Math 151 , Day 5 Monday, Sept.4, Fall 2006 Hit
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class
HW Day 5
Reading: pp.47-50- standard deviation, Check 2.19, 20, 21,
22:
PLEASE read ahead in Ch.3, 64-9 density curves, & ahead
Normal
Distributions 70-84:
There's a lot there, and I will cover a good chunk Friday
Hand in
A. You are driving on the thruway from Syracuse to Rochester and
keep track of how many vehicles you pass and how many pass you.
You find that these 2 numbers are the same. Your speed on the
thruway is: (a) the Mean spead of the cars, (b) the Median speed of the
cars, (c) the Modal speed of the cars. Choose one, and justify
your choice.
Standard deviation
**Where it says to use SPSS, you
may use SPSS(Didn't get handout?
Link), or a statistical calculator
if you have one, or the Applet, One Variable Statistical Calculator, on
the web http://www.whfreeman.com/bps or on the CD in your book.
B. Find the mean and standard
deviation of 2, 2, 4, 8 by hand.
p. 50, 2.9 Blood phosphate Do a and b by hand. Use SPSS**
to do c. Write your answers from screen to paper.
Also (re)make a dotplot of the data, mark the mean with a wedge, and
indicate the standard deviation s with <----> lines from
the mean to both sides, s long. (like the sketch below)
p. 51, 2.10 xbar=7.50, s = 2.03
the
same for both dist's. Don't do the calculations--just make stemplots
&
compare their shapes!
ALSO, type the data for Dataset B into SPSS**, excluding the
outlier of
12.50. Find and write down the mean and s.d. now. Compare
to xbar=7.50, s = 2.03 .
p.55, 2.12 Rainforest logging. Use the
4-step process, see below,
p. 53-5&/or inside front cover. Note that "state", the
first step, is usually "done"=the textbook statement of the problem.
The data are probably suitable for mean&
standard deviation, but we don't have the SPSS power to do them easily
yet, so use your hand methods. This is one where working together
with others can have real benefits, since it's pretty open-ended.
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Postpone to Friday: Complete
Handout on
Densities (get from outside my door if you missed class)
|
Read, to discuss
|
Optional
p. 62, 2.40, 2.43 Play with summary numbers. Use the
Applet, One variable statistical calculator; type data in at the Data
tab.
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+Next time (Wednesday) meet in Mac 101. SPSS
Bring Text, + whatever you store files on..
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Quartiles, five
number summary, boxplot, IQR
(Optional: Mechanical way to "decide" to boxplot an
outlier) Day
4
HW Questions?
Summaries
of Middle & Spread continued--"Systems:"
-- (Midrange, Range Very
sensitive to outliers--they use only the max and min!)
-- Median, IQR (+
Quartiles Q1, Q3,5-number summary), based on percentiles (j'th
percentile is > j% of the data)
-- Mean, StandardDeviation "y-bar"
(or "x-bar"), "s" (good for symmetric unimodal, no outliers)
Standard
deviation
(measure of Spread that goes with mean)
Variance s2:
(almost) average
of squared deviations from the mean.
(Divide by (n-1)
"degrees of freedom")
s :
Standard deviation is the square
root of
the
variance.
Computation: I will require you to know how to do it by hand for
4 or 5 observations
(see BPS4e p. 48-9 for formula & computation example.
Demo: 1,1,2,4, mean = 2, sum of squared deviations
= 6, variance = 2, s = 1.41
1,1,2,4,12, mean = 4, sum of squared deviations = 86, variance =
21.5, s = 4.64.
(Midcomputation check: Sum of deviations from the mean (before
squaring
each) always = 0 )
--Always > 0 (0 only if all observations are =)
--s units the same as those of the observations
(squared and squarerooted).
Physics: angular momemtum (spinning ice skater)
Not so weird: High school geometry?
Remember
Pythagorean theorem: c2
= a2 + b2:
hypotenuse of right triangle is also square root of a sum of squares.
Very
sensitive
to outliers (the outliers contribute much more than their
share to the Sum of
Squared Deviations from the Mean)
Mean and Standard Deviation are for
Symmetric
Unimodal distributions without big outliers.
(ideally "Bell-shaped" = Normal)
Got to here, more or less
SPSS, for simple computation: Handout
Organizing a statistical problem:
Four-step process (pp. 53-5, & inside front
cover)
State: the issue to be explored, question to be addressed
(real-world) (In hw problems, often already stated.)
Formulate: What statistical tools, measures,
analyses should we use to answer the question?
Solve: Carry out the process. (May need to
back up. Decide on mean, s.d., but stemplot shows badly
skewed: go back and decide on 5#summary instead.)
Conclude: Give the conclusion as it addresses the
real-world question/issue.
Begin p. 55, 2.12 in class in pairs. Decide what analyses to do;
start doing them (make a copy for each)
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * *
Start here Friday. Will quickly review
the above, then. Go to Day 7
Density curves,
BPS4e pp.64-69
GET handout HW sheet: "Density curves"
(When values can take on any of a continuous interval
of numbers)
Example: Spinner: Label edge with continuous values from
0 to 1. Spinning should produce 1/10 of all spins in each colored
sector.
Simulations of 500, 3000 spins show roughly true. More spins would get
closer.
Abstraction, idealized histogram ("Mathematical model") = Density
curve. Describes a theoretical distribution of data.
Any density curve: is a curve
--always on or above the horizontal axis
--has area exactly 1 underneath it.
This allows area to represent proportion of "histogram" between
specified values.
(We will assume the proportion of observations precisely
equal to a value is 0. "So proportion less than 2" is the
same
number as "proportion less than or equal to 2.")
Many, many density curves are possible, modeling many phenomena.
For the spinner, the density curve is "Uniform on 0 to
1".
If you have two spinners like this, spin both at once and add the
results--the
corresponding density curve is "triangular, symmetric, on 0 to 2"
A more complicated mechanism will produce data corresponding to
the
density
curve I have called "trapezoid, -1 to 2"
A very important one is the "normal" distribution family.
Median, mean, percentiles, standard deviation are defined for a
density curve in analogy to those for a histogram.
-- median has half of area below and half above.
-- mean is balance point. On the long-tail side of median if
distribution is skewed. Same as median if symmetric.
--First quartile has 1/4 of area below, 3/4 above. Etc. for others.
Many densities have tables to describe them.
Especially
tables showing area to the left of (below) a given value.
You will make and use tables for the simple distributions on the
handout.
These are similar to the table we will use to describe the normal
distribution.
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