3-3 Measures of Variation

Empirical Rule

A concept helpful in interpreting the value of a standard deviation is the __________. This 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.

Standard Deviation of a Population

A slightly different formula is used to calculate the __________, denoted by σ (lowercase sigma): Instead of dividing by n − 1, we divide by the size N, as shown here: σ = [(∑( x − μ )^2)/N]^1/2

Notation

A summary of __________ for the standard deviation and variance includes: s = sample standard deviation s^2 = sample variance σ = population standard deviation σ 2 = population variance

Deviation

For a particular data value x, the amount of __________ is x - x-bar, which is the difference between the individual x value and the mean.

Properties of Standard Deviation (s)

Important __________ are: • It is a measure of how much data values deviate away from the mean. • The value is usually positive. It is zero only when all of the data values are the same number. (It is never negative.) Also, larger values of s indicate greater amounts of variation. • The value can increase dramatically with the inclusion of one or more outliers. • The units (such as minutes, feet, pounds, and so on) are the same as the units of the original data values. • The sample is a biased estimator of the population standard deviation σ.

Properties of Variance

Important __________ are: • The units are the squares of the units of the original data values. (If the original data values are in feet, it will have units of ft^2 ; if the original data values are in seconds, it will have units of sec^2.) • The value can increase dramatically with the inclusion of one or more outliers. • The value is usually positive. It is zero only when all of the data values are the same number. (It is never negative.) • The sample variance s^2 is an unbiased estimator of the population variance σ^2.

Range Rule of Thumb

The __________ is based on the principle that for many data sets, the vast majority (such as 95%) of sample values lie within 2 standard deviations of the mean.

Range

The __________ of a set of data values is the difference between the maximum data value and the minimum data value. X = (maximum data value) - (minimum data value)

Variance

The __________ of a set of values is a measure of variation equal to the square of the standard deviation.

Standard Deviation of a Sample

The __________, denoted by s, is a measure of how much data values deviate away from the mean.

Coefficient of Variation (CV)

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.

Sample (s)

The coefficient of variation for a __________ is given by CV = s/x-bar * 100%.

Population

The coefficient of variation for a __________ is given by CV = σ/μ * 100%.

Biased Estimator

The sample standard deviation s is a __________ of the population standard deviation σ. This means that values of the sample standard deviation s do not target the value of the population standard deviation σ.

Unbiased Estimator

The sample variance s^2 is an unbiased estimator of the population variance σ ^2, which means that values of s^2 tend to target the value of σ^2 instead of systematically tending to overestimate or underestimate σ^2.

Mean Absolute Deviation (MAD)

To get a statistic that measures variation (instead of always being zero), we need to avoid the canceling out of negative and positive numbers. One simple and natural approach is to add absolute values, as in ∑ | x − x-bar |. If we find the mean of that sum, we get the __________, which is the mean distance of the data from the mean: X = ∑ | x − x-bar | n

Population Variance

__________ is equal to the square of the population standard deviation σ. X = σ^2

Sample Variance

__________ is equal to the square of the standard deviation s. X = s^2

Chebyshev's Theorem

__________ states that the proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 − 1 / K 2 , 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