Psychology 203

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Psychology 203

• Semester 1, 2007
• Week 4
• Lecture7

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Who am I?

• Linda Jeffery
• Postdoctoral Research Fellow
• What do I study?
• Face perception

3
Contact Details

• Room 109 - 1st Floor, Main Psych Building
• Email linda_at_psy.uwa.edu.au
• put Re 203 in Subject
• Phone [phone]

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The Power Wonder of z-Scores

• Everything you ever wanted to know about
  Standardized Distributions, but were afraid to
  ask

Gravetter Wallnau, Chpt 5
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Standardization
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Introducing the z-Score

• Are you hot?
• Or not?
• Find out at www.hotornot.com
• Submit your photo people will rate you

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How hot is hot?

• Thats gotta be ok!
• Or is it?
• What if I found out that the average rating is
  8?!
• I would soooo not be super hot any more -(

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Hotness depends on how people use the scale
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Like this?
No - Very few people use this end of the scale
People (10 000s)
1
5
10
1
HOT
NOT
Rating
9
Hotness depends on how people use the scale

• Its actually more like this
• 7.5
• range 3 - 10

Population Mean
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Am I hot or not?

• Depends on the mean hotness rating
• If ? 9.0, then Im less hot than average
• If ? 4.0, then I am WAAY hot!
• Since the mean rating is actually 7.5
• Im hotter than average! Awesome!
• Or am I?

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Am I hot or not?

• How are the ratings spread?

b)
Population Standard Deviation
a)
?1
?.2
?7.5
7.7
?7.5
8.5
X 8.0
X 8.0
Verdict - Im pretty average
Verdict - Im HOT!
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Raw scores

• Often not meaningful by themselves
• To interpret them we need to know the mean and
  the standard deviation of the distribution from
  which they came
• But we can transform them into new scores that
  are stronger, more powerful, and more meaningful!
  We call them

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Superhero Scores?
Z-Scores
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Why z-Scores rock

• Transforming a raw score to a z-score tells us
  exactly where the score is located in the
  distribution - all in one easy number!
• We can also use z-scores to standardize an entire
  distribution
• This means we can compare scores that come from
  completely different scales, tests etc. (like
  buying size 8 shoes from different companies)

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Behold! A z-Score!

• -1.0

A sign
A number
Its so much more, Ben! It slices, it dices, let
me show you how!
Im sceptical. It just looks like a regular ol
number to me!
Sceptical 203 student
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Heres how we pack a tonne of info into one
little number!

• The sign of the z-score tells you whether it is
  above or below the mean
• is above
• - is below
• The number tells you how far it is from the mean
• Big number far from mean
• Little number close to mean

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z-Scores the Distribution
z-score of 1 is one standard deviation from the
mean
Mean z-score of zero
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Hotness in z-Scores
Calculating a z-score
a)
?1
z
?7.5
1
0
8.5
z 0.5
X 8.0
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Hotness in z-Scores
Calculating a z-score
b)
?.2
?7.5
7.7
z
1
0
X 8.0
z 2.5
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Hot or Not?
pretty average
Super hot!!!
?
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From z-Score to Raw Score

• What if I only have the z-score but want to know
  what the raw score really was?

X 8.0
My original rating
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Problem Who is the hottest?
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Standardizing a whole distribution using z-scores

• If we convert every raw score in a distribution
  to a z-score
• The new distribution of z-scores will have
  exactly the same shape as the original,
• The mean of the new distribution will always be
  0,
• The standard deviation of the new distribution
  will always be 1.

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How does this help us compare ratings?

• Convert Vances rating to a z-score

X 3.6, ? 3.0, ? 1.5

• Compare z-scores
• Lindas z-score of 0.5
• Vances z-score of 0.67!! Vance is hotter!

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Other Standardized Distributions

• Can standardize a distribution to any mean and
  standard deviation, not just 0 and 1
• Best known example?
• IQ, mean 100, sd 15
• What would be a good way to standardize
  hotness?

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Standardized Hotness
Mean 5.0, sd 2.0
3

• So most people would fall between 3 7.
• Anything above 7 would clearly be pretty hot!
• And 10 would be pretty special and rare

People (100 000s)
1
5
10
1
HOT
NOT
7
3
Rating
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Calculating Linda Vances Standardized Hotness
ratings
Calculate SH score by converting it to a new
score using the z-score to raw score formula X
? z?
Lindas z score 0.5
Vances z score 0.67
Use our SH mean of 5, instead of ?
Use our SH sd of 2, instead of ?
X 5 (0.5 x 2)
X 5 (0.67x 2) X 6.34 Vances SH Score is
6.34
X 6.0 Lindas SH Score is 6.0
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z-Scores for samples

• Weve been using z-scores to work out where
  someone is in a population.
• Just as easy to find out their position in a
  sample (a subset of the population)
• Substitute ? (population mean) for M (sample
  mean) and ? (population sd) for s (sample sd)
  into the same formula

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Time for an Extreme Makeover!

• Imaginary Experiment
• I go under the knife
• After 6 months recovery I put a photo of the new
  me on hotornot.com
• My new rating is 9.5!
• Money well spent!
• But what if my new rating was 8.5

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Effect of surgery

• z-scores can help us decide if treatments have an
  effect
• by telling is whether a treated score is now
  extreme
• i.e. unsual or rare if no treatment had occured

My score is not extreme and quite typical. Not
really worth the ,pain and suffering?
?1
My score is now extreme i.e. pretty rare.
Meaning I am now unusually hot! well spent!
?7.5
Xafter 8.5
Xbefore 8.0
Xafter 9.5
z 2.0
z 1.0
z 0.5
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z-scores for fun profit
www.massucci.com/Articles/Statistics/statistics.ht
ml
http//Store.traders.com/-v16-c01-002zsco-pdf.html
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Disclaimer Some ratings of hotness in this
lecture have been altered by the producers, to
protect the innocent, and may not reflect the
true ratings obtained. The mean and standard
deviation of ratings on hotornot.com reported in
the lecture have been modified for teaching
purposes. The producers have also made up a whole
bunch of other stuff but swear that all the
statistics in this lecture are true.