LIR 493:
Quantitative Methods
Professor Wallace Hendricks


ANOVA Table (One way or Single Factor)

Est. Variance
Sum of Squares

df

(mean square)

F

Total

N - 1

Between ss

Between ss / k - 1
Between

k - 1

k - 1

Within ss/ N - k
Within

N - k

Within ss

N - k

Example

Scores

Test Type 1 18 25 21 24 22
  2 20 23 24 25  
  3 28 32 33    
  4 22 24 27 25 32

i = individual score on test

j = index of 1 - 4 for test type

  x i1   x i12   x i2   x i22   x i3   xi32   x i4   x i42
  18   324   20   400   28   784   22   484
  25   625   23   529   32   1024   24   576
  21   441   24   576   33   1089   27   729
  24   576   25   625           25   625
  22   484                   32   1024
i j2 = 110   2450   92   2130   93   2897   130   3438
22 23 31 26
Nj 5 4 3 5

Thus, an ANOVA Table

Sum of Squares df Est. Variance F
Total 290 16
Between 174 3 58 6.5
Within 116 13 8.92

EXCEL OUTPUT FOR THIS PROBLEM

Sum of Squares

indiv - grand indiv - sample j sample j-grand

Thus


"Error" SS

"Residual" SS

"Unexplained" SS

Degrees of Freedom


FORMULAS FOR SUM OF SQUARES




COMPUTATION FOR OUR EXAMPLE

I. VARIANCE METHOD

Total

WITHIN = WEIGHTED AVERAGE OF SAMPLE VARIANCES

USING D.F. AS WEIGHTS


Each sample variance is


Example


Therefore








II Sums of Squares Method

TOTAL =
=
=
BETWEEN =
=
=
=
WITHIN =
Between Variance =
Within Variance =
F =