Communicating Quantitative
Information
Everybody to take the PSAT
Homework: Look up reports on school test
scores, especially trends. Assess reports.
Quiz Wednesday – Short Answer on
Definitions
Quick Review
• Statistics must always be analyzed
Qualitatively as well as Quantitatively –
what, if anything can they tell us.
• Batting Statistics (38 Separate Ones)-http://en.wikipedia.org/wiki/Baseball_Statis
tics#Batting_statistics
Quick Review II
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Purchase College Potential
Growth Ann. Increase 0.05 Year Population
2001 4000
2002 4200
2003 4410
2004 4631
2005 4862
2006 5105
2007 5360
2008 5628
2009 5910
2010 6205
2011 6516
2012 6841
2013 7183
2014 7543
2015 7920
Quick Review III
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Purchase College Potential Growth Ann. (show formulae -- <ctrl ~>
Increase 0.05 Year Population
2001 4000
=A6+1 =B6*1.05
=A7+1 =B7*1.05
=A8+1 =B8*1.05
=A9+1 =B9*1.05
=A10+1 =B10*1.05
=A11+1 =B11*1.05
=A12+1 =B12*1.05
=A13+1 =B13*1.05
=A14+1 =B14*1.05
=A15+1 =B15*1.05
=A16+1 =B16*1.05
=A17+1 =B17*1.05
=A18+1 =B18*1.05
=A19+1 =B19*1.05
Real story
• Fox Lane High School (Bedford Central School
District) meeting
• Principal announces: all juniors will take the
PSAT
– fee paid by school
– done during school hours
• [ A parent] says, "nice and everyone will accept
'scores' going down.
– What did Dr. Meyer mean? Why did she assume this
to be true?
Background: Measures of
Centrality
How to talk about a set of numbers? How to
compare sets of numbers?
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Mean
Median
Mode
Standard Deviation
Other ways, including charts
Mode
• value that occurs the most times
2, 4, 4, 4, 6, 7, 8, 9
The mode is 4
• can have multiple values
2, 2, 4, 4, 6, 7, 8, 9
modes 2 and 4
• Our little examples may not have a unique
mode—no instance repeated means each value
is a mode.
Mean (average)
… of N numbers is the sum / N
sum = t1 + t2 + …. tN
mean = sum/n
n * mean = sum
As if you had N occurrences of the mean
Examples
• What is the mean of: 30, 66, 78, 90?
• Same as the mean of:
• 60, 66, 78, 60
• 66, 66, 66, 66
• Do same for 48, 55, 75, 92
Mean, continued
• 30, 66, 78, 90 situation (mean is 66)
• If these are class grades (assuming equal
weighting) and you make 70 on the next
project, will your average go up, down or
stay the same? By how much?
Median
• Put the numbers in order
• If odd number of numbers, the median is the
middle number
• If even, the median is the mean of the two
middle numbers.
• The median is the number such that half the
numbers are >= and half the numbers are <=.
• It is the number in the middle
• Think of the median line strip in a road.
Median calculations
• 30, 66, 78, 90?
• The median is
– mean (average) of 66 and 78 is 72 (72 is 6
more than 66 and 6 less than 78)
• Median of 30, 66, 70, 78, 90 is 70
• Median of
– 66000, 120000, 450000, 478000, 1000000
– 800000, 450000, 1000000, 356000, 560000
Median vs Mean vs Mode
• No fixed relationship
• In so-called normal distribution, median,
mean and mode are the same
– The value that occurs the most (mode) is the
average value and is the value in the middle
when the values are sorted.
– Normal distribution also is a certain shape
Housing prices
• Median is preferred measurement.
• Why?
Housing, continued
• Typical situation is Westchester
– many months, there is one house sale of a
very expensive house. If sales other than this
sale are
– 300000, 350000, 410000, 420000, 550000,
600000, 660000
• Mean is ?
• Median is ?
Measures of centrality for example
• Mean is 470000
• Median is 420000
• Now, say one house sold for 3000000
300000, 350000, 410000, 420000, 550000, 600000, 660000, 3000000
• What is new mean and new median?
– median is average of 420000 and 550000
– mean is…. (3000000-470000)/8 + 470000
Housing, continued
• Mean is 786250
• Median is 485000
• What is a better indicator of sale prices of
houses?
[Young] Women earning more than
men in NYC
• Study was on MEDIANs
– Similar point can be made with mean, but not as
simple
• Underlying issue is that there are 2 times 2
populations (at least)
– Female college graduates, male college graduates,
other females, other males
• Posting opportunity: find original article by
Andrew A. Beveridge, Gotham Gazette,
summarize, explain, comment.
Standard deviation
• Measure of spread of data
• The range is the highest – lowest. The range also is a
measure of the spread.
– doesn't distinguish between one 'outlier' and many
• SD is roughly, the average distance from the mean
• Take the difference between each item and the mean.
Square it. Add. Divide by the number of items.
• Variance = (Σ(xi-m)2)/n)
– Squaring the difference makes entries less than mean contribute
the same as entries greater than mean.
• Standard Deviation is the square root of the variance
• Variance and Standard Deviation are each single
numbers telling us something about the data.
Standard Deviation Example
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Two Sets of Data
1,2,3,4,5
1,3,3,3,5
Range (highest – lowest)
5–1=4
5–1=4
Median
15 / 5 = 3
15 / 5 = 3
Yet the data is quite different
Standard Deviation Example II
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1,2,3,4,5
1,3,3,3,5
Subtract the data points from the Median
1 - 3 = -2
1 - 3 = -2
2 – 3 = -1
3 -3 = 0
3–3=0
3 -3 = 0
4–3=1
3 -3 = 0
5–3=2
5–3=2
The sum of the answers is 0
Standard Deviation Example III
• So we need another way – we square the
answers
• 1 - 3 = -2 = 4
1 - 3 = -2 = 4
• 2 – 3 = -1 = 1
3 -3 = 0 = 0
• 3–3=0=0
3 -3 = 0 = 0
• 4–3=1=1
3 -3 = 0 = 0
• 5–3=2=4
5–3=2=4
Standard Deviation Example IV
• The “Sum of the Squares” / n (the number
of items) is the Variance
10 / 5 = 2
8 / 5 = 1.6
• The Standard Deviation is the Square
Root of the Variance (Excel function:
=SQRT (cell reference)
• Square Root of 2 = 1.414213562;
of 1.6 = 1.264911064
Standard Deviation Example V
• The Variance formula used (The “Sum of
the Squares” / n (the number of items)) is
when the entire population is being
analyzed; if it is a sample (more on that
later), the formula is The “Sum of the
Squares” / (n – 1) (one less than the
number of items)
Standard Deviation Calculation
• Calculating Standard Deviation -http://www.youtube.com/watch?v=pFGcMI
L2NVo
• With Excel -http://www.youtube.com/watch?v=efdRmG
qCYBk&feature=related
Standard Distributions
• In a “Normal Distribution” of a lot of data,
68% of the data will fall between 1
Standard Deviation (+ / -) of the mean;
95% of the data will fall between 2
Standard Deviations (+ / -) of the mean;
99.7% of the data will fall between 3
Standard Deviations (+ / -) of the mean;
Distributions
• go back to looking at all the data
A distribution keeps track of how many
occurrences of each number (or each of a
set of ranges).
Class exercise?
• Height
• Hours watching TV since last class
– Reporting error?
• Number of CDs
• Change
• ??
Class exercise, continued
• Determine unit or range
• Chart graph
Normal distribution
• Many things but not everything! are
distributed normally
Dip
(inflection
point)
Median is
mean is
mode
Normal distributions
• Can be fat or thin ….
Smaller
variance/std
deviation
Larger
variance/std
deviation
These are
continuous
curves as if
there were
quantities at
every X point
Distributions
What does the set of numbers … look like?
• Normal
• Uniform = every value occurs the same
number of times
• Bi-modal = 2 normals next to each other
• Bath tub = upside down normal
• Or something else or nothing in particular
Two distributions
• 100000, 100000, 100000, 500000,
500000, 500000
– What is mean? What is median?
• 300000, 300000, 300000, 300000,
300000, 300000
– What is mean? What is median?
….two distributions
• Same 2 measures but very different
distributions
Back to the PSAT story
• Why did I say…scores would go down?
• It is/was not definite, but pretty likely…
• Population that chose to take the PSAT
when it was harder to take was more likely
to be better prepared.
• New procedure added….more at low end,
more 'low outliers'….
General principle
• Sample versus (whole) population
• The juniors who took the test when it was
an individual choice were a sample of the
population. The new policy was to include
the whole population.
• How can you characterize the (old)
sample?
– more partial to going to college….
Questions to ask
• What is the denominator: Is this a whole
population or a sample?
• If a sample, what are factors controlling
the sample?
• More on this later….
Real story: Library bond
• In the run-up to election day (Nov 8, 2005) for a
bond resolution
– Village of Mt. Kisco to borrow money to build new
library
• Issue: what will the cost be to the taxpayer to repay the bond?
• Answer: depends on [your home] assessment
– For example, home assessed at $33,000 (which is
the median assessment) would pay $124.
– If your home is assessed at more, you would pay
proportionately more, if less, you would pay less
Problem
• Definition and context
• Mt. Kisco has two systems of assessments:
Village and Town.
• Village is less than Town and both are much,
much less than 'market value'
• The $33,000 figure caused real confusion!
• Attempted to get reporter to either omit the
number OR say more in the article. Not
sucessful.
• Did succeed in getting costs of the status quo
(not building a new library) into news, publicity.
• www.mtklibraryfoundation.org
Puzzles
• 1) A bat and a ball cost $1.10 in total. The bat
costs $1 more than the ball. How much does the
ball cost?
• 2) If it takes five machines five minutes to make
five widgets, how long would it take 100
machines to make 100 widgets?
• 3) In a lake, there is a patch of lily pads. Every
day, the patch doubles in size. If it takes 48 days
for the patch to cover the entire lake, how long
would it take for the patch to cover half the lake?
Puzzles from study on risk
• New York Times article by Virginia Postrel
http://www.nytimes.com/2006/01/26/business/26sc
ene.html?_r=1
about study by Shane Frederick
mit.edu/people/shanefre/publications.htm
Getting answers right (in one study among college
students) correlated with willingness to take risk
Not clear if distinction was made regarding level
of risk.
Puzzle
• Sock drawer holds: 10 white socks, 10
black socks and 1 gray sock. What is the
maximum number of socks that can be
removed until getting a matching pair?
(a sample of size X guarantees a pair)
Puzzle
• 32 cards are dealt from a well-shuffled
deck of 52 cards. The deck contains 26
red and 26 black cards. What is the
difference between the number of black
cards among the 32 dealt and the red
cards remaining in the deck?
Homework
• Study Definitions for Quiz
• Keep up with postings.
– Find multiple sources on same topic
• Look up SAT or PSAT or other educational
tests and comment.