# Percentals and Z-scores

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Published on October 27, 2008

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Z scores & percentiles : Z scores & percentiles headlessprofessor When to use? : When to use? You have an interval or ratio scale: discrete or continuous When to use? : When to use? You have an interval or ratio scale: discrete or continuous You have a score for a specific subject When to use? : When to use? You have an interval / ratio scale: discrete / continuous You have a score for a specific subject You have population norms When to use? : When to use? You have an interval / ratio scale: discrete / continuous You have a score for a specific subject You have population norms mean & SD How to do it : How to do it Subtract mean from raw score How to do it : How to do it Subtract mean from raw score Divide by standard deviation How to do it : How to do it Subtract mean from raw score Divide by standard deviation Quotient is the Z score Error Check : Error Check If raw score was above the mean, the Z score should be positive. Error Check : Error Check If raw score was below the mean, the Z score should be negative. Error Check : Error Check If raw score was equal to the mean, the Z score should be zero. Error Check : Error Check It is very rare for a Z score to be below -3 or above +3. Example : Example What is the Z score of someone who scores 130 on an IQ test? Here is what you need : Here is what you need Raw score = 130 Mean IQ = 100 SD IQ = 15 Here is how you calculate it : Here is how you calculate it 130 – 100 = 30 Here is how you calculate it : Here is how you calculate it 130 – 100 = 30 That means 30 points above average Here is how you calculate it : Here is how you calculate it 130 – 100 = 30 30 / 15 = 2 Here is how you calculate it : Here is how you calculate it 130 – 100 = 30 30 / 15 = 2 That says two SDs above the mean Error Check : Error Check If raw score was above the mean, the Z score should be positive. Yes, we got a positive 2. Error Check : Error Check It is very rare for a Z score to be below -3 or above +3. Yes, even though the subject had a high IQ, he is not above z of three. Example : Example What is the Z score of someone who scores 85 on an IQ test? Here is what you need : Here is what you need Raw score = 85 Mean IQ = 100 SD IQ = 15 Here is how you calculate it : Here is how you calculate it 85 – 100 = -15 Here is how you calculate it : Here is how you calculate it 85 – 100 = -15 That means 15 points below average Here is how you calculate it : Here is how you calculate it 85 – 100 = -15 -15 / 15 = -1 Here is how you calculate it : Here is how you calculate it 85 – 100 = -15 -15 / 15 = -1 That’s one SD below the mean Error Check : Error Check If raw score was below the mean, the Z score should be negative. Yes, we got a negative, -1 Error Check : Error Check It is very rare for a Z score to be below -3 or above +3. Yes, even though the subject had a low IQ, he is not below a Z of -3. Convert the Z to a percentile : Convert the Z to a percentile Go to a Z table Standard Normal Gauss Using the Z table : Using the Z table Find the row for the ones and tenths place Using the Z table : Using the Z table Find the row for the ones and tenths place Find the column for the hundredths place. Using the Z table : Using the Z table Take that number and add it to .5 (for positive Z) Using the Z table : Using the Z table Take that number and subtract it from .5 (for negative Z) Calculating the percentile : Calculating the percentile Take that p value and multiply by a hundred to get the percentile. Calculating the percentile : Calculating the percentile Take that p value and multiple by a hundred to get the percentile. Round off to a whole percentile Example : Example Convert the Z score of +2 to a percentile Calculating the percentile : Calculating the percentile .4772 + .5 = .9772 Calculating the percentile : Calculating the percentile .4772 + .5 = .9772 .9772 X 100 = 97.72 Rounoff to the nearest percentile : Rounoff to the nearest percentile .4772 + .5 = .9772 .9772 X 100 = 97.72 98th percentile Genius IQ : Genius IQ .4772 + .5 = .9772 .9772 X 100 = 97.72 98th percentile Error Check : Error Check If the raw score is greater than the mean, the percentile is greater than 50 Error Check : Error Check If the raw score is greater than the mean, the percentile is greater than 50 But not over 100 Error Check : Error Check If the raw score is greater than the mean, the percentile is greater than 50 If the raw score is less than the mean, the percentile is less than 50 but cannot be negative Error Check : Error Check If the raw score is greater than the mean, the percentile is greater than 50 If the raw score is less than the mean, the percentile is less than 50 but cannot be negative If you get a negative, you subtracted .5 from the number Error Check : Error Check If the raw score is greater than the mean, the percentile is greater than 50 If the raw score is less than the mean, the percentile is less than 50 If the raw score is equal to the mean, the percentile is 50 Example : Example Convert the Z score of -1 to a percentile Calculating the percentile : Calculating the percentile .5 - .3413 = .1587 Calculating the percentile : Calculating the percentile .5 - .3413 = .1587 .1587 X 100 = 15.87 Calculating the percentile : Calculating the percentile .5 - .3413 = .1587 .1587 X 100 = 15.87 16th percentile Normal Range IQ : Normal Range IQ .5 - .3413 = .1583 .1587 X 100 = 15.87 16th percentile Error Check : Error Check If the raw score is greater than the mean, the percentile is greater than 50 If the raw score is less than the mean, the percentile is less than 50 but cannot be negative Error Check : Error Check If the raw score is greater than the mean, the percentile is greater than 50 If the raw score is less than the mean, the percentile is less than 50 but cannot be negative If you get a negative, you subtracted .5 from the number Error Check : Error Check If the raw score is greater than the mean, the percentile is greater than 50 If the raw score is less than the mean, the percentile is less than 50 If the raw score is equal to the mean, the percentile is 50 Z scores & percentiles : Z scores & percentiles headlessprofessor

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