Standard Scores and The Normal Curve 9162020 HK

Z-Score • Just like percentiles have a known basis of comparison (range 0 to

Z-score Equation • x is a raw score to be standardized • σ is

Z-score for 300 lb Squat • Assume a population of weight lifters had a

What about a 335 lb Squat • How many standard deviation is a 335

From z-score to raw score • A squat that is 1 standard deviation below

Z-score for 10. 0 s 100 m • Assume a population of sprinters had

Converting Z-scores to Percentiles • The cumulative area under the standard normal curve at

Histogram of Male 100 m 300 Frequency 250 200 150 250 100 250 150

The Histogram • Each bar in the histogram represents a range of sprint times.

Proportion • For example, 50 sprinters ran less than 10. 0 s. • That

Male NCAA 100 m Sprint 300 Frequency 250 200 150 250 100 300 250

Male NCAA 100 m Sprint 300 Frequency 250 200 150 100 50 0 9.

Normal Distribution • If the data are normally distributed, then the raw scores can

Frequency Male NCAA 100 m Sprint -3 -2 -1 0 1 2 3 z-score

Male NCAA 100 m Sprint 50% Frequency 15. 9% 84. 1% Cumulative % 34.

Excel • The function NORMSDIST() calculates the cumulative area under the standard normal curve.

Z-score and Percentile Agreement • Converting a z-score to a percentage will yield that

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Standard Scores and The Normal Curve 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 1

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-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. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 2

Z-score Equation • x is a raw score to be standardized • σ is the standard deviation of the population • μ is the mean of the population 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 3

Z-score for 300 lb Squat • 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. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 4

What about a 335 lb Squat • 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. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 5

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? • What would you know about the raw score if it had a z-score of 0 (zero)? 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 6

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. • 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. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 7

Converting Z-scores to Percentiles • The cumulative area under the standard normal curve at a particular z-score is equal to that score’s percentile. • The total area under the standard normal curve is 1. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 8

Histogram of Male 100 m 300 Frequency 250 200 150 250 100 250 150 50 0 300 150 50 <10. 0 50 10. 1 to 10. 5 10. 6 to 11. 0 11. 1 to 11. 5 11. 6 to 12. 0 12. 1 to 12. 6 >12. 7 Time (s) 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 9

The Histogram • Each bar in the histogram represents a range of sprint times. • 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. • Dividing by the total number of sprinters yields the proportion of sprinters that have run faster. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 10

Proportion • For example, 50 sprinters ran less than 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. • If there a sufficient number of scores, the bars can be replaced by a smooth line. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 11

Male NCAA 100 m Sprint 300 Frequency 250 200 150 250 100 300 250 150 50 0 9. 8 10. 2 50 10. 6 11. 0 11. 4 11. 8 12. 2 12. 6 13. 0 Time (s) 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 12

Male NCAA 100 m Sprint 300 Frequency 250 200 150 100 50 0 9. 8 10. 2 10. 6 11. 0 11. 4 11. 8 12. 2 30 16 12. 6 13. 0 Time (s) 97 90 84 70 50 10 3 Percentile 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 13

Normal Distribution • If the data are normally distributed, then the raw scores can be converted into zscores. • This yields a standard normal curve with a mean of zero instead of 11. 4 s. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 14

Frequency Male NCAA 100 m Sprint -3 -2 -1 0 1 2 3 z-score (standard deviations) 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 15

Male NCAA 100 m Sprint 50% Frequency 15. 9% 84. 1% Cumulative % 34. 1% 2. 3% 97. 7% 13. 6% 0. 14% -3 0. 1% 9/16/2020 13. 6% 2. 2% -2 2. 2% -1 0 1 z-score (standard deviations) HK 396 - Dr. Sasho Mac. Kenzie 2 99. 9% 3 0. 1% 16

Excel • 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. • NORMDIST() and NORMINV() perform the same calculations for scores that have not been standardized. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 17

Z-score and Percentile Agreement • Converting a z-score to a percentage will yield that score’s percentile. • However, the population must be normally distributed. • The less normal the population the greater discrepancy between the converted z-score and the percentile. 9/16/2020 HK 396 - Dr. Sasho Mac. Kenzie 18