| 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 7): 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))
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.
|
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) See
Day 7
Very sensitive to
outliers--they contribute much more than their share to the Sum of
Squared Deviations from the Mean.
Mean and Standard Deviation are for
Symmetric
Unimodal distributions without big outliers.
(ideally "Bell-shaped" = Normal)
- - - - - - - - - - - - - - - - - - - - - - -
- -
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.
Start here Wednesday
&&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.
Optional
SPSS handout to create new computed variables.
= = = = = = = = = = = = = = = = = = = = = = =
= = = = = = =
GET handout HW sheet: "Tables for simple
models (densities)"
Models for quantitative variables
(AS6-2 ¶1)
(When values can take on any of a continuous interval of numbers)
Example: Spinner: Label edge with continuous values from
0 to 1. Spinning should produce 1/10 of all spins in each colored
sector.
Simulations of 500, 3000 spins show roughly true. More spins would get
closer to Uniform shape.
Abstraction, idealized histogram ("Probability Model") =
Density
curve. Describes a theoretical distribution of
data.
Any such model is a curve
--always on or above the horizontal axis
--has area exactly 1 underneath it.
Numerical summary:
(D&Vp.86)
Statistic from
data:
xbar
s
Q1
Median Q3
Parameter
for model :
µ
sigma Q1
Median
Q3
Many models have tables to describe them. Especially
percentiles
tables showing area to the left of (below) a given value
= theoretical proportion of observations below the value. 30%
below x, x is the 30th percentile).
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