Math 151 , Spring 2006, Day 7  Mon. Feb. 13 Hit reload...After class

Errors in my book: Ch4p43 middle; "for men...a narrower broader peak..."
    Ch5p65top: Needs square root sign over formula (later printings are ok(?))
Day 7 (Mon. Feb. 13): Reading:  (Re)read D&V Ch5 (Re-expressing paragraph p.68 is optional, but don't miss any of What can go Wrong pp. 67-8) Re(do) AS Ch5 .  Changing units and Normal dist:  Start D&V Ch6 pp. 82-98.  (Normal Prob. Plots p. 94-95 is  Optional, but don't miss What Can Go Wrong, p95 bottom).  & AS Ch. 6, in order.Changing units D&V 84-5, AS 6-1 ¶activities 1&2
Hand in Wednesday:  SPSS work (Day 5:  See clarification, #D)  and these:
Ch5, p. 72 
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Mean/Median. 
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
+ + + + + + + + + + 
Continue the work you started last time on a separate page:  Keep this page if (B) below is not assigned tonight. (Hand in Wed.)
19, 20 (no computations needed.  19 d may not be decidable from pictures.  Don't worry about it.)
5 Mistake 
#9 Standard deviation First, make  dot plots of each pair on axes with the same unit size, find the mean of each set and mark it with a little ^ (like fig. 5.6 p. 64).  Notice this looks like a good balance point. Leave space to calculate  some standard deviations next time.  Also, make a dot plot of  #10b set 2 (10, 50, 60, 70, 110).  Which of the data sets in problem 9 does it most resemble?

#9 Standard deviation, finishing.   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 part a;  check b and c in the back of the book.  Also, dotplot and calculate the mean and  standard deviation for #10b set 2 (10, 50, 60, 70, 110).  Verify that Each number w in #10b set 2 is the number x in #9bset1 less 9, multiplied by 10.  (w=10(x-9))
+ + + + + + + + + + + + +

(Hand in the rest Wed.)
p110 ( Review probs)  #28 Pay
p. 77 #33 (use SPSS)
= = = = = = = = = = = = = = = = = 
Do  these, using shift & rescale(D&V 84-85) 
A) The U.S. is almost the only country left that uses Fahrenheit to measure temperatures. To change F to C (Celsius), you subtract 32, and divide by 1.8.  HANDOUT with both scales ("Alias").  Keep the handout. 
a)  The high temperature a few days ago was  500 F.  Calculate the temperature in C. (Check your calculation on the handout scale) 
b)  If the mean high temperature in Ithaca during  Feb. is 40o, and the standard deviation is 100 F, and you want those in Celsius instead, what do you do? Calculate  the results.  (Check your results on the handout scale.)
 B) 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,  be able to discuss
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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? 

p. 72 ff., #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.

38 Holes What is the problem here? The slow method looks better, mostly, but the summary values are worse! (Examine the data.) 

p.99 2e(effect of outlier) 

Demo:  AS5-3, Dotplot (see bottom of Std. dev. section, this page.)  Experiment.
 

 Op
tion
al 
 
Cluster:  Tell everyone your name, even if you think they know it.   
Check for SPSS problems?  Don't postpone...
    Other HW?

Ch.5 Summarizing distribution info with numbers
  Median, Quartiles, IQR, 5-number summary, Boxplot     Day 6
  HW questions?
Mean vs. Median  Day 6
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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 D&V p. 65 for formula; computation in a table in sidebar.  Demo:  1,1,2,4
                          AS6-2 2nd activity 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 (the outliers  contribute much more than their share to the Sum of Squared Deviations from the Mean)
  
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)
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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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