Pearson International Edition

Chapter 3

Answers

3.56

Range is the simplest measure of variation. It is the difference between the largest and the smallest values:

Range = Xlargest – Xsmallest

The advantage of range is that it is very simple but the disadvantage is that it ignores the way in which data are distributed.

If we square the difference between each value and the mean and sum the squared differences, we find the sum of squares and if we divide this sum by the number of values minus 1, we get variance. It is the approximate average of squared deviations of values from the mean. The square root of variance is the standard deviation. Standard deviation is the most commonly used measure of deviation because it is always a number that is in the same units as the original sample data. For almost all sets of data, the majority of the observed values lie within an interval of plus and minus one standard deviation above and below the mean. So it helps to define where at least majority of the data values are clustering.

The coefficient of variation is a relative measure of variation, a percentage. It shows variation relative to mean. It is equal to the standard deviation divided by the mean, multiplied by 100%. The advantage of the coefficient of variation is that you may compare the variability of two or more data sets measured in different units.

3.58

The empirical rule approximates the variation of data in a bell-shaped distribution.

Approximately 68% of the data in a bell shaped distribution is within 1 standard deviation of the mean or .

Approximately 95% of the data in a bell shaped distribution is within 1 standard deviation of the mean or .

Approximately 99.7% of the data in a bell shaped distribution is within 1 standard deviation of the mean or .

The Chebyshev rule states that for any data set, regardless of how data are distributed, at least

(1 - 1/k2) x 100% of the values will