3.2

standard deviation

A different formula is used to calculate the standard deviation s of a population: Instead of dividing by n - 1 for a sample, we divide by the population size N.

Round-off Rule for the Coefficient of Variation

Round the coefficient of variation to one decimal place (such as 25.3%).

Coefficient of Variation

The coefficient of variation (or CV) for a set of nonnegative sample or population data, expressed as a percent, describes the standard deviation relative to the mean, and is given by the following:

Empirical Rule for Data with a Bell-Shaped Distribution

The empirical rule states that for data sets having a distribution that is approximately bell-shaped, the following properties apply. •About 68% of all values fall within 1 standard deviation of the mean. •About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean

Chebyshev's Theorem

The proportion of any set of data lying within K standard deviations of the mean is always at least 1 - 1/K2, where K is any positive number greater than 1. For K = 2 and K = 3, we get the following statements: •At least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean. •At least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean.

Chebyshev's Theorem

The proportion of any set of data lying within K standard deviations of the mean is always at least 1 - 1/K2, where K is any positive number greater than 1. For K = 2 and K = 3, we get the following statements: •At least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean. •At least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean.

range rule of thumb

The range rule of thumb is a crude but simple tool for understanding and interpreting standard deviation. The vast majority (such as 95%) of sample values lie within 2 standard deviations of the mean.

Round-off Rule for Measures of Variation

When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data.

notation

s = sample standard deviation s2 = sample variance s = population standard deviation s2 = population variance

Range

the difference between the highest and lowest scores in a distribution

standard deviation

the square root of the variance