Probability Distributions

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Probability Distributions

• What proportion of a group of kittens lie in any
  selected part of a pile of kittens?

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Probability Distributions

• Sometimes we want to know the chances that
  something will occur?
• For example
• What are the odds that I will win the lottery?
• What are my chances of getting an A?
• If a person is young, what are the chances that
  he or she will be in poverty?
• What chances do poor people have of graduating
  from college?
• To answer questions such as these, we turn to
  probability.

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Probability Distributions

• Probability
• Out of all possible outcomes, the proportionate
  expectation of a given outcome. Values for
  statistical probability range from 0 (never) to 1
  (always) or from 0 chance to 100 chance.
• For example
• 12 of 25 students in an engineering class are
  women. The probability that a randomly selected
  student in that engineering class will be a woman
  is 12/25 .48 or 48.

13
12
F M
4
Probability Distributions

• What is the probability that a student will get a
  C in Statistics?
• What about a C or Higher?

10 5 0
3
5
12
7
5
F D C B A
5
Probability Distributions

• What is the probability that a student will get a
  C in Statistics?
• 12/32 .375
• What about a C or Higher? 24/32 .75
• What is the probability that a person in the
  class got a grade? 32/32 1

10 5 0
3
5
12
7
5
F D C B A
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Probability Distributions

• Empirical probability distribution
• All the outcomes in a distribution of research
  results and each of their probabilitieswhat
  actually happened
• The probability distribution of a variable lists
  the possible outcomes together with their
  probabilities

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Probability Distributions

• What is the probability that a student will get a
  C in Statistics?
• 12/32 .375
• What about a C or Higher? 24/32 .75

.375 .219 0
F D C B A
P1 100 of cases
P.25 or 25
F D C B A
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Empirical Rule

• Many naturally occurring variables have
  bell-shaped distributions. That is, their
  histograms take a symmetrical and unimodal shape.
• When this is true, you can be sure that the
  empirical rule will hold.
• Empirical rule If the histogram of data is
  approximately bell-shaped, then
• About 68 of the cases fall between Y-bar s.d.
  and Y-bar s.d.
• About 95 of the data fall between Y-bar 2s.d.
  and Y-bar 2s.d.
• All or nearly all the data fall between Y-bar
  3s.d. and Y-bar 3s.d.

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Empirical Rule

• Empirical rule If the histogram of data is
  approximately bell-shaped, then
• About 68 of the cases fall between Y-bar s.d.
  and Y-bar s.d.
• About 95 of the cases fall between Y-bar 2s.d.
  and Y-bar 2s.d.
• All or nearly all the cases fall between Y-bar
  3s.d. and Y-bar 3s.d.

Body Pile 100 of Cases
s.d.
15
15
15
s.d.
15
M 100 s.d. 15
85
55
70
115
130
145
or 1 s.d.
or 2 s.d.
or 3 s.d.
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Probability Distributions

• The Normal Probability Distribution
• A continuous probability distribution in which
  the horizontal axis represents all possible
  values of a variable and the vertical axis
  represents the probability of those values
  occurring. Values are clustered around the mean
  in a symmetrical, unimodal pattern known as the
  bell-shaped curve or normal curve.

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Probability Distributions

• The Normal Probability Distribution
• No matter what the actual s.d. (?) value is, the
• proportion of cases under the curve that
  corresponds
• with the mean (?)/- 1s.d. is the same (68).
• The same is true of mean/- 2s.d. (?95)
• And mean /- 3s.d. (almost all cases)
• Because of the equivalence of all
• Normal Distributions, these are often
• described in terms of the Standard Normal Curve
• where mean 0 and s.d. 1 and is called z

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Probability Distributions

• The Normal Probability Distribution
• No matter what the actual s.d. (?) value is, the
• proportion of cases under the curve that
  corresponds
• with the mean (?)/- 1s.d. is the same (68).
• The same is true of mean/- 2s.d. (?95)
• And mean /- 3s.d. (almost all cases)
• Because of the equivalence of all
• Normal Distributions, these are often
• described in terms of the Standard Normal Curve
• where mean 0 and s.d. 1 and is called z
• Z of standard deviations away from the mean

68
68
Z -3 -2 -1 0 1 2 3
Z-3 -2 -1 0 1 2 3
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Probability Distributions

• Converting to z-scores
• To compare different normal curves, it is helpful
  to know how to convert data values into z-scores.
• It is like have two rulers beneath each normal
  curve. One for data values, the second for
  z-scores.

IQ ? 100 ? 15
Values 55 70 85 100 115 130
145
Z-scores -3 -2 -1 0 1
2 3
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Probability Distributions

• Converting to z-scores
• Z Y ?
• ?

Z 100 100 / 15 0 Z 145 100 / 15 45/15
3 Z 70 100 / 15 -30/15 -2 Z 105 100
/ 15 5/15 .33
IQ ? 100 ? 15
Values 55 70 85 100 115 130
145
Z-scores -3 -2 -1 0 1
2 3
15
Probability Distributions

• Engagement Ring Example
• Mean cost of an engagement ring is 500, and the
  standard deviation is 100.
• Z Y ?
• ?

Z 500 500 / 100 0 Z 600 500 / 100
100/100 1 Z 200 500 / 100 -300/100 -3 Z
550 500 / 100 50/100 .5
Ring ? 100 ? 15
Values 200 300 400 500 600 700
800
Z-scores -3 -2 -1 0 1
2 3
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Probability Distributions

• Engagement Ring Example
• Mean cost of an engagement ring is 500, and the
  standard deviation is 100.

Now, use the empirical rule What percentage of
people will be above or below my preferred ring
price of 300?
Ring ? 500 ? 100
2.5
2.5
68
Values 200 300 400 500 600 700
800
Z-scores -3 -2 -1 0 1
2 3
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Probability Distributions

• Comparing two distributions by Z-score
• Imagine that your partner didnt get you a ring,
  but took you on a trip to express their love for
  you. You could convert the trips price into a
  ring price using z-scores.
• Your trip cost 2,000. The average love trip
  costs 1,500 with a s.d. of 250. What is the
  equivalent ring price?

Trips
Rings
200 300 400 500 600 700 800
750 1000 1250 1500 1750 2000 2250
-3 -2 -1 0 1 2
3
-3 -2 -1 0 1 2
3
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Probability Distributions

• Comparing two distributions by Z-score
• Your trip cost 2,000. The average love trip
  costs 1,500 with a s.d. of 250. What is the
  equivalent ring price?
• What percentage of persons got a trip that cost
  less than yours?

Trips
Rings
200 300 400 500 600 700 800
750 1000 1250 1500 1750 2000 2250
-3 -2 -1 0 1 2
3
-3 -2 -1 0 1 2
3
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Probability Distributions

• Comparing two distributions by Z-score
• What about ACT versus SAT scores?
• NOTE This is a helpful process, but can be
  illogical at times. Remember that you are
  comparing scores on a population base or
  percent of people above or below each score. Is
  it logical to compare SAT score to self-esteem
  this way? No.

SAT
ACT
15 18 21 24 27 30
33
400 600 800 1000 1200 1400 1600
-3 -2 -1 0 1 2
3
-3 -2 -1 0 1 2
3
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Probability Distributions

• How to use a z-score table. (I could use some z
  z z zs).
• F-NL-G Appendix B has reports from the literal
  measurements of area under normal curves. The
  table gives you the percent of values above,
  below, or between particular z-scores ( of s.d.s
  away from the mean).
• Left column z (out to two decimals)
• Second column is areaproportion of
  distributionfrom mean to z
• Right column is areaproportion of
  distributionfrom z to the end of the line.
• Can work in reverse to find z-scores too.
• Other tables will use different layouts, online
  you can get automatic answers without using a
  table.

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Probability Distributions

• Theoretical probability distribution
• The proportion of times we would expect to get a
  particular outcome in a large number of
  trialswhat would happen if we had the time to
  observe it.
• Q Why are these important?
• A Sociologists usually get only one chance to
  draw a sample from a population. Therefore, if
  we know what kind of variation in measurement we
  would see if we repeatedly sampled
  (theoretically), we can judge the chance that
  numbers produced by our sample are accurate (this
  will make sense later).

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Probability Distributions

• Theoretical probability distribution
• The number of times we would expect to get a
  particular outcome in a large number of trials.
• For Example Lets say the mean GPA at SJSU is
  2.5.
• Randomly take 100 SJSU students GPAs.
• Record it.
• Now, take 100 more SJSU students GPAs.
• Record that.
• Now, repeat the above.
• Record again.
• Now, lather, rinse, repeat.
• Again.
• Again. And on and on.
• What might you see?

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Probability Distributions

• Theoretical probability distribution
• The number of times we would expect to get a
  particular outcome in a large number of trials.

50 of samples would have a mean GPA greater than
2.5
1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1
3.3 3.5 3.7 3.9
a samples mean
2.5 Overall Mean