Array: Quiz Scores 14.75 12 11.5 13.5 14.75 14.75 13
12.5 13.5 14.75 14.75 14.75 13.50 13.50 13.00 12.50 12.00 11.50
Frequency Example 1: Weights
X Tallies f cf 95 1 1 9 71 1 1 8 56 1 1 7 32 1 1 6 23 1 1 2 5 19 1 1 3 17 1 1 2 2
Frequency Example 2: Quiz scores
X Tallies f cf 14.75 1 1 1 3 9 13.50 1 1 2 6 13.00 1 1 4 12.50 1 1 3 12.00 1 1 2 11.5 1 1 1
SUMMATION Weights Quiz scores 95 14.75 71 14.75 56 14.75 32 13.50 23 13.50 23 13.00 19 12.50 17 12.00 17 11.50 353 120.25
Putting it All Together Weights
X f cf fx 95 1 9 95 71 1 8 71 56 1 7 56 32 1 6 32 23 2 5 46 19 1 3 19 17 2 2 34 n = 9 = 353
Putting it All Together Quiz scores
X f cf fx 14.75 3 9 44.25 13.50 2 6 27.00 13.00 1 4 13.00 12.50 1 3 12.50 12.00 1 2 12.00 11.50 1 1 11.50 n = 9 = 120.25
Situations where calculating the median will NOT be so easy. Consider:
7 7 7 8 8 8 9 9 10 10
Mdn = L +[ ( n / 2 - cfb) / fw) } i
7.5 + { ( 10 / 2 - 3 ) / 3 } 1 = 7.5 + (5 - 3) / 3} 1 = 7.5 + (2 / 3) 1 = 8.17
E. Measures of Variability 1. Range
R = Xh - Xl Example 1: Weights R = 95 - 17 = 78 Example 2: Quiz Scores R = 14.75 - 11.5 = 3.25
Score X X - Mean x2 95 55.78 3111.41 71 31.78 1009.97 56 16.78 281.57 32 -7.22 52.13 23 -16.22 263.09 23 -16.22 263.09 19 -20.22 408.85 17 -22.22 493.73 17 -22.22 493.73 n = 9 x2=6377.57 Sum = 353 Mean = 39.22
Example 2: Quiz Scores X X - Mean x2 14.75 1.39 1.93 14.75 1.39 1.93 14.75 1.39 1.93 13.50 0.14 0.02 13.50 0.14 0.02 13.00 -0.36 0.13 12.50 -0.86 0.74 12.00 -1.36 1.85 11.50 -1.86 3.46 n = 9 = 0 x2 =
12.01 Mean = 13.36
Example 1: Weights sigma = square root { 63377.63 /
9} = square root {708. 63} = 26.62
Example 2: Quiz Scores sigma = square root {1.33} = 1.15
Standard Scores
z-scores Mean = 0 Standard Deviation
= 1 Equation: z = (X - Mean) /
sigma Mean of raw score distribution sigma = SD of raw score
distribution
b. T-scores Mean = 50 SD = 10 Equation: T = 50 + 10 (z) Example: Let's suppose that a teacher
wants to compare the results of an English and of an Algebra test:
Test Score Mean Highest SD
English 84 110 180 26
Algebra 40 47 60 5
English z-score = ( 84 - 110) / 26
= - 26 / 26 z = - 1.00 Algebra z-score = ( 40 - 47) /
5 = -7 / 5 z = -1.4
English T-score = 50 + 10 (-1.00)
T = 50 + -10 T = 40.00 Algebra T-score = 50 + 10 (-
1.4) T = 50 + -
14.00 T = 36.00