Math 151 , Spring 2004, Monday, Feb. 9, Day 4 Hit reload to get most current version..After class

Changed Clinic hours for Faherty
HW Day 4: Reading: pp. 33-37 for 5-number summary, boxplots.  Pp. 37-40 for S.D.  Next, 1.3 (Normal Distributions), pp.46-62.  It'll take some getting used to--start now!
Hand in 
A.  You are driving down the freeway and note that the number of cars that pass you is the same as the number of cars that you pass.  Is your speed the median, the mean, or the midrange speed?  Justify your answer. 

p. 36 1.31 (C-sec, 5#, boxplots) also give the IQR for each set of doctors. 
1.33, SAT's, and 1.36 (p.41) boxplot, stemplot
p. 43ff  1.39 ( hotdogs, box)
p. 72, 1.79 (SUV's &midsize)
1.80 (guinea pigs) make a boxplot only, to hand in. Be able to discuss the rest .

Put on separate paper and keep for Day 5 HW
p. 40, 1.34 a. Graph the data with a dotplot.   Next time, use SPSS to  do c.  Write your answers from screen to paper.  I'll assign b next time.  (Lost handout? Link) 
1.35 (Maris HR) Graph the ten values with a dotplot. Use SPSS to do the calculations.  Just delete the outlier and repeat the analysis.

 Read, to discuss 
 
 
 
 
 
 
 

p. 74, 1.81a,b, 82(be sure you could compute it, don't bother to do it), &83.  (Walmart)Note how outliers are listed separately ("Low" and "High" ) rather than taking huge space for them in the stemplot.  A common practice. The outlier rule is also common for computer packages. 
(SPSS stemplot cheats--tells you they exist but not what they are!)

 Optional 
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--Handout for p. 35  (given out last time, get if you were absent)
Friday in class time you'll go to the lab and work intensively with SPSS.
--Handout for finding mean and standard deviation in SPSS
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Sec. 1.1:  Bimodal: often two important influences/populations.  Unimodal: sum of small independent influences.  Cf. M/F heights vs F heights.  Galton board--balls and pins (bell shaped "normal" distribution)
 Review Time plot. (pp. 17-19)Time on horiz. axis, values on vertical.
 -- Research data: time, or order of taking measurements, is often a lurking variable.  (Learning. Run-in time. Fatigue.)
                  Always do a time plot.

--HW questions?
   P. 69, 1.74. (Hospital discharge)
--Review measures of center
   In a skewed distribution, what is the relationship between mean and median?
       P. 45, 1.46, 1.47, 1.48
Measures of Spread (dispersion, variability)  distributions with different spreads
    Range:  largest - smallest.   Resistant?  NO!  Two observations carry all the info; the rest could be anywhere.
  Quartiles, five-number summaries, boxplot, InterQuartile Range. (HANDOUT)
   (Variance), Standard deviation.

Quartiles Divide data into quarters: 1st quartile Q1: 1/4 below, 3/4 above. = 25th percentile.
             (2nd quartile= median = 50th percentile)
            3rd quartile Q3: 3/4 below, 1/4 above.  = 75th percentile.

Computation of quartiles:  Different texts, packages use different methods.
By hand: We'll use Tukey's quick and dirty: (he called them "hinges")
Take the two halves of the data you got from finding the median.  Find the median of each half, using the same rule as before.  (Detail.  IF you had an even number of observations to start with, the data divides evenly into an upper and a lower half.  IF you had an odd number to start with, you have one in the middle, the median. In this case only, you throw the median away, and use the remaining halves.)
1 3 5 6 8 8 11 20, are n=8 observations.
    Median at (8+1)/2= 9/2=4 1/2th ;  1 3 5 6 | 8 8 11 20, M = 7
 8/2 = 4 in each half: Halves are 1 3 5 6, and 8 8 11 15.  The quartiles are the medians of each half; count in (4+1)/2= 2 1/2.  1 3 | 5 6,  Q1=(3+5)/2= 4.
8 8 |11 15. Q3= (8+11)/2= 9.5                                             1 3 | 5 6 | 8 8 | 11 20

1 3 5 6 6 8 8 11 20, are n=9 observations.
     Median at (9+1)/2=10/2=5th ; 1 3 5 6 6 8 8 11 20, M = 6
 Throw away the median.  Now we have an even number again, 8 numbers
8/2 = 4 in each half: Halves are 1 3 5 6, and 8 8 11 15.  Continue as before. (This is a  dirty method because it gives the same quartiles for both these data sets.  Quick because computation is minimal and simple.)
1 3 | 5 6 6 8 8 | 11 20

Five-number summary:  min, Q1, Median, Q3, max.  (1, 4, 7, 9.5, 20  for the set of 8 above)
INTERQUARTILE RANGE = IQR= Q3 - Q1. (9.5 - 4 = 5.5for both sets above)
      =The range of the middle half of the observations.  Resistant to outliers!
Box (and whisker) plot:  Graphical form of five number summary.
    Especially good for comparing sets of data.
"Plain vanilla" Draw and label the numerical scale first.  Then mark the five numbers. Finish the picture.
The box spreads over the middle half, the whiskers over the smallest and largest quarter.  Each section shows the spread of 1/4 of the data: the longer the section the thinner the data must be spread in there.
Demonstration.  Direction of boxplot?  Vertical or horizontal is a matter of taste.  I do horizontal, usually.
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Other measures of middle exist, tho are less common.  Midrange:  halfway between min and max.  Mode(Modal class): category with the most individuals.  Trimmed mean: Trim (throw away) a small proportion of largest and smallest.
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START HERE WED: Standard deviation (goes with mean)
Mean/standard deviation pair useful for symmetric, unimodal (one-humped), no outliers. ("Normal" dist.)
Handout : SPSS for mean and standard deviation
     Variance:  (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.  (We'll compute by hand next time.)
        Physics: angular momemtum (spinning ice skater)
        Not so weird: High school geometry?
   Very sensitive to outliers (squared  deviations do it)
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