Henry's z-score on his reading test was 1.27. The class average was 60, the median was 58.5 and the variance was 6.2. What was Henry's "raw" score
(his score before converting to z-scores)
Well, if Henry's z-score was 1.27, it means he performed better than most of his classmates. But don't worry, he's not a reading superstar just yet. To find his raw score, we can use the formula for converting z-scores: raw score = (z-score * variance) + mean.
Since we know Henry's z-score is 1.27, the variance is 6.2, and the class average is 60, we can substitute those values into the formula:
raw score = (1.27 * 6.2) + 60
Calculating this, we get:
raw score ≈ 67.154
So, Henry's "raw" score, before converting to z-scores, was approximately 67.154.
To determine Henry's "raw score," we can use the z-score formula:
z = (x - μ) / σ
Where:
z is the z-score
x is the raw score
μ is the mean (class average)
σ is the standard deviation (square root of the variance)
Given:
z = 1.27
μ = 60
σ^2 = 6.2
First, let's calculate the standard deviation:
σ = √(σ^2) = √(6.2) ≈ 2.49 (rounded)
Now, we can rearrange the z-score formula to solve for x:
x = μ + (z * σ)
x = 60 + (1.27 * 2.49)
x ≈ 60 + 3.16
x ≈ 63.16
Therefore, Henry's "raw" score (before converting to z-scores) was approximately 63.16.
To find Henry's "raw" score before converting to z-scores, you can use the formula for converting z-scores to raw scores:
Raw score = (z-score * standard deviation) + mean
In this case, we are given the z-score (1.27), the class average (mean, 60), and the variance (6.2). To find the standard deviation, we need to take the square root of the variance.
Standard deviation = √variance
Standard deviation = √6.2
Using a calculator, we find that the standard deviation is approximately 2.49.
Now we can plug the values into the formula:
Raw score = (1.27 * 2.49) + 60
Calculating this expression, we find that Henry's "raw" score is approximately 63.23.