Russel-Research-Method-in-Anthropology

Univariate Analysis 581
Both groups have an average age of 35, but one of them obviously has a lot
more variation than the other. Consider MFRATIO in table 19.8. The mean
ratio of females to males in the world was 100.2 in 2000. One measure of
variation in this mean across the 50 countries in our sample is the range.
Inspecting the column for MFRATIO in table 19.8, we see that Latvia had the
lowest ratio, with 83 males for every 100 females, and the United Arab Emirates
(UAE) had the highest ratio, with 172. The range, then, is 172 83
91. The range is a useful statistic, but it is affected strongly by extreme scores.
Without the UAR, the range is 27, a drop of nearly 70%.
The interquartile range avoids extreme scores, either high or low. The
75th percentile for MFRATIO is 101 and the 25th percentile is 96, so the interquartile
range is 101 96 5. This tightens the range of scores, but sometimes
it is the extreme scores that are of interest. The interquartile range of
freshmen SAT scores at major universities tells you about the middle 50% of
the incoming class. It doesn’t tell you if the university is recruiting athletes
whose SAT scores are in the bottom 25%, the middle 50%, or the top 25% of
scores at those universities (see Klein 1999).
Measures of Dispersion II: Variance and the Standard Deviation
The best-known, and most-useful measure of dispersion for a sample of
interval data is the standard deviation, usually written just s or sd. The sd is
a measure of how much, on average, the scores in a distribution deviate from
the mean score. It is gives you a feel for how homogeneous or heterogeneous
a population is. (We use s or sd for the standard deviation of a sample; we use
the lower-case Greek sigma, , for the standard deviation of a population.)
The sd is calculated from the variance, written s 2 , which is the average
squared deviation from the mean of the measures in a set of data. To find the
variance in a distribution: (1) Subtract each observation from the mean of the
set of observations; (2) Square the difference, thus getting rid of negative
numbers; (3) Sum the differences; and (4) Divide that sum by the sample size.
Here is the formula for calculating the variance:
2
xx
s 2
Formula 19.4
n1
where s 2 is the variance, x represents the raw scores in a distribution of interval-level
observations, x is the mean of the distribution of raw scores, and n is
the total number of observations.