CHUCK TUCKER 970421
I am behind on my reading but I did notice comments on this paper
(does anyone mind posting the paper on CSGNET- I don't have all
those fancy browers - still have gas powered computer!) but I
wondered how what is said by Kennaway that is not implied by
the stuff below that I have posting on all the lists that talks
about statistics (especially sociology lists). BTW, if you want
some examples of the misuse of statistics just examine ANY sociology
sociology journal. Regards, Chuck
A TABLE SHOWING THE RELATIONSHIP BETWEEN
SEVERAL DESCRIPTIVE STATISTICS*
···
____________________________________________
r r2 k2 k E
____________________________________________
1.00 1.00 .00 .00 100 %
.9995 .999 .001 .032 97 %
.9987 .997 .003 .054 95 %
.995 .99 .01 .099 90 %
.954 .91 .09 .299 70 %
.90 .81 .19 .435 56 %
.87 .756 .244 .493 51 %
.865 .748 .252 .50 50 %
.80 .64 .36 .60 40 %
.75 .56 .44 .66 34 %
.71 .50 .50 .70 30 %
.65 .42 .58 .76 24 %
.60 .36 .64 .80 20 %
.55 .30 .70 .83 17 %
.50 .25 .75 .87 13 %
.45 .20 .80 .89 11 %
.40 .16 .84 .92 8 %
.35 .12 .88 .94 6 %
.30 .09 .91 .95 5 %
.25 .06 .94 .97 3 %
.20 .04 .96 .98 2 %
.15 .02 .98 .99 1 %
.10 .01 .99 .995 0 %
.00 .00 1.00 1.00 0 %
DEFINITION AND INTERPRETATION OF THESE STATISTICS**
All of these measures describe two variables (X, Y) within a particular
sample:
r is a correlation (or coefficient of correlation) which
describes the linear association of one variable with another.
It can also be characterized as "... a relative measure of the
degree of association between two series " of values for two
variables. It varies between 1 (perfect positive correlation) to
-1 (perfect negative correlation). The closer this measure is to
a perfect correlation the more confidence one has in "predicting"
the values of one variable from another variable.
r2 is a measure of "explained" variance (or coefficient of
determination) which describes "shared" variation or the amount
of variance that one variable is "explained" by the other
variable or the proportion of the sum of y2 that is dependent on
the regression of Y on X. The larger the numerical value of this
measure the more confidence one has in "predicting" the values of
one variable from another.
k2 is a measure of "unexplained" variance (or coefficient of
nondetermination) which describes "unshared" variation or the
amount of variance that one variable is NOT "explained" by the
other variable or the proportion of the sum of y2 that is
independent of the regression of Y on X. The smaller the
numerical value of this measure the more confidence that one has
in "predicting" the values of one variable from another.
k is a measure (called coefficient of alienation) which
describes the lack of linear association of one variable with
another or the ratio of the standard error of estimate to the
standard deviation of the variable. The smaller the numerical
value of this measure the more confidence one has in "predicting"
the values of one variable from another.
E this measure is computed by (1-k)100 and is called an "index
of forecasting efficiency" (Downie and Heath, 1965: 226) and
indicates the "improvement" for a prediction by knowing the
coefficient of correlation (r) for two variables as contrasted
with knowing nothing about the linear association of the two
variables. For example, with a coefficient of correlation of .71
one can "predict" the values of one variable from another 30%
better (on the average) than one could "predict" those values
WITHOUT any knowledge of the relationship between the two
variables OR one has decreased the size of the "error of
prediction" by 30% (on the average) by knowing that the
correlation of the two variables is .71.
REFERENCES
Arkin, Herbert and Raymond R. Colton. 1956. Statistical
Methods. College Outline series, Fourth Edition, Revised.
Downie, N. M. and R. W. Heath. 1965. Basic Statistical
Methods. Second Edition. New York: Harper and Row.
______________________________
*compiled by Charles W. Tucker with the encouragment and
assistance of the Control Systems Group (especially Gary Cziko)
and the comments of Jimy Sanders. Other comments appreciated -
N050024 AT UNIVSCVM.BITNET
**It should be noted that these descriptions and interpretations,
especially those involving "predictions" are limited to a
particular sample; if another sample is not a random sample from
the same population then predictions about the other variable
("Y") will be unpredictably worse than the original sample.