how do i calculate the SAT score that defines the bottom 3% of students with a mean of 500 and standard deviation of 100
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.0300) related to the Z score. Insert the values into the above equation to find the score.
To calculate the SAT score that corresponds to the bottom 3% of students, you would use the z-score formula. The z-score measures the number of standard deviations a particular value is away from the mean.
Here are the steps to calculate it:
1. Determine the z-score corresponding to the desired percentile. Since you want to find the score that defines the bottom 3%, subtract 3% from 100% to get 97%. This is the percentage of students above the desired threshold.
- The z-score corresponding to 97% can be found using a standard normal distribution table or a statistical calculator. In this case, a z-score of approximately -1.8808 corresponds to 97%.
2. Apply the z-score formula to convert the z-score to the actual SAT score.
- Z = (X - μ) / σ, where Z is the z-score, X is the SAT score, μ is the mean, and σ is the standard deviation.
Rearranging the formula, we get:
X = Z * σ + μ
3. Plug in the values into the formula:
- Z = -1.8808
- μ = 500 (mean)
- σ = 100 (standard deviation)
Calculating:
X = -1.8808 * 100 + 500
X ≈ 314.92 + 500
X ≈ 814.92
Therefore, the SAT score that defines the bottom 3% of students is approximately 814.92.