Percentile to z score procedure PDF

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Percentile projections...

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Percentiles, Z Scores, and IQ Tests Sample problem: What is the z score that corresponds to the 85th percentile? How to find the answer: The 85th percentile means a score that is equal to or better than 85% of all the scores on a particular test or measure. If you visualize a graph that colors in all the scores that fall at or below the 85th percentile, that would mean include all the scores below the mean (which is 50% of the scores, right?), plus 35% of the scores above the mean. We can see on this graph that the 85th percentile is slightly more than 1 standard deviation above the mean. This means that the 85th percentile is slightly more than a z score of +1. Imagine coloring in the entire left side of the graph up to the 85th percentile. The 85th percentile means a score that is equal to or greater than the scores in the colored area.

Now that we have visualized where the 85th percentile falls, and now that we know that the z score that corresponds to the 85th percentile should be slightly more than 1, here is how to get the a closer estimate of that z score: 

The 85th percentile is a score that is 85% above all others. So, we can break the 85th percentile into 2 parts: 0.50 = half of the scores fall below the mean on a normal distribution; the 85th percentile is above all of those, of course. 0.35 = the 85th percentile also is 35% above the mean on a normal distribution. This is the same as saying: 0.50 of the scores + 0.35 of the scores = 0.85 of the scores

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Now look at Column 5 in Appendix B in your textbook. Column 5 tells you about the area that is above the mean. We want to find a value of 0.35 in column 5—because we want the z score that lies 35% above the mean (0.35 above the mean), which is the same as the 85th percentile.

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You will see that the closest value to 0.35 in column 5 is 0.3508.

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Look at the z score (in column 1) that corresponds to 0.3505 in column 5. The z score (in column 1) is 1.04 – so that would be the rounded up answer, or the estimate, because 0.3505 is slightly more than 0.35.

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So we know that the z score that corresponds to the 85th percentile is more than 1.03, but not quite 1.04.

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We also know that the z score is closer to 1.04 than 1.03, because 0.85 (the 85th percentile) is closer to 0.3508 in column 5 than to 0.3485.

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So our estimated (rounded up) z score for the 85th percentile = 1.04

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The calculation for the actual z score is complex, so we can double-check our estimate by consulting this online calculator found here: http://www.measuringusability.com/zcalcp.php The online calculator tells us that that the actual z score for the 85th percentile = 1.0364 If we round up 1.0364 to two decimal places, we get: 1.04 So the rounded up estimate that we found in Appendix B is correct: 1.04

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So why would we use appendix B to get an estimate when we have an online calculator? We should use both to ensure that our answer is correct—and also so that we have a good understanding of what a percentile means on the normal distribution graph! The calculator gives you the correct answer, but Appendix B helps us visualize what the percentile means in regard to the normal distribution graph. Also, often the estimated z score at 2 decimal places is good enough. The exact z score is not always needed.

Practical use of this information! We can use the above information to transform a percentile on a standard test to a score on the test. Go to the next page for an example using IQ scores.

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Percentiles and the IQ Test The mean of IQ tests is 100, and the standard deviation is 15. This means that the average score is 100. The mean is always at the 50th percentile, as we can see on the normal bellshaped graph below. So an IQ of 100 means that your IQ is equal to or greater than 50% of the population. Sample question: If your IQ is at the 85th percentile, what is your IQ? We multiple the z score for the 85th percentile times the standard deviation of the IQ test. On the previous page, we found that the z score for the 85th percentile is 1.0364. We can use the rounded up value, which was 1.04: 1.04 x 15 = 15.6 Now we add that to the mean of the IQ test, because the 85th percentile is above the mean: 15.6 + 100 = 115.6 We can round that up to 116. If you have an IQ of 116, you are pretty smart! It means your IQ is equal to or higher than 85% of the population. Congratulations! 

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