Transcript Relative standing

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Outline
1. Chebyshev’s Theorem
2. The Empirical Rule
3. Measures of Relative Standing
4. Examples
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Chebyshev’s Theorem
• Applies to any data set.
• At least ¾ of the observations in any data set will fall
within 2s (2 standard deviations) of the mean:
x
• ( x – 2s, x + 2s).
• At least 8/9 of the observations in any data set will
fall within 3s of the mean:
• ( x – 3s, x + 3s).
•  k >1, at least 1 – (1/k2) of the observations will fall
within ks of the mean. (Works with ,  too)
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The Empirical Rule
• Applies to distributions that are mound-shaped and
symmetric
• 68% of the observations will fall within +/- 1s of the
mean
• 95% of the observations will fall within +/- 2s of the
mean
• 99.7% (essentially all) of the observations will fall
within +/- 3s of the mean
• Works with ,  too.
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The Empirical Rule
• Example: IQ scores
 = 100
 = 15
o Population mean:
o Population std. dev:
o The empirical rule tells us that 99.7% of IQ scores
fall in the range 55 to 145.
o Do you see why?
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Measures of Relative Standing
A. Percentiles
B. Standard scores (Z scores)
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Measures of Relative Standing
• Measures of relative standing tell us
something about a given score by reporting
how it relates to other scores.
• For example, one such measure tells us what
proportion of scores in a data set are smaller
than a given score.
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Measures of Relative Standing
A. Percentiles
• With observations in a set arranged in order
(smallest to largest), Pth percentile is a number such
that P% of the observations fall below it.
• In a set of 200 observations, if a number X is larger
than 150 of the observations, then X is at the 75th
percentile (150/200 = 75%).
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Measures of Relative Standing
B. Standard Scores (aka Z scores)
i. For a sample:
Z = X –x
s
ii. For a population:
Z=X–

• Z-scores can be positive or negative.
• A (raw) score below the mean has a negative Z
score.
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2 Important qualities of the Z score
1. Z-scores are like a ruler – they measure
distances.
o Z scores give the distance between any score X
and the mean, expressed in standard deviation
units.
2. Values expressed in Z scores can be
compared, regardless of their original units.
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Z scores are like a ruler
• Compare these situations:
 = 100 and  = 10
120 =  + 2
 = 100 and  = 40
120 =  + ½
A score of 120 is more impressive on the left,
where it is 2 standard deviations above the
mean (vs. one-half s on the right).
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Z-Scores are unit-free
• Values expressed in Z scores can be
compared, regardless of their original units.
o For example, suppose you had exam grades for 100
students in Psych 281 and also knew how many hours
each student studied for that exam.
o You could compare any student’s Z-score for their grade
with the Z-score for their # of hours of studying.
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Z-Scores
• Suppose we have the following data for our
class:
Mean grade for the exam:
Standard deviation:
70
10
Mean # hours studying for exam: 8
Standard deviation:
2
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Z-Scores
Grade
Z
Bill
80
+1
Bob
80
+1
Ben
60
-1
# hours
Z
6
-1
12
+2
6
-1
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Example – Assignment 2, Q.1
• The distribution of a sample of 100 test
scores is symmetrical and mound-shaped,
with a mean of 50 and a variance of 144.
a. Approximately how many scores are equal to or
greater than 74?
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Example – Assignment 2, Q.1
Since s2 = 144, s = 12, and 74 =
x + 2s
What percentage
of scores falls
above the red
line?
50
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62
74
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Example – Assignment 2, Q. 1
• Since 74 = x + 2s, we have p = .975.
• 2.5% of scores are ≥ 74.
• Since there are 100 scores, 2.5% = approx. 3
scores.
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Example – Assignment 2, Q. 1
b. What score corresponds to the 75th
percentile?
• To answer this, we use interpolation.
o A score at the 84th percentile is 1 s above the
mean.
o x = 50 and s = 12, so 62 is one s above the mean –
which is the 84th percentile. (Why?)
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Example – Assignment 2, Q.1
• Remember that the distribution is moundshaped and symmetric.
• By the Empirical Rule, 68% of the distribution
is between -1 s and +1 s around the mean – so
half of that (34%) is between the mean and 1 s
above the mean.
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Example – Assignment 2, Q. 1
• The 84th percentile is 1 s above the mean by
the Empirical Rule.
• A score of 62 is 1 s above the mean because
the mean is 50 and s = 12.
• Therefore 84th percentile = 62
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Example – Assignment 2, Q.1
50% + 34% = 84%
34%
50%
84th
percentile
X = 50
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X = 62
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Example – Assignment 2, Q. 1
• Now we know which score is at the 50th
percentile (the mean score – by definition in a
symmetric distribution).
• We also know which score is at the 84th
percentile.
• Now we can answer our question: which score
is at the 75th percentile?
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Example – Assignment 2, Q. 1
X
%ile
50
50
∆
12
25
a
75
62
84
34
∆ = 25
12
34
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Example – Assignment 2, Q. 1
∆
12
=
25
34
Interpolation
∆ = 25 (12) = 8.82
34
75th percentile:
X = 50 + 8.82 = 58.82 ~ 59
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Example – Assignment 2, Q. 1
c. If the lowest and highest scores in this sample are 14
and 89, respectively, what is the range of the scores
in standard deviation units?
Z1 = 14 – 50
12
= -3 .0
Z2 = 89 – 50
12
= 3.25
Range = -3 .0 to 3.25 = 6.25
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Example – Assignment 2, Q. 1
d. Assume the 100 scores have same mean and
variance but now have a strongly skewed
distribution. At least how many of the scores fall
between 32 and 68 in this distribution?
By Chebyshev’s Theorem:
Z1 = 32 – 50 = –1.5
12
Z2 = 68 – 50 = 1.5
12
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Example – Assignment 2, Q. 1
k = 1.5
1 – (1/k2) = 1 – (1/1.52) = 1 – 1/2.25
= .555.
55.5% or ~ 56% of scores lie between 32 and 68.
Lecture 3