Analysis of Variance ANOVA- formula and example

Analysis of Variance, ANOVA, is used to analyze data from experiments which have two or more groups. It is used in order to decide whether the difference between group 1 and group 2 is real or a chance event. Another way of saying this is: the ANOVA is used in order to decide whether the difference between mean of group 1 and mean of group 2 is reliable. In statistical jargon we say that we test to see whether the difference is significant.

Analysis of Variance ANOVA -formula

The formula for ANOVA t is: $$F={{MS_{between}} \over {MS_{within}}}$$ We read this as follows: Mean square between over mean square within. What is mean square, you ask. It is the mean of squares. What is squares, you ask. Squares is the statistical term for squared deviations (of squared differences) of each score X from the mean. What are the squared differences, you ask. Remember the formula for variance? $$s^2 ={\sum{({X}-{\bar{X})}}^2 \over {n}}$$ Look at the numerator $${\sum({X}-\bar{X})}^2$$ These are the squared differences or summed. To complete our reasoning, we go back to where we started, the F formula, or F ratio, the formula for ANOVA. Why mean sums of squares? Simple because like all average, we divide by the number of scores. If you are observant, you will notice that the F formula is a modified t formula.

Analysis of variance (ANOVA) -practice example

Group 1 Group 2 Group 3

n1: 5
df1 = n-1= 5- 1= 4
Mean1: 199.2
SS1: 122.8
\(s^2_1\)= SS1/(n - 1) = 122.8/(5-1) = 30.7

n2: 5
df2 = n - 1 = 5 - 1 = 4
Mean2: 211.6
SS2: 121.2
\(s^2_2\) = SS2/(n - 1) = 121.2/(5-1) = 30.3
n3: 5
df3 = n - 1 = 5 - 1 = 4
Mean3: 221.6
SS3: 129.2
\(s^2_3\) = SS3/(b - 1) = 129.2/(5-1) = 32.3

What is df? df stands for degrees of freedom. The simplest way to grasp this concept is to think of a game in which a list of numbers is hidden except for the n (how many) and the mean. You are asked to draw one number and asked to guess the number. Of course, you cannot guess it. When you draw the last number, you are asked to guess it, and of course you can guess it. At this point, thee are no more degrees of freedom What you should remember is:

Every time you calculate a mean, you lose 1 degree of freedom. df= n-1

Source SS df MS F p
Berween 1259 2 629.6 20.24 <0.0001
Within 373.2 12 31.10
Total 1632 14

After we calculate the F, we go to the F tables and enter with the degrees of freedom we have, in this case 2 and 12. We first check the 0.05 level (level of significance). If our F is greater than the one in the F table, we say p<0.05, p less than 0.05. It has been accepted among scientist that at the 0.05 level we are allowed to say that we have significance, that the finding of our experiment is reliable.
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Why learn ANOVA, run t-tests.

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no, that is not allowed

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If you have more than two groups, you must run ANOVA. The t value in the t-table has been calculated for one draw.

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You cannot run repeated t tests.

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Agree. Some do it though but ....

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