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12/5/2015HK 396 - Dr. Sasho MacKenzie1 Standard Scores and The Normal Curve Slide 2 12/5/2015HK 396 - Dr. Sasho MacKenzie2 Z-Score Just like percentiles have a known basis of comparison (range 0 to 100 with 50 in the middle), so does the z-score.Just like percentiles have a known basis of comparison (range 0 to 100 with 50 in the middle), so does the z-score.z-score Z-scores are centered around 0 and indicate how many standard deviations the raw score is from the mean.Z-scores are centered around 0 and indicate how many standard deviations the raw score is from the mean. Z-scores are calculated by subtracting the population mean from the raw score and dividing by the population standard deviation.Z-scores are calculated by subtracting the population mean from the raw score and dividing by the population standard deviation. Slide 3 12/5/2015HK 396 - Dr. Sasho MacKenzie3 Z-score Equation x is a raw score to be standardized x is a raw score to be standardized is the standard deviation of the population is the standard deviation of the populationstandard deviationstandard deviation is the mean of the population is the mean of the populationmean Slide 4 12/5/2015HK 396 - Dr. Sasho MacKenzie4 Z-score for 300lb Squat Assume a population of weight lifters had a mean squat of 295 19.7 lbs.Assume a population of weight lifters had a mean squat of 295 19.7 lbs. That means that a squat of 300 lb is.25 standard deviation above the mean. This would be equivalent to the 60 th percentile.That means that a squat of 300 lb is.25 standard deviation above the mean. This would be equivalent to the 60 th percentile. Slide 5 12/5/2015HK 396 - Dr. Sasho MacKenzie5 What about a 335 lb Squat How many standard deviation is a 335 lb squat above the mean?How many standard deviation is a 335 lb squat above the mean? That means that a squat of 335 lb is 2 standard deviation above the mean. This would be equivalent to the 97.7 th percentile.That means that a squat of 335 lb is 2 standard deviation above the mean. This would be equivalent to the 97.7 th percentile. Slide 6 12/5/2015HK 396 - Dr. Sasho MacKenzie6 From z-score to raw score A squat that is 1 standard deviation below the mean (-1 z-score) would have a raw score of?A squat that is 1 standard deviation below the mean (-1 z-score) would have a raw score of? What would you know about the raw score if it had a z-score of 0 (zero)? Right on the mean.What would you know about the raw score if it had a z-score of 0 (zero)? Right on the mean. Slide 7 12/5/2015HK 396 - Dr. Sasho MacKenzie7 Z-score for 10.0 s 100 m Assume a population of sprinters had a mean 100 m time of 11.4 0.5 s.Assume a population of sprinters had a mean 100 m time of 11.4 0.5 s. That means that a sprint time of 10 s is 2.8 standard deviations below the mean. This would be equivalent to the 99.7 th percentile.That means that a sprint time of 10 s is 2.8 standard deviations below the mean. This would be equivalent to the 99.7 th percentile. Slide 8 12/5/2015HK 396 - Dr. Sasho MacKenzie8 Converting Z-scores to Percentiles The cumulative area under the standard normal curve at a particular z-score is equal to that scores percentile.The cumulative area under the standard normal curve at a particular z-score is equal to that scores percentile. The total area under the standard normal curve is 1.The total area under the standard normal curve is 1. Slide 9 12/5/2015HK 396 - Dr. Sasho MacKenzie9 Histogram of Male 100 m Time (s) 11.1to11.5 Frequency 50 100 150 200 2503000 300 150 250 50 10.6to11.012.7 150 50 250 Slide 10 12/5/2015HK 396 - Dr. Sasho MacKenzie10 The Histogram Each bar in the histogram represents a range of sprint times.Each bar in the histogram represents a range of sprint times. The height of each bar represents the number of sprinters in that range.The height of each bar represents the number of sprinters in that range. We can add the numbers in each bar moving from left to right to determine the number of sprinters that have run faster than the current point on the x-axis.We can add the numbers in each bar moving from left to right to determine the number of sprinters that have run faster than the current point on the x-axis. Dividing by the total number of sprinters yields the proportion of sprinters that have run faster.Dividing by the total number of sprinters yields the proportion of sprinters that have run faster. Slide 11 12/5/2015HK 396 - Dr. Sasho MacKenzie11 Proportion For example, 50 sprinters ran less than 10.0 s.For example, 50 sprinters ran less than 10.0 s. That means that, (50/1200)*100 = 4%, of the sprinter ran < 10.0 s.That means that, (50/1200)*100 = 4%, of the sprinter ran < 10.0 s. Notice that the area of each bar reflects the number of scores in that range. Therefore, we could just look at the amount of area.Notice that the area of each bar reflects the number of scores in that range. Therefore, we could just look at the amount of area. If there are a sufficient number of scores, the bars can be replaced by a smooth line.If there are a sufficient number of scores, the bars can be replaced by a smooth line. Slide 12 12/5/2015HK 396 - Dr. Sasho MacKenzie12 Male NCAA 100 m Sprint Time (s) 9.811.4 10.6 11.010.211.8 12.6 13.012.2 Frequency 50 100 150 200 2503000 300 150 250 50 150 50 250 Slide 13 12/5/2015HK 396 - Dr. Sasho MacKenzie13 Male NCAA 100 m Sprint Time (s) 9.811.4 10.6 11.010.211.8 12.6 13.012.2 97508470903010316Percentile Frequency 50 100 150 200 2503000 Slide 14 12/5/2015HK 396 - Dr. Sasho MacKenzie14 Normal Distribution If the data are normally distributed, then the raw scores can be converted into z- scores.If the data are normally distributed, then the raw scores can be converted into z- scores. This yields a standard normal curve with a mean of zero instead of 11.4 s.This yields a standard normal curve with a mean of zero instead of 11.4 s. Slide 15 12/5/2015HK 396 - Dr. Sasho MacKenzie15 Male NCAA 100 m Sprint z-score (standard deviations) -30-2 Frequency 13 2 Slide 16 12/5/2015HK 396 - Dr. Sasho MacKenzie16 Male NCAA 100 m Sprint z-score (standard deviations) -30-2 Frequency 13 2 0.1% 2.2% 2.2% 13.6%13.6% 34.1% 34.1% 0.1% Cumulative % 0.14% 2.3% 15.9% 50% 84.1% 97.7% 99.9% Slide 17 12/5/2015HK 396 - Dr. Sasho MacKenzie17 Excel The function NORMSDIST() calculates the cumulative area under the standard normal curve.The function NORMSDIST() calculates the cumulative area under the standard normal curve. The function NORMSINV() performs the opposite calculation and reports the z-score for a given proportion.The function NORMSINV() performs the opposite calculation and reports the z-score for a given proportion. NORMDIST() and NORMINV() perform the same calculations for scores that have not been standardized.NORMDIST() and NORMINV() perform the same calculations for scores that have not been standardized. Slide 18 12/5/2015HK 396 - Dr. Sasho MacKenzie18 Z-score and Percentile Agreement Converting a z-score to a percentage will yield that scores percentile.Converting a z-score to a percentage will yield that scores percentile. However, the population must be normally distributed.However, the population must be normally distributed. The less normal the population the greater discrepancy between the converted z-score and the percentile.The less normal the population the greater discrepancy between the converted z-score and the percentile.