sample data from all compared groups |
Source | Degrees of Freedom | Sum of Squares | Mean Square | F statistic | p-value |
---|---|---|---|---|---|
Groups (between groups) |
k - 1
Number of groups minus one |
Represents the difference between the groups [Sum of Squares Between Groups, the squared differences of each groups' average from the overall average] |
MSG = SSG / (k - 1)
The bigger MSG is, the higher the chance that not all the group's averages equal |
F = MSG / MSE
The bigger F is, the higher the chance that not all the group's averages equal |
1 - P(x ≤ F)
The smaller p-vlaue is, the higher the chance that not all the groups' averages equal |
Error (within groups) |
n - k
Overall sample side minus number of groups |
Represents the differences within the groups [The squared differences of each observation average from the overall average] |
MSE = SSE / (n - k)
The smaller MSE is, the higher the chance that not all the groups' averages equal |
||
Total | n - 1
Overall sample side minus one, includes observations in all the groups |
SS(total) = SSG + SSE
Sum of Squares for all observations, not group related |
Sample Variance = SS(total) / (n - 1)
Regular sample variance for all observations, not group related |
H_{0}: μ_{1} = .. = μ_{k} | |
H_{1}: not(μ_{1} = .. = μ_{k}) | |
Header:
You may change groups' name to the real names.
Data:
When entering data, press Enter after each value.
You may copy data from Excel, Google sheets or any tool that separate data with Tab and Line Feed.
Copy the data, one block of consecutive columns includes the header, and paste below.
. click to see example: