Math 151 , Day 5 Mon., Sept. 8, Fall '08 .. Hit reload to get most current version
HW Day 5: (Re)Read Ch.2 thru p. 47 Check: 2.15,17,18
(5#summary/boxplot). Read 53-55, "Organizing...".. New: pp.47-50-
standard
deviation, Check 2.19(don't calculate. It's not #a), 20, 21,
22.
Read in Ch.3,
64-9 density curves, & ahead
Normal
Distributions 70-84:
There's a lot there, and I will cover a good chunk Wed. or Friday
(?).
Hand
in:
Standard deviation
B. Find the mean and standard
deviation of these 4 numbers: 2, 2, 4, 8 by hand.
p. 50, 2.9 Blood phosphate Do a and b by hand. Use
SPSS or some other tool**
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 or other**,
excluding the outlier of 12.50. Find and write down the mean and
s.d. now. Compare to xbar=7.50, s
= 2.03 .
Ch 3, beginning:
Complete the Densities Handout (Link,
or get from outside my door if you missed
class) Solutions
p. 66, 3.1 Sketch density curves
p. 69, 3.2 & 3.3Uniform distribution This is the
same density as A on the Handout on Densities.
p. 69 3.4 means and medians
|
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.
|
**Where
it says to use SPSS, you may use SPSS (preferred) (Didn't get
handout? Link ), or a statistical
calculator if you have one, or the Applet, One Variable Statistical
Calculator, on the web http://bcs.whfreeman.com/bps4e
or on the CD in your book.
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Sign in on the clipboard. . Find someone you
don't know and introduce yourself. Introduce yourself to at least
one person you know, in case they've forgotten who you are.Note Class members is up. Check that
you're correct.
Compare HW with others, tell me unanswered questions, write #s on the
board.
Email me if you didn't
get an email from me Friday about the Canada/US problem.
Math Clinic General schedule is
up.
Handouts today: SPSS--Mean and SD
We'll have a whole day on SPSS, in the Mac 101 computer
lab, probably next class.
Next time (Wednesday) meet in Mac 101,
probably. SPSS! Bring
Text, + whatever you store files on.
(maybe Fri--watch this page,
+ email.)
Tables for Simple Models
(Densities))
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- -
- - - - - -
Another timeplot, on overhead, or Beer1.pdf
Homework
questions? Day 4
Quartiles, five number summary, boxplot, IQR.
Notes Day 3
4-step process (Day 4,
bottom)
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 "x-bar"
(or "y-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 )
--s is 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) Note contribution
of 12 -->
Mean and Standard Deviation are for
Symmetric
Unimodal distributions without big outliers.
(ideally "Bell-shaped" = Normal)
SPSS, for simple computation: Handout
Ch. 3, Density curves, BPS4e
pp.64-69
GET handout HW sheet:
"Tables for Simple Models (Densities)"
Handout )
(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.
(Histograms of simulations)
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
(familiar "bell-curve").
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 ("Cumulative Proportion").
You will make and use "Cumulative Proportion" tables for the
simple distributions on the
handout.
These are similar to the table we will use to describe the Normal
distribution.
Ahead:
Applet:
Normal Density Curve http://bcs.whfreeman.com/bps4e
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