Math 151 , Spring 2001, Monday, Feb. 5, Day 4 Hit reload to get most current version..

Range: (Max -min) (see day 3)

Quartiles  Divide data into quarters: 1st quartile: 1/4 below, 3/4 above.  (2nd quartile= median) 3rd quartile: 1/4 above, 3/4 below.

Computation of quartiles:  Different texts, packages use different methods.  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 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
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.)
 

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 =5.5 for 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.

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.

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Standard deviation (goes with mean)
Handout for SPSS
     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)
Mean/standard deviation pair useful for symmetric, unimodal (one-humped), no outliers. ("Normal" dist.)