| Hand in (All D&VCh5
p. 72ff except as noted)
Mean/Median. (from Day4) 7a,b,c,Payroll Also, with c: What measure would be most useful if you wanted to use it to figure the total weekly payroll cost? 6 Sick days + + + + + + + + + + (Problems Continued from Day 4): 19, 20 (no computations needed. 19 d may not be decidable from pictures. Don't worry about it.) 5 Mistake (finish it) 9 Standard deviation You made dot plots of each pair on axes with the same unit size, found the mean of each set and marked it with a little ^ (like fig. 5.6 p. 64). Note this is the balance point. Which of each pair has the bigger "spread"? Calculate standard deviations by hand for parts a & & & & & & & & more: Review probs p110#28 Pay 33 (use SPSS) Use the Dotplot Tool (AS5-3--see below for details.) = = = = = = = = = = = = = = = = = Postpone to Day 7 asst., but Try these, using shift & rescale(D&V 84-85) A) See Ch.5 p.72 #9and10, above. Pair #9 c is a shift. Check that the mean shifts correctly and that the s.d. stays the same (use the back of the book and your picture.) Pair #9bset1 and #10b set 2 is a shift followed by a rescale. (w=10(x-9)). Check that the mean undergoes the shift and rescale, but the s.d. undergoes just the rescale. Ch6 p. 99: 1 Payroll (hint for d: each employee gets 110% of previous) 3 (SATtoACT) |
Read,
to discuss
http://www.whfreeman.com/scc or http://bcs.whfreeman.com/ips5e. Under Student Categories or Student tools, choose "Statistical Applets", Mean &Median . (50 points max.)Check out symmetric, skewed, distributions with outliers. How far apart can you get the mean and median? 13 Marriage age. Ithaca Journal Jan 22, '05 had quiz answers:
"How old is the average bride? 24.5 years.... How old is the average groom?
26.5 years." Give some reasons that could account for the big difference
between these numbers and the graphed numbers
p.99 2e(effect of outlier)
|
Op-
tion- al |
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 D&V p. 65 for formula; computation in a table in sidebar.
AS6-2 2nd "paragraph" models computation, not in a table ).
(Midcomputation check: Sum of deviations from the mean (before
squaring
each) always = 0 )
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 (squared deviations do it)
Mean/standard deviation pair useful
for symmetric, unimodal (one-humped), no outliers. ("Normal" dist.)
Demo: AS5-3,
click on top righmost button (red dots on yellow background, "Tool1")
After the Dotplot Tool opens, right-click to get menu. In it,
Click on (choose) Show Buttons, Centers, Spreads, Mean Graphically,
Standard Deviation Graphically. Then you can click in the yellow
area to add points. You can drag points. Check that the dark band
shown goes from 1 s.d. below the mean to 1 s.d. above the mean. Experiment,
especially with the results of outliers on the S.D. & IQR (You
can show only one middle, one spread graphically at a time but you can
right click and change which one at any time.)
Mean and Standard Deviation are for Symmetric
Unimodal distributions without big outliers.
(ideally "Bell-shaped" = Normal)
- - - - - - - - - - - - - - - - - - - - - - -
- -
Start here Monday D&V
Ch. 6, AS 6
Standardizing an observation
or value. New ruler:
Make the mean the baseline (0) and measure
in units one standard deviation wide.
Standardized value = "z-score" = # of standard deviations
above the mean
"raw" y becomes z = (y -ybar)/s
p. 83
Find z: Subtract the mean from
y . Now you know how far "above" the mean y is, in "raw" units. (If
it's below the mean, the number will be negative.)We "shifted" it.
Find how far this is in "standard deviations" by dividing by the standard
deviation. (We "rescaled" it. That's the z-score.
Changing units: (D&V 84-85,
AS 6-1 ¶paragraphs 1&2)
Variable: your Heights. Units = inches. Change
this:
1) Shift: Take 5 feet = 60" as the new baseline:
60" =0 inches above 5 feet. How? Subtract 60 from each value.
y-60.
2) Rescale: Change to cm. How?
1" = 2.54cm. Multiply each value by 2.54. y*2.54 (x
or /? Need more centimeters for the same length, so multiply.
Or a non-American might know 1 cm = .394 inches, and divide by .394,
the length of a cm measured in inches.)
( +
shift ) Measures of middle should shift along with the
raw data. Measures of spread are unaffected by +
( x/
rescale) Measures of
middle and of spread
should stretch or shrink along with raw data (We assume
we only multiply/divide by positive numbers.)
To recalculate: Do the same thing to measures of middle
as you do to raw data.
To spreads, just do the multiplying or dividing part.
Shapes (skewness, humps, clumps, outliers) are not
affected by shifts and rescaling.
&&Alias/alibi: When you change units of measurement
for all your data values, you can think of the result 2 different ways:
"Alias (other name)": The data distribution
sits still. You have just changed the ruler stick you measure by.
(in/cm ruler. Thermometer)
"Alibi (other place)" : The ruler stick keeps
the 0 the same and 1 the same width, and the data distribution with "new"
values moves to the new location. D&V pp. 84-5.
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