data can be pictured as a histogram—a bar graph from the frequency distribution—or as
afrequency polygon—a line graph that replaces the bars with single points and connects
the points with a line. With a very large number of data points, the frequency polygon
approaches a smooth curve. Frequency polygons are shown in Figure 6.1.Measures of Central Tendency
Measures of central tendencydescribe the average or most typical scores for a set of
research data or distribution. Measures of central tendency include the mode, median, and
mean. The modeis the most frequently occurring score in a set of research data. If two
scores appear most frequently, the distribution is bimodal;if three or more scores appear
most frequently, the distribution is multimodal.Themedianis the middle score when the
set of data is ordered by size. For an odd number of scores, the median is the middle one.
For an even number of scores, the median lies halfway between the two middle scores. The
meanis the arithmetic average of the set of scores. The mean is determined by adding up all
of the scores and then dividing by the number of scores. For the set of quiz scores 5, 6, 7,
7, 7, 8, 8, 9, 9, 10; the mode is 7; the median is 7.5; the mean is 7.6. The mode is the least
used measure of central tendency, but can be useful to provide a “quick and dirty” measure
of central tendency especially when the set of data has not been ordered. The mean is gen-
erally the preferred measure of central tendency because it takes into account the informa-
tion in all of the data points; however, it is very sensitive to extremes. The mean is pulled in
the direction of extreme data points. The advantage of the median is that it is less sensitive to
extremes, but it doesn’t take into account all of the information in the data points. The mean,
mode, and median turn out to be the same score in symmetrical distributions. The two sides
of the frequency polygon are mirror images as shown in Figure 6.1a. The normal distribu-
tionor normal curve is a symmetric, bell-shaped curve that represents data about how many
human characteristics are dispersed in the population. Distributions where most of the
scores are squeezed into one end are skewed.A few of the scores stretch out away from the
group like a tail. The skew is named for the direction of the tail. Figure 6.1b pictures a neg-
atively skewed distribution, and Figure 6.1c shows a positively skewed distribution. The
mean is pulled in the direction of the tails, so the mean is lower than the median in a neg-
atively skewed distribution, and higher than the median in a positively skewed distribution.
In very skewed distributions, the median is a better measure of central tendency than the
mean.Measures of Variability
Variabilitydescribes the spread or dispersion of scores for a set of research data or distribu-
tion. Measures of variability include the range, variance, and standard deviation. The range
is the largest score minus the smallest score. It is a rough measure of dispersion. For the
same set of quiz scores (5, 6, 7, 7, 7, 8, 8, 9, 9, 10), the range is 5. Variance andstandard
deviation (SD)indicate the degree to which scores differ from each other and vary around
the mean value for the set. Variance and standard deviation indicate both how much scores
group together and how dispersed they are. Variance is determined by computing the dif-
ference between each value and the mean, squaring the difference between each value and
the mean (to eliminate negative signs), summing the squared differences and then taking
the average of the sum of squared differences. The standard deviation of the distribution is
the square root of the variance. For a different set of quiz scores (6, 7, 8, 8, 8, 8, 8, 8, 9,
10), the variance is 1 and the SD is 1. Standard deviation must fall between 0 and half the
value of the range. If the standard deviation approaches 0, scores are very similar to each
other and very close to the mean. If the standard deviation approaches half the value of the
range, scores vary greatly from the mean. Frequency polygons with the same mean and the56 STEP 4. Review the Knowledge You Need to Score High
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