STA301 GDB Solution Spring 2012

Explain how measure of central tendency and measure of dispersion are complementary to each other to describe the data.


When data are described by a measure of central tendency (mean, median, or mode), all the scores are summarized by a single value. Reports of central tendency are commonly supplemented and complemented by including a measure of dispersion. The measures of dispersion you have just seen differ in ways that will help determine which one is most useful in a particular situation.Range Of all the measures of dispersion, the range is the easiest to determine. It is commonly used as a preliminary indicator of dispersion. However, because it takes into account only the scores that lie at the two extremes, it is of limited use.
Quartile Scores are based on more information than the range and, unlike the range, are not affected by outliers. However, they are only infrequently used to describe dispersion because they are not as easy to calculate as the range and they do not have the mathematical properties that make them so useful as standard deviation and variance.
The standard deviation and variance are more complete measures of dispersion which take into account every score in a distribution. The other measures of dispersion we have discussed are based on considerably less information. However, because variance relies on the squared differences of scores from the mean, a single outlier has greater impact on the size of the variance than does a single score near the mean. Some statisticians view this property as a shortcoming of variance as a measure of dispersion, especially when there is reason to doubt the reliability of some of the extreme scores. For example, a researcher might believe that a person who reports watching television an average of 24 hours per day may have misunderstood the question. Just one such extreme score might result in an appreciably larger standard deviation, especially if the sample is small. Fortunately, since all scores are used in the calculation of variance, the many non-extreme scores (those closer to the mean) will tend to offset the misleading impact of any extreme scores.
The standard deviation and variance are the most commonly used measures of dispersion in the social sciences because:
Both take into account the precise difference between each score and the mean. Consequently, these measures are based on a maximum amount of information.
The standard deviation is the baseline for defining the concept of standardized score or “z-score”.
Variance in a set of scores on some dependent variable is a baseline for measuring the correlation between two or more variables (the degree to which they are related).