SAT I scores around the nation tend to have a mean scale score around 500, a standard deviation of about 100 points, and are approximately normally distributed. What SAT I score within the population would have a percentile rank of approximately 16?
Go through the reverse of the same process as indicated in previous post. Find the Z score in the table related to .16 and put that value in the equation to solve for the score.
To determine the SAT I score with a percentile rank of approximately 16, we need to use the concept of z-scores and the standard normal distribution.
Step 1: Calculate the z-score.
The z-score represents the number of standard deviations away from the mean that a particular score is. It is calculated using the formula:
z = (X - μ) / σ
Where:
X is the observed score,
μ is the mean,
σ is the standard deviation.
Given that the mean (μ) is 500 points, the standard deviation (σ) is 100 points, and we want to find the score (X) with a percentile rank of 16:
Step 2: Lookup the z-score in the standard normal distribution table.
The standard normal distribution table provides the cumulative probability (percentile rank) for each z-score. In this case, we want to find the z-score that corresponds to the 16th percentile.
Looking up the 16th percentile in the standard normal distribution table, we find that the z-score corresponding to a percentile of 0.16 is approximately -0.9945.
Step 3: Substitute the calculated z-score back into the z-score formula to find the score (X).
-0.9945 = (X - 500) / 100
Now solve for X:
-0.9945 * 100 = X - 500
-99.45 = X - 500
X = -99.45 + 500
X = 400.55
Therefore, an SAT I score of approximately 400.55 would have a percentile rank of 16 within the population.