68 ❯ STEP 4. Review the Knowledge You Need to Score High
orderly arrangement of scores indicating the frequency of each score or group of scores. The
data can be pictured as a histogram—a bar graph from the frequency distribution—or as
a frequency 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 tendency describe 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 mode is 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. The median is 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 mean is
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 generally
the preferred measure of central tendency because it takes into account the information in
all of the data points; however, it is very sensitive to extremes. The mean is pulled in the direc-
tion 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 distribution or
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 negatively 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 negatively skewed dis-
tribution, 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
Variability describes 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 and standard
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 the