A distribution with μ=65 and σ=6 is being standardized to produce a new mean of μ=50 and a new standard deviation of σ=10. Find the new standardized value for each of the following scores from the original distribution: 71, 62, 74, and 53.
Step 1:
The z-score for each score can be calculated using:
z = (x – μ) / σ
For the first score given, 71:
z = (71 – 65) / 6
z = 6/6
z = 1
This means that the score 71 is 1 standard deviation above the mean.
Step2:
Use the formula again to figure out what the equivalent score would be in the new grading system of mean 50 and standard deviation 10:
z = (x – μ) / σ
1 = (x - 50) / 10
10 = x - 50
x = 60
So the original score of 71 would be equivalent to a score of 60 in the new system.
Repeat steps 1 and 2 for each of the remaining three scores.
To standardize a value, you need to calculate its z-score using the formula:
z = (x - μ) / σ
Where:
z is the standardized value (z-score)
x is the original value
μ is the mean of the distribution
σ is the standard deviation of the distribution
Given:
μ1 = 65 (original mean)
σ1 = 6 (original standard deviation)
μ2 = 50 (new mean)
σ2 = 10 (new standard deviation)
Let's calculate the z-scores for each of the given original values:
For x = 71:
z1 = (71 - 65) / 6 = 1
For x = 62:
z2 = (62 - 65) / 6 = -0.5
For x = 74:
z3 = (74 - 65) / 6 = 1.5
For x = 53:
z4 = (53 - 65) / 6 = -2
To find the new standardized values, we need to use the z-scores with the new mean and standard deviation:
For z1 = 1:
x1 = (1 * 10) + 50 = 60
For z2 = -0.5:
x2 = (-0.5 * 10) + 50 = 45
For z3 = 1.5:
x3 = (1.5 * 10) + 50 = 65
For z4 = -2:
x4 = (-2 * 10) + 50 = 30
So, the new standardized values for the scores 71, 62, 74, and 53 from the original distribution are 60, 45, 65, and 30, respectively.