One Way ANOVA test Determine whether there are any statistically significant differences between the means of three or more independent groups

There are no significant differences between the means Null Hypothesis There are no significant differences between the means A = B = C = D = E

Alternative Hypothesis If you reject the null hypothesis and accept the alternative: there are at least two groups with a significant difference Cannot pinpoint which ones are significantly different just with this test

Another way to look at how spread data is around the mean ANOVA Analysis of variance Another way to look at how spread data is around the mean

Remember Standard Deviation

Take away the square root and you get variance -1 -1

Each part of the equation ∑ is sigma (sum of) x is each number x is the mean N is how many numbers are in the set

Steps to work out variance Work out the mean Find (x – x)2 for each value Add the numbers together Divide the total by the number of values minus 1

Daily coffee consumption Italian French American n (sample size) 70 M (mean) 4.0 3.7 3.4 S^2 (variance) 4.4 5.2 6.1 (N) Total sample size = 210

Null Hypothesis There are no significant differences between the means French = Italian = American

Alternative Hypothesis At least one group is different to another French = Italian = American

Degrees of freedom (there are two for this test!) BETWEEN Total number of groups – 1 3-1 DF between = 2 (numerator DF) WITHIN Sum of individual degrees of freedoms for each group French: 70 – 1 = 69 Italian: 70 – 1 = 69 American: 70 – 1 = 69 Sum = 207 (denominator DF)

This is what you will compare your ANOVA value to Numerator DF = 2 Denominator DF = 207 Critical value is 3.0718 This is what you will compare your ANOVA value to

Calculating the ANOVA Step 1: The grand mean Sum of all individual means divided by total number of groups (4.0 + 3.7 + 3.4)/3 = 3.7

Step 2: Variance of the means ∑ is sigma (sum of) x is each number x is the mean N is how many numbers are in the set

Step 2: Variance of means (4.0 – 3.7)2 + (3.7 – 3.7)2 + (3.4 – 3.7)2 (3 – 1)

Step 2: Variance of means (4.0 – 3.7)2 + (3.7 – 3.7)2 + (3.4 – 3.7)2 (3 – 1) = 0.09

Step 3: Calculate variance BETWEEN groups Variance of means x n (n is the sample size for each group) 0.09 x 70 = 6.3 s^2BETWEEN

Step 4: Calculate variance WITHIN groups (Sum of variances of each group)/number of groups 4.4 + 5.2 + 6.1 3 5.233 s^2WITHIN

F = s^BETWEEN/s^WITHIN Step 5: ANOVA value (F) F = s^BETWEEN/s^WITHIN = 6.3/5.233 = 1.20

Step 6: compare F value with critical value Reject the null hypothesis if F>critical value What do we do?

Step 6: compare F value with critical value Accept the null as F value is lower than critical value Cannot say with confidence there is a difference between how French, Italian and American people drink coffee