Question

calculate the sum of the squares within groups
sum of the squares within table
the sum of the squares within groups is
14
25
44
72

Explanation:

Step1: Calculate the means

• For \(x\): \(\bar{x}=\frac{1 + 2+3+3+6}{5}=\frac{15}{5} = 3\)
• For \(y\): \(\bar{y}=\frac{7 + 8+9+7+4}{5}=\frac{35}{5}=7\)
• For \(z\): \(\bar{z}=\frac{10 + 4+1+6+4}{5}=\frac{25}{5} = 5\)

Step2: Calculate \((x-\bar{x})\), \((y - \bar{y})\), \((z-\bar{z})\)

• For \(x = 1\): \((1 - 3)=- 2\); for \(x = 2\): \((2 - 3)=-1\); for \(x = 3\): \((3 - 3)=0\); for \(x = 3\): \((3 - 3)=0\); for \(x = 6\): \((6 - 3)=3\)
• For \(y = 7\): \((7 - 7)=0\); for \(y = 8\): \((8 - 7)=1\); for \(y = 9\): \((9 - 7)=2\); for \(y = 7\): \((7 - 7)=0\); for \(y = 4\): \((4 - 7)=-3\)
• For \(z = 10\): \((10 - 5)=5\); for \(z = 4\): \((4 - 5)=-1\); for \(z = 1\): \((1 - 5)=-4\); for \(z = 6\): \((6 - 5)=1\); for \(z = 4\): \((4 - 5)=-1\)

Step3: Calculate \((x-\bar{x})^2\), \((y - \bar{y})^2\), \((z-\bar{z})^2\)

• \((x-\bar{x})^2\): \((-2)^2 = 4\), \((-1)^2=1\), \(0^2 = 0\), \(0^2=0\), \(3^2 = 9\). \(\sum(x-\bar{x})^2=4 + 1+0+0+9=14\)
• \((y - \bar{y})^2\): \(0^2 = 0\), \(1^2=1\), \(2^2 = 4\), \(0^2=0\), \((-3)^2 = 9\). \(\sum(y - \bar{y})^2=0 + 1+4+0+9=14\)
• \((z-\bar{z})^2\): \(5^2 = 25\), \((-1)^2=1\), \((-4)^2 = 16\), \(1^2=1\), \((-1)^2 = 1\). \(\sum(z-\bar{z})^2=25+1 + 16+1+1=44\)

Step4: Calculate SSW

\(SSW=\sum(x-\bar{x})^2+\sum(y - \bar{y})^2+\sum(z-\bar{z})^2=14 + 14+44=72\)

Answer:

72