For the following set of scores, fill in the cells. The mean is 70 and the standard deviation is 8.
Raw score Z score
68.0 ?
? –1.6
82.0 ?
? 1.8
69.0 ?
? –0.5
85.0 ?
? 1.7
72.0 ?
z score = (your data value - mean)/sd
I will do the first and fourth, follow the same procedure for the others
68, ?
z = (68-70)/8 = -2/8 = -.25
? 1.8
1.8 = (x - 70)/8
multiply by 8
14.4 = x-70
add 70
84.4 = x
notice the z value may be ±
but your data value must be positive for your problem
To fill in the missing cells, we need to calculate the corresponding z-scores for each raw score.
Z scores can be calculated using the formula:
Z = (X - μ) / σ
Where:
X = Raw score
μ = Mean
σ = Standard deviation
Let's calculate the missing z-scores:
For the first missing z-score:
X = 68.0
μ = 70
σ = 8
Using the formula:
Z = (68.0 - 70) / 8
Calculating the value:
Z = -0.25
So, the first missing z-score is -0.25.
For the second missing z-score:
Z = -1.6
μ = 70
σ = 8
Using the formula:
-1.6 = (X - 70) / 8
Rearranging the formula:
-12.8 = X - 70
Calculating the value:
X = 57.2
So, the second missing raw score is 57.2.
For the third missing z-score:
X = 82.0
μ = 70
σ = 8
Using the formula:
Z = (82.0 - 70) / 8
Calculating the value:
Z = 1.5
So, the third missing z-score is 1.5.
For the fourth missing z-score:
Z = 1.8
μ = 70
σ = 8
Using the formula:
1.8 = (X - 70) / 8
Rearranging the formula:
14.4 = X - 70
Calculating the value:
X = 84.4
So, the fourth missing raw score is 84.4.
For the fifth missing z-score:
X = 69.0
μ = 70
σ = 8
Using the formula:
Z = (69.0 - 70) / 8
Calculating the value:
Z = -0.125
So, the fifth missing z-score is -0.125.
For the sixth missing z-score:
Z = -0.5
μ = 70
σ = 8
Using the formula:
-0.5 = (X - 70) / 8
Rearranging the formula:
-4 = X - 70
Calculating the value:
X = 66
So, the sixth missing raw score is 66.
For the seventh missing z-score:
X = 85.0
μ = 70
σ = 8
Using the formula:
Z = (85.0 - 70) / 8
Calculating the value:
Z = 1.875
So, the seventh missing z-score is 1.875.
For the eighth missing z-score:
Z = 1.7
μ = 70
σ = 8
Using the formula:
1.7 = (X - 70) / 8
Rearranging the formula:
13.6 = X - 70
Calculating the value:
X = 83.6
So, the eighth missing raw score is 83.6.
The filled-in table is as follows:
Raw score Z score
68.0 -0.25
57.2 -1.6
82.0 1.5
84.4 1.8
69.0 -0.125
66.0 -0.5
85.0 1.875
83.6 1.7