On a standardized test where the results are distributed normally, Elliete scored 246. If the mean score was 182 with a standard deviation of 36, then what is Elliete’s percentile rank rounded to the nearest whole percent?
The Correct Answer and Explanation is:
To determine Elliete’s percentile rank, we need to calculate her z-score, which measures how many standard deviations her score is away from the mean, and then use that to find the percentile rank.
Step 1: Calculate the Z-score
The formula for the z-score is:z=X−μσz = \frac{X – \mu}{\sigma}z=σX−μ
Where:
- XXX is Elliete’s score (246)
- μ\muμ is the mean (182)
- σ\sigmaσ is the standard deviation (36)
Substituting the values:z=246−18236=6436≈1.78z = \frac{246 – 182}{36} = \frac{64}{36} \approx 1.78z=36246−182=3664≈1.78
So, Elliete’s z-score is approximately 1.78.
Step 2: Find the percentile corresponding to the z-score
A z-score of 1.78 indicates how far Elliete’s score is above the mean. To find the percentile, we refer to the standard normal distribution table (or use a calculator that provides the cumulative probability for a z-score).
A z-score of 1.78 corresponds to a cumulative probability of approximately 0.9625. This means that Elliete scored higher than about 96.25% of the people who took the test.
Step 3: Round to the nearest whole percent
The percentile rank is rounded to the nearest whole number:Percentile rank=96%\text{Percentile rank} = 96\%Percentile rank=96%
Conclusion:
Elliete’s percentile rank is 96%, meaning she scored higher than 96% of the people who took the test.
