the SAT mean is 1092 and standard deviation is 197.
the ACT mean is 21.6 and the standard deviation is 4.4.
the 49th percentile means that 49% of all scores are below that.
under the normal distribution curve, that would be the area to the left of the equivalent z-score.
you would use the z-score table or a z-score calculator to find the z-score associated with an area of .49 to the left of it.
the calculator that i used can be found at https://davidmlane.com/hyperstat/z_table.html
the calculator is set to getting a z-score with an area to the left of it equal to .49.
the results of that is shown below.
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the calculator says that the z-score is -.025.
you can use the z-score formula to find the equivalent raw scores for both the SAT scores and the ACT scores.
the z-score formula is z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard deviation.
for the SAT, you get z = (x - m) / s becomes -.025 = (x - 1092) / 197.
solve for x to get x = -.025 * 197 + 1092 = 1087.075.
for the ACT, you get z = (x - m) / s becomes -.025 = (x - 21.6) / 4.4.
solve for x to get x = -.025 * 4.4 + 21.6 = 21.49.
the z-score for a SAT score of 1367.8 is found by solving the z-score formula of z = (x - m) / s becoming z = (1367.8 - 1092) / 197 = 1.4.
that z-score formula is used to find the comparable ACT score.
z = (x - m) / s becomes 1.4 = (x - 21.6) / 4.4.
solve for x to get x = 1.4 * 4.4 + 21.6 = 27.76.
answers to your questions are:
If a student gets an SAT score that is the 49-percentile, find the actual SAT score.
the SAT score is 1087.1.
What would be the equivalent ACT score for the student?
the equivalent ACT score is 21.5.
If a student gets an SAT score of 1367.8, find the equivalent ACT score.
the equivalent ACT score is 27.8.
the calculator at was used to calculate the z-scores.
it can also be used to calculate the raw scores directly.
with z-score, you set the mean and standard deviation to 0 and 1 resprecively.
with raw scores, you set the mean and standard deviation to what they are.
the z-score analysis results are shown below:
the comparable raw score results are shown below.
let me know if you have any questions.
theo