Describing Data

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Daily Check

• The following data set shows the number of points
  scored by the Hawks in each game this year.
• 108, 91, 107, 111, 113, 92, 86, 120, 100, 110,
  97, 113, 83, 125, 114
• Find the sample standard deviation.
• Find the variance.

12.55
157.50
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Math II
UNIT QUESTION Can real world data be modeled by
algebraic functions? Standard MM2D1, D2 Todays
Question How do you use a score to find
probabilities of events? Standard MM2D1d
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The Standard Deviation is a Ruler
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Who should get the Gold?
In the heptathlon, women compete in 7 events the
200-m and 800-m runs, 100-m high hurdles, shot
put, javelin, high jump, and long jump. To
determine the Gold medal winner, all of the
performances in the 7 events need to be combined
into one single score. Races are measured in
seconds. Distances are measured in meters (or
centimeters). How can these events be compared?
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Who should get the Gold?
2000 Sydney Olympics
Gertrud Bacher from Italy
Won 800 meter run in 129 seconds. The mean of all
runners was 137 seconds with a standard deviation
of 5 seconds.
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Who should get the Gold?
2000 Sydney Olympics
Yelena Prokhorova (Russia)
Won the long jump with a jump of 660 cm. The mean
jump was 600 cm with a standard deviation of 30
cm.
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Whos result was more unusual?
Unusual results are far from the mean. We need
to compare distance from the mean using the same
scale. Use the standard deviation as the scale,
that is, as a ruler.
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How far from the mean?
The z value tells you how many standard
deviations you are away from the mean.
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Who should get the Gold?
Bacher (Italy)
Prokhorova (Russia)
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Who should get the Gold?
Bacher (Italy)
Prokhorova (Russia)
Prokhorova is further from the mean, so she
deserves the Gold.
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Not exactly
We only considered two events The heptathlon
has 7 events
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Not exactly
Yelena Prokhorova actually won the Silver medal.
(She won the Gold at the 2001 World
Championships.) Gertrud Bacher placed
14th. The Gold was won by Denise Lewis of Great
Britain.
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Example
A towns (not Atlanta) January high temperatures
average 36F with a standard deviation of 10,
while in July the mean high temperature is 74F
and the standard deviation is 8. In which month
is it more unusual to have a day with a high
temperature of 55F
July
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Standardized Test Score Example

• Marys ACT score is 26. Jasons SAT score is
  900. Who did better?
• The mean SAT score is 1000 with a standard
  deviation of 100 SAT points. The mean ACT score
  is 22 with a standard deviation of 2 ACT points.
• Who did better?

Mary
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The Standard Normal Curve

• Theoretically perfect normal curve
• Use to determine the relative frequency of
  z-scores and raw scores
• Proportion of the area under the curve is the
  relative frequency of the z-score
• Rarely have z-scores greater than 3 (.26 of
  scores above 3, 99.74 between /- 3)

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The Standard Normal Curve

• Mean is 0, Standard Deviation is 1.

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Z scores
A z score is a raw score expressed in standard
deviation units. It is the number of standard
deviations a number is from the mean.
What is a z-score?
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Why z-scores?

• Transforming scores in order to make comparisons,
  especially when using different scales
• Gives information about the relative standing of
  a score in relation to the characteristics of the
  sample or population
• Location relative to mean
• Relative frequency and percentile

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The Standard Normal Table

• A table of areas under the standard normal
  density curve. The table entry for each value z
  is the area under the curve to the left of z.

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The Standard Normal Table

• The table can be used to find the proportion of
  observations of a variable which fall to the left
  of a specific value z if the variable follows a
  normal distribution.

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The Standard Normal Table
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Example 1 Calibrating Your GRE Score
GRE Exams between 10/1/89 and 9/30/92 had mean
verbal score of 497 and a standard deviation of
115. (ETS, 1993)

• Suppose your score was 650 and scores were
  bell-shaped.
• Standardized score (650 497)/115 1.33.
• Table, z 1.33 is between the 90th and 91st
  percentile.
• Your score was higher than about 90 of the
  population.

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Example 2 Removing Moles
Company Molegon remove unwanted moles from
gardens.
Weights of moles are approximately normal with a
mean of 150 grams and a standard deviation of 56
grams.
Only moles between 68 and 211 grams can be
legally caught.

• Standardized score (68 150)/56 1.46, and
  Standardized score (211 150)/56 1.09.
• Table 86 weigh 211 or less 7 weigh 68 or
  less.
• About 86 7 79 are within the legal
  limits.

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Your Turn!
Suppose it is known that verbal SAT scores are
normally distributed with a mean of 500 and a
standard deviation of 100.
Find the proportion of the population of SAT
scores are less than or equal to 600.
First we need to find the standardized
score Z-score(observed value-mean)/(standard
deviation) (600-500)/100 1
From Table 8.1 we see that a z-score of 1 is the
84th percentile and the proportion of population
SAT scores that are less than or equal to 600 is
0.84.
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SAT SCORES
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Standardized Scores (Z-Scores)
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Estimate the proportion of population SAT scores
that are between 400 and 600.
An observed value of 400 has a z-score of -1 and
represents the 16th percentile (proportion below
z -1 is 0.16). An observed value of 600 has a
z-score of 1 and represents the 84th percentile
(proportion below z 1 is 0.84). Lets draw a
picture.
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So the proportion with scores between 400 and
600 Proportion below 600 Proportion below
400 0.84 - 0.16 0.68