Transcript 003 The Normal Curve

Standard Scores
and
The Normal Curve
0.45
Standard Normal Curve
0.4
0.35
Frequency
0.3
0.25
0.2
0.15
0.1
0.05
0
-3
-2
-1
0
1
2
3
Z Score
3/27/2016
HK 396 - Dr. Sasho MacKenzie
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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.
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HK 396 - Dr. Sasho MacKenzie
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Z-score Equation
z
x

• x is a raw score to be standardized
• σ is the standard deviation of the population
• μ is the mean of the population
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HK 396 - Dr. Sasho MacKenzie
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Z-score for 300lb Squat
• Assume a population of weight lifters had a
mean squat of 295 ± 19.7 lbs.
z
x

300  295
z
 .25
19.7
• That means that a squat of 300 lb is .25
standard deviation above the mean. This
would be equivalent to the 60th percentile.
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HK 396 - Dr. Sasho MacKenzie
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What about a 335 lb Squat
• How many standard deviation is a 335 lb
squat above the mean?
z
x

335  295
z
2
19.7
• That means that a squat of 335 lb is 2
standard deviation above the mean. This
would be equivalent to the 97.7th percentile.
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HK 396 - Dr. Sasho MacKenzie
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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?
x  295  (1)(19.7)  275.3 lbs
• What would you know about the raw score if it
had a z-score of 0 (zero)? Right on the mean.
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HK 396 - Dr. Sasho MacKenzie
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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.
z
x

10  11.4
z
 2.8
0.5
• 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.
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HK 396 - Dr. Sasho MacKenzie
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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.
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HK 396 - Dr. Sasho MacKenzie
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Histogram of Male 100 m
300
Frequency
250
200
150
300
250
100
150
50
0
250
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)
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HK 396 - Dr. Sasho MacKenzie
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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.
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HK 396 - Dr. Sasho MacKenzie
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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.
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HK 396 - Dr. Sasho MacKenzie
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Male NCAA 100 m Sprint
300
Frequency
250
200
150
300
250
100
250
150
50
150
50
0
9.8
10.2
50
10.6
11.0
11.4
11.8
12.2
12.6
13.0
Time (s)
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HK 396 - Dr. Sasho MacKenzie
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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
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HK 396 - Dr. Sasho MacKenzie
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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.
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HK 396 - Dr. Sasho MacKenzie
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Frequency
Male NCAA 100 m Sprint
-3
-2
-1
0
1
2
3
z-score
(standard deviations)
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HK 396 - Dr. Sasho MacKenzie
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Male NCAA 100 m Sprint
50%
Frequency
15.9%
84.1%
Cumulative %
34.1%
34.1%
2.3%
97.7%
13.6%
0.14%
-3
0.1%
3/27/2016
13.6%
2.2%
-2
2.2%
-1
0
1
z-score
(standard deviations)
HK 396 - Dr. Sasho MacKenzie
2
99.9%
3
0.1%
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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.
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HK 396 - Dr. Sasho MacKenzie
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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.
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