Chapter 4 Flashcards
Measures of variability
Numbers that describe diversity or variability in the distribution of a variable.
. The concept of variability has implications not only for describing the diversity
of social groups such as Asian American women but also for
issues that are important in
your everyday life
These
changes challenge us to rethink every conceptualization of society based solely on the
experiences of European populations and force us to ask questions that focus on the
experiences of different racial/ethnic groups. For instance,
we may want to compare the
racial/ethnic diversity in different cities, regions, or states or may want to find out if a
group has become more racially and ethnically diverse over time
Index of qualitative variation (IQV)
A measure of variability for nominal variables. It is based on the ratio
of the total number of differences in the distribution to the maximum number of possible differences within
the same distribution.
The index of qualitative variation (IQV) can vary from
0.00 to 1.00
It is important to remember that the IQV is partially a function of the number of
In this example, there were four and five racial/ethnic categories in Maine and
Hawaii, respectively. Had we used more categories, the IQV for both states would have
been considerably more (pg 184)
To summarize, these are the steps we follow to calculate the IQV:
- Construct a percentage distribution.
- Square the percentages for each category.
- Sum the squared percentages.
- Calculate the IQV using the formula
The IQV can also be expressed as a percentage rather than a proportion: how
Simply multiply
the IQV by 100. Expressed as a percentage, the IQV would reflect the percentage of
racial/ethnic differences relative to the maximum possible differences in each distribution.
t is estimated that by 2044, the United
States will be a minority-majority nation - explain
While the non-Hispanic white population will
still be the largest group, no racial or ethnic group will have the majority share of the
population.
for ex, We can use the IQV to measure the amount of
diversity in different regions
range
A measure of variation for interval-ratio variables. It is the difference between the highest (maximum)
and the lowest (minimum) scores in the distribution.
the range can also be found in
percentages
IQR d and purpose
To remedy the limitation of the range, we can employ an alternative—the interquartile
range. The interquartile range (IQR), a measure of variation for interval-ratio and ordinal
variables, is the width of the middle 50% of the distribution. It is defined as the difference
between the lower and upper quartiles (Q1 and Q3
).
IQR = Q3 − Q1
what is the first quartile?
Recall that the first quartile (Q1
) is the 25th percentile, the point at which 25% of the cases
fall below it and 75% above it.
box plot
A graphic device called the box plot can visually present the range, the IQR, the median, the lowest (minimum) score, and the highest (maximum) score.
The box plot provides us
with …
a way to visually examine the center, the variation, and the shape of distributions of
interval-ratio variables.
What can we learn from creating a box plot? (first two things)
We can obtain a visual impression of the
following properties: First, the center of the distribution is easily identified by the solid line
inside the box. Second, since the box is drawn between the lower and upper quartiles, the
IQR is reflected in the height of the box.
Similarly, the length of the vertical lines drawn
outside the box (on both ends) represents (in regards to a box plot)
the range of the distribution
Both the IQR
and the range give us… aso what does the relative position of the box and the position of the median within the box tell us ?
a visual impression of the spread in the distribution. Finally, the
relative position of the box and the position of the median within the box tell us whether
the distribution is symmetrical or skewed. A perfectly symmetrical distribution would have
the box at the center of the range as well as the median in the center of the box.
When the
distribution departs from symmetry,
the box and/or the median will not be centered; it will
be closer to the lower quartile when there are more cases with lower scores or to the upper
quartile when there are more cases with higher scores.
box plots are particularly useful for…
comparing distributions
IQR (illustrated by the height of the box) are much wider in the West (range =
36.2%; IQR = 14.85%) than in the Northeast (range = 18.9%; IQR = 8.7%), indicating
that there is more variability among states in the West than among those in the Northeast
Another reason for using
the mean as a reference point is
s that more advanced measures of variation require the use of
algebraic properties that can be assumed only by using the arithmetic mean
Variance
A measure of variation for interval-ratio and ordinal variables; it is the average of the squared
deviations from the mean.
Standard deviation
A measure of variation for interval-ratio and ordinal variables; it is equal to the square
root of the varianc
variance formula
The formula for the variance
can be stated as
This formula means
that the variance is equal to
the average of the squared deviations from the mean.
One problem with the variance is that it is based on
squared deviations and therefore is no
longer expressed in the original units of measurement. For instance, it is difficult to
interpret the variance of 36.27, which represents the distribution of the percentage change
in the elderly population, because this figure is expressed in squared percentages. Thus, we
often take the square root of the variance and interpret it instead. This gives us the standard
deviation, symbolized as s, is the square root of the variance
stdev
equal to the square
root of the average of the squared deviations from the mean.
what does the stdev mean
The actual number tells us very little by itself, but it allows us to
evaluate the dispersion of the scores around the mean
when is there no variation or dispersion in the scores
In a distribution where all the scores are identical, the standard deviation is zero (0). Zero is
the lowest possible value for the standard deviation; in an identical distribution, all the
points would be the same, with the same mean, mode, and median.
how many measures of variatioan to we tend to use
1
what are the five measures of variation
: (1) the IQV, (2) the range, (3) the
IQR, (4) the variance, and (5) the standard deviation
Another way to interpret the standard deviation is
to compare it with another distribution
a relatively low standard deviation for anything indicates
that this group is relatively homogenous
in wtv ur looking at (can see pg 206 for an example i think)
nominal level measures of vblility
With nominal variables, your choice is restricted to the IQV as a measure of
variability.
ordinal level measures of vbility- also which are preferred?
The choice of measure of variation for ordinal variables is more problematic.
The IQV can be used to reflect variability in the distributions of ordinal variables, but
because IQV is sensitive to the rank-ordering of values implied in ordinal variables, it loses
some information. Another possibility is to use the IQR, interpreting the IQR as the range
of rank-ordered values that includes the middle 50% of the observations.
14 For example, if
the IQR for income categories begins with the category $50,000 to $70,500 and ends with
the category $100,000 to $120,500, the IQR can be reported as between $50,000 and
$120,500. However, in most instances, social science researchers treat ordinal variables as
interval-ratio measures, preferring to calculate variance and standard deviation
interval ratio levels of measures of vibility- also which are preferred?
For interval-ratio variables, you can choose the variance, standard
deviation, the range, or the IQR. Because the range, and to a lesser extent the IQR, is based
on only two scores in the distribution (and therefore tends to be sensitive if either of the
two points is extreme), the variance and/or standard deviation is usually preferred.
However, if a distribution is extremely skewed so that the mean is no longer representative
of the central tendency in the distribution, the range and the IQR can be used. The range
and the IQR will also be useful when you are reading tables or quickly scanning data to get
a rough idea of the extent of dispersion in the distribution.
theres a good chart on page 208
ok lol
Pope’s analysis about mentoring is based on
a survey measuring student
perceptions about campus climate, institutional diversity, mentoring, and administrative
support of diversity
we present statements that measure mentoring from three different sources:
(1) staff, (2) peers, and (3) students themselves. The mean agreement scores and standard
deviations vary by each group, that is to say, perceptions about mentoring differ by racial or
ethnic identity.
The standard deviations indicate the variability of responses
A smaller
standard deviation reveals more consistency of responses clustered around the mean score,
whereas a larger standard deviation indicates more variation, more spread from the mean.
results of myron popes survey
The standard
deviations are all above 1.0, with the most variation in responses for the group of African
American students
To find the range,
subtract the lowest from the highest score in a distribution. For an ordinal variable
report the
lowest and the highest values without subtracting.
The box plot is a graphical device that visually presents the
e range, the interquartile range, the
median, the lowest (minimum) score, and the highest (maximum) score. The box plot provi
The variance and the standard deviation are
two closely related measures of variation for intervalratio and ordinal variables that increase or decrease based on how closely the scores cluster around
the mean. The variance is the average of the squared deviations from the center (mean) of the
distribution; the standard deviation is the square root of the variance.
