5'RATIO VARIABLES

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5. RATIO VARIABLES   An interval variable is
called a ratio variable when it has no negative
levels and has a true zero. Such a variable
measures a quantity of something which is totally
absent when the score is zero. Both ratios of
scores and percentage statements are meaningful.
Also we can compute logarithms and square roots
of ratio variables.   Some simple examples have
to do with temperature scales. Fahrenheit and
Celsius temperature scales are interval, because
the same amount of heat is required to raise an
object's temperature one degree, regardless of
its initial temperature. But these temperature
scales are not ratio. Not only do they have
negative scores, but they do not measure a
quantity of something which is totally absent
when the score is zero. On the other hand,
absolute temperature scales measure the average
kinetic energy per molecule, when applied to a
gas. The customary absolute temperatures use the
same units as some common scale, but have the
zero set so cold that a crystal would no longer
vibrate. The formulas for Rankine and Kelvin
temperature are   Rankine Fahrenheit
491.69 Kelvin Celsius 273.16
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6. CHOOSING THE RIGHT METHOD   Different
graphical displays are appropriate for variables
on different scales. Pie charts and bar graphs
are appropriate for nominal variables that have
no more than about ten levels. A box plot works
only for an ordinal variable. A histogram is a
bar graph with bars listed in an increasing order
from left to right, so it works only for an
ordinal (or interval) variable. (A histogram can
be understood with a large number of
bars.)   Different summary statistics are
appropriate for variables on different scales.
With a nominal variable, we can report the mode
(most common level). With an ordinal variable,
we can also report the percentiles in a data set
(including the median, quartiles, minimum, and
maximum). With an interval variable, we can also
report the range, inter-quartile range, midrange,
mean, variance, and standard deviation.
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The range and standard deviation are both
nonnegative, and if either is zero, then all the
scores in the data set are the same. And the
bigger the range or standard deviation, the more
spread out the scores are. But the range
considers only the minimum and maximum scores,
while the standard deviation considers all scores
in a data set.   The histogram of an interval
variable is called symmetric about a level L
when it can be folded at L so the two parts of
the scale of levels come together and the two
sides of the histogram match. If a histogram is
symmetric, it is symmetric about a level which is
simultaneously the median and the mean the
median because half the subjects are below it and
half above it, and the mean because scores in the
histogram occur in paired strips equidistant from
the center L of symmetry.
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A histogram which is not symmetric is called
skewed. If the mean is higher than the median,
we say the histogram is skewed positively or
high. But if the mean is lower than the median,
we say the histogram is skewed negatively or low.
However, the mean cannot be farther from the
median than the amount of the standard deviation.
In any set of scores (data), whether it is a
population or a sample, we can find a mean and a
standard deviation. Then each raw score x has
a standard score   for a population, or
for a sample.   You may be asked about the
standard score for a population or for a sample.
If you are given the standard score z and the
appropriate mean and standard deviation, you
should be able to solve for the raw score x .
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Chebychev's rule says that the fraction of
scores within k standard deviations of the mean
(i.e., for which ) is at least
. The number k
is intended to be more than 1, but need not be an
integer. (Be especially familiar with the cases
k2 and 3.)