Question

The purpose of this activity is to show how the sample statistics for each group relate to information provided in the ANOVA summary table. For this exercise, you should run a one-factor ANOVA with the data provided below. Additionally, use a spreadsheet to generate the sample statistics for each group. At a minimum, this should include the group means and standard deviations.

Group 1	Group 2	Group 3	Group 4
84	68.4	38.6	78.5
55.3	66.7	32.8	84.7
73.4	28	43.5	60.5
67.8	51.2	44.8	54.9
72.5	67.3	63.5	68.2
70.6	43	74.3	60.5
46.4	47.8	42.6	57.2
45.5	46.3	45.9	63
68	53.8	48.9	37.1
73	60.9	61.7	67.2

For this activity, we will be focusing on the Sum of Squares portion of the ANOVA summary table. Please report the following values from the table (report all numbers for this exercise accurate to 3 decimal places).
SSbetween=SSbetween=  
SSwithin=SSwithin=  
SStotal=SStotal=

This next activity requires you to treat the data set as one large group. Use a spreadsheet to find the standard deviation of all the dependent variables (as one data set).
sy=sy=  
Now, square this value to obtain the variance:
s2y=sy2=  
Now, multiple the variance by one less than the entire sample size:
(n⋅g−1)⋅s2y=(n⋅g-1)⋅sy2=  
If you obtained the correct value, this should be one of the SS values from the summary table. Question for reflection: Why did this process produce this value in the table?  (Hint: What “spread” is being measured by these values?)

This next activity requires you to work with the means of each group as a new (much smaller) data set. First, please report the group means:
Group 1:  M1=M1=  
Group 2:  M2=M2=  
Group 3:  M3=M3=  
Group 4:  M4=M4=  
Now, calculate the standard deviation of these four sample means:
sM=sM=  
Now, square this value to obtain the variance:
s2M=sM2=  
Finally, multiple the variance by one less the number of groups and then by the number of subjects per group:
n⋅(g−1)⋅s2M=n⋅(g-1)⋅sM2=  
If you obtained the correct value, this should be one of the SS values from the summary table. Question for reflection: Why did this process produce this value in the table?  (Hint: What “spread” is being measured by these values?)

This final activity requires you to first calculate the sample standard deviations for each group. Please report the values here:
Group 1:  s1=s1=  
Group 2:  s2=s2=  
Group 3:  s3=s3=  
Group 4:  s4=s4=  
For the first group, square the standard deviation to obtain the sample variance:
s21=s12=  
Now multiply the variance by one less than the number of subjects in that group:
(n−1)⋅s21=(n-1)⋅s12=  
Now, repeat this procedure for the other three groups. Finally, add these 4 values together:
∑(n−1)⋅s2j=∑(n-1)⋅sj2=  
If you obtained the correct value, this should be one of the SS values from the summary table. Question for reflection: Why did this process produce this value in the table?  (Hint: What “spread” is being measured by these values?)

Answer 1

Treatments
	Group 1	Group 2	Group 3	Group 4	Total
N	10	10	10	10		40
∑X	656.5	533.4	496.6	631.8		2318.3
Mean	65.65	53.34	49.66	63.18		57.958
∑X2	44515.31	29942.96	26137.5	41455.18		142050.95
Std.Dev.	12.5436	12.8729	12.8077	13.0727		14.0403

By using ANOVA calculator:

Source	SS	df	MS
Between-treatments	1766.1888	3	588.7296	F = 3.57897
Within-treatments	5921.889	36	164.4969
Total	7688.0778	39

The f-ratio value is 3.57897.

The p-value is .023091. The result is significant at p < .05.

the standard deviation of all the dependent variables (as one data set).

Sy = Sqrt(197.13019871795)

= 14.04030621881

(n.g - 1) = (9*4 -1) 197.13019871795

= 6899.55695513

The sample mean for each group.

Group 1 M1 = 65.65

Group 2 M2 = 53.34

Group 3 M3 = 49.66

Group 4 M4 = 63.18

The sample standard deviations for each group.

Standard deviation for group 1 = 12.5436

Standard deviation for group 2 = 12.8729

Standard deviation for group 3 = 12.8077

Standard deviation for group 4 = 13.0727

The sample variance for Group 1 = 157.34190096

Now multiply the variance by one less than the number of subjects in that group:
(n−1)⋅ = (9 - 1)157.34190096 = 1258.73520768

Now, repeat this procedure for the other three groups. Finally, add these 4 values together:
∑(n−1)⋅ = (9 -1) (157.34190096 + 165.71155441 + 164.03717929 + 170.89548529)

∑(n−1)⋅ = 5263.8889596