Download In order to discuss the variables in the study, they must first

1
In order to discuss the variables in the study, they must first be examined individually by
measures of Univariate Descriptive Statistics. Univariate statistics examines the measures of
central tendency, dispersion, as well as the shape of the curve. These measures allow us to
discuss how each case in the variable compares to itself.
First, the measures of central tendency of the curve made by the data will be discussed.
The measures of central tendency include the mean, median and mode of the curve. The mean is
the average response given in a data set. In a perfect bell shaped curve this will be in the exact
middle and the rest of the responses are around it, 50% above and below. As seen in Table 1, the
variable regarding whether or not one was for or opposed to preferential hiring, the average
responds was 1.57. This means the average was between opposed and strongly opposed to
preferential hiring to blacks. The average income bracket of the respondents in this set was
13.46. Therefore the average income of the respondents was around $22,500. For the
respondents in this data set, the average total years of education were 13.91 years, or essentially
2 years of college education.
For qualitative variables, only the mean is taken into account. The mean for each part of
the qualitative variable is its percent of the total respondents. With regards to the gender of the
respondents, 48.5% were male, and 51.5% were female. The majority of the respondents were
white, with 81.22%, and only 18.78% being non white.
The next kind of measure of central tendency is the median. The median is similar to that
of mean, but it is the value in the exact middle of the values weather it be perfectly bell or not. If
the curve is perfectly bell shaped, then this will be the same value as mean. But if the shape is off
to one side more then the other, these will no longer be the same. Something that could affect
this would be outliers in a data set. The median of feelings towards preferential hiring of blacks
2
over whites is 1.00, as seen in Table 1. As we can see, the median is less then that of the mean.
The mean and median could be used as a comparison to see how skewed the data set is in one
direction or the other. In this case, the mean is higher then the median, therefore the set can be
said to be positively skewed, or more to the left. The median income for the respondents was in
the 14th bracket. This means that from $22500 to 24999, are the middle values in this case.
Because median bracket is higher then the mean, it can be said to be negatively skewed. The
median years of education were 14 years, or roughly 2 years of education. Similarly to income,
because the median is lower then the mode, it can be said to have negative skewness.
The final measure is the mode. This value is simply the most frequently occurring value
in the data set. Once again, if this is a perfectly distributed data set, this will be with the mean
and median. But as in most cases, there will be some skewness. Therefore this will be to the left
of the mean if it is positively skewed, and to the right if negatively skewed. The most occurring
response for the feelings toward preferential hiring of blacks over whites was strongly opposed,
or 1 (see Table 1). With regards to income, the most occurring income bracket was $40000 to
49999. This is so slightly different from the mean and median, which could mean that it is
skewed to the right of the mean. Finally, the mode years of education are 12 years, or roughly 1
year in college.
The second type Univariate Statistics discussed will be the dispersion of the data. This
includes the range and the standard deviation. The range is the difference between the minimum
value and the maximum value. This has limited utility because it does not really show how
spread out the cases are. The problem with the range is it could be greatly affected by outliers if
the cases are not limited to a certain range in the beginning. The maximum response for the
3
feelings towards preferential hiring is strongly for it, and the minimum is strongly opposed as
shown in Table 1. The range income bracket is 22, which ranges from under $1,000, to over
$110,000. This allows for a wide range of response brackets in-between. The range of years of
education for the respondents is 0 to 20 years of education, giving it a range of 20 years. This
range covers from no years of formal education through graduate school.
The second measure of dispersion is the standard deviation. Essentially the standard
deviation is the square root of the sum of the squared range, divided by the sample size. After
doing this calculation, the standard deviation describes, on average, how far the scores from the
mean are. This can be used to determine its significance of the values to the set. The standard
deviation is measured in sections where the first set contains 68% of the values, then 95%, and
finally 99.7% The standard deviations can continue to expand but after three standard deviations,
there is minimal use for it. With regards to the feelings towards preferential hiring, the standard
deviation is 0.87 (see Table 1). This means that 68% of the data falls between 2.44 and 0.7, or
oppose and strongly opposed. For income, 68% of the respondents fall between 18.6 bracket and
8.32 bracket, or $50,000 and $8,000, with a standard deviation bracket of 5.14. Finally the
standard deviation of the education of the respondents is 2.62 years. This makes 68% of the
respondents between 16.53 years and 11.29 years, or junior year in high school through 4 and a
half years of college.
The final type of measure used for Univariate Statistics is the shape of the curve for each
variable. The shape refers to the skewness and kurtosis of a curve. Skewness refers to weather
the values in the data set are shifted more to the right or left with elongate tails in the opposing
direction. If the curve is positively skewed, it will have more data on the left side and a long
4
thinner tail to the right. This can be caused by outliers on the right. Similarly, negative skewness
is where the tail extends to the left. As discussed above, the difference between the mean and the
median is a simple way to determine general skewness of a curve. Skewness will be determined
with the variable of 0.8 to -0.8 as calculated by the data in this study. The feelings towards
preferential hiring of blacks over whites seem to be positively skewed as stated prior. This is
confirmed because the calculated skewness for this variable is 1.72, which is greater then 0.8 and
therefore has a significant positive skewness (see Table 1). For income of the respondents, the
skewness is -0.55, which is between -0.8 and 0.8, so is therefore not significantly skewed but is
still slightly more negative. As for education, the skewness is very minimal with -0.006.
Therefore education is not show to be significantly skewed.
The second determination of shape is the kurtosis the curve. Kurtosis refers to the
measure of the proportion of the distribution in the tails. Essentially kurtosis refers to how flat or
spiked the curve is. With regards to the feelings towards preferential hiring, this curve is said to
be more leptokurtic because its kurtosis is measured to be 2.280, as show in Table 1. Leptokurtic
is if the data clusters around the mean, giving it little variation. A variable is said to be
leptokurtic if its measured value is greater then 0.8. On the opposite end of the scale, Platykurtic
refers to a curve that is much more evenly distributed and is much more flat in shape. If the
calculated measure is less then -0.8 then the curve is more Platykurtic and has more variation.
The shape of income is said to have no significant kurtosis because it is only -0.28, which is
between 0.8 and -0.8. It is slightly more Platykurtic but still not significantly. Finally, the shape
of the education variable is more significantly more Leptokurtic with a value of 1.1 with respect
to kurtosis. This means it has much less variation around the mean.
5
Table 1: Univariate Descriptive Statistics
Variable
Y AFFIRMAC
Mean Median Mode
S.D.
Range Skewness Kurtosis Minimum Maximum
1.57
1.00
1
0.87
3
1.718
2.28
1
4
X1 Income
13.46
14.00
18
5.14
22
-0.548
-0.28
1
23
X2 Education
13.91
14.00
12
2.62
20
-0.006
1.10
0
20
X3 Male
0.490
X4 Female
0.515
X5 White
0.812
X6 Non White
0.187
Source: General Social Survey, 1998.