MATH 251, Probability and Statistics I, Fall 2001, Day 4

Standard deviation (goes with mean)
            Variance:  (almost) average of squared deviations from the mean.
                   (deviations sum to 0)
                 (Divide by (n-1) "degrees of freedom"--dimension of vector space)
         s : Standard deviation  is the square root of the variance.
                Computation:  I will require you to know how to do it by hand for up to 7 observations (use a table).
             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.)
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Linear transformations do not change the shape of a distribution :   A "good" measure of center or spread should "act naturally" if you change units of measurement by shifting (translating) or by changing scale (stretching or shrinking).
Measures of spread are unaffected by translating.
Page 57 gives the rules explicitly.  Problem A has you prove them for mean and standard deviation.
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1.3 Normal distributions.  Start with idea of Density function or curve: idealized histogram.  Area = relative frequency.
Any curve that is above the x-axis and has area exactly 1 under it can be thought of as the idealization of some set of observations, and can be called a Density curve.  We carry over our terms for shape, and our summary measures.

Day 4(Fri. Sept 7) Assigned: ( std. dev., linear transformations, densities)  Covers end of ch. 2, + pp. 66-9
Read for Wednesday's class, rest of 1.3 (Normal distributions), up to Normal quantile plots.
Meet in Mac 101 lab Monday.  Bring SPSS manual, and a disk.

Use SPSS handout for computation of mean and std.

and in:  1.53 (golf scores, s; use SPSS) 1.54 (Do xbar and s by hand.  Then put them in SPSS & do them.) p.96 1.113abc (ed scores "standardize")    (hint for a: make 2 equations and solve for a and b) Problem A below  p.85 1.69, 1.70, 1.71, 1.72 (unif. density)

Read, discuss  p. 94 1.107 (hosp. discharge) 1.55 (s.d. contest) B. In problem A below, you need b > 0. Where does this come in to the computation--what would happen if you used a b that was negative?.

Optional    1.65 (lin. transf)

A.  You have a data set x₁, x₂,... , x_n,  which has mean xbar and standard deviation s.
You make a linear transformation x_i*= a+b x_i, on each data point.  (The book uses x_new  instead of x*, p. 56)
a can be + or - , but b should be positive.  (In practical terms, negative b would "flip" the data, reversing the order.)
Show that the mean xbar* of the transformed data set = a + b xbar,  (last sentence of p. 57)
and that the standard deviation of the transformed data set,  s* = bs .
(If you are not skilled at working with big sigmas, do it for n = 3 and write out all the sums with +'s.)
Do the proof by starting with the formula for the mean expressed in the x_i*'s, e.g.  xbar*= (x₁*+x₁*+x₃*)/3.
Plug in  x_i* = a+b x_i, and work the algebra to arrive at the desired expression (a+b xbar) involving the mean of the x_i's.  Repeat for the  standard deviation formula.  Hint:  xbar* appears in the standard deviation expression:  substitute a +b xbar for it, since you already proved they were equal.