1. Standard Scores and The Normal Curve
7/15/2019
HK Dr. Sasho MacKenzie

2. 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.
7/15/2019
HK Dr. Sasho MacKenzie

3. 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
7/15/2019
HK Dr. Sasho MacKenzie

4. Z-score for 300lb 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 60th percentile.
7/15/2019
HK Dr. Sasho MacKenzie

5. 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 97.7th percentile.
7/15/2019
HK Dr. Sasho MacKenzie

6. 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)?
7/15/2019
HK Dr. Sasho MacKenzie

7. 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 99.7th percentile.
7/15/2019
HK Dr. Sasho MacKenzie

8. 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.
7/15/2019
HK Dr. Sasho MacKenzie

9. Histogram of Male 100 m 300 250 250 150 150 50 50 Frequency Time (s)
200
250
300
300
250
250
Frequency
150
150 50 50
<10.0
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)
7/15/2019
HK Dr. Sasho MacKenzie

10. 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.
7/15/2019
HK Dr. Sasho MacKenzie

11. 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 are a sufficient number of scores, the bars can be replaced by a smooth line.
7/15/2019
HK Dr. Sasho MacKenzie

12. Male NCAA 100 m Sprint 300 250 250 150 150 50 50 Frequency Time (s)
9.8
11.4
10.6
11.0
10.2
11.8
12.6
13.0
12.2 50
100
150
200
250
300
300
250
250
Frequency
150
150 50 50
7/15/2019
HK Dr. Sasho MacKenzie

13. Male NCAA 100 m Sprint Frequency Time (s) Percentile 9.8 11.4 10.6
11.0
10.2
11.8
12.6
13.0
12.2 50
100
150
200
250
300
Frequency 97 50 84 70 90 30 10 3 16
Percentile
7/15/2019
HK Dr. Sasho MacKenzie

14. Normal Distribution
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.
7/15/2019
HK Dr. Sasho MacKenzie

15. (standard deviations)
Male NCAA 100 m Sprint
Frequency -3 -2 -1 1 2 3
z-score
(standard deviations)
7/15/2019
HK Dr. Sasho MacKenzie

16. (standard deviations)
Male NCAA 100 m Sprint
50%
15.9%
84.1%
Cumulative %
Frequency
34.1%
34.1%
2.3%
97.7%
13.6%
13.6%
0.14%
99.9%
2.2%
2.2% -3 -2 -1 1 2 3
z-score
(standard deviations)
0.1%
0.1%
7/15/2019
HK Dr. Sasho MacKenzie

17. 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.
7/15/2019
HK Dr. Sasho MacKenzie

18. 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.
7/15/2019
HK Dr. Sasho MacKenzie