Measures of Variation

Introduction to Measures of Variation

Measures of variation describe how data values are spread out or dispersed within a data set. Understanding variation is essential for interpreting the reliability and consistency of data, especially when comparing different data sets with similar measures of central tendency.

• Range: The simplest measure of variation, representing the difference between the maximum and minimum values.

• Variance and Standard Deviation: Quantify the average squared and average absolute deviation from the mean, respectively.

• Coefficient of Variation: Expresses standard deviation as a percentage of the mean, allowing for comparison between data sets with different units or means.

Range

The range is the difference between the largest and smallest values in a quantitative data set.

• Formula:

• Example: If the starting salaries for Corporation A are and , then .

Deviation, Variance, and Standard Deviation

These measures provide more detailed information about how data values differ from the mean.

• Deviation: The difference between a data entry and the mean (population) or (sample).

• Population Variance (): The average of the squared deviations from the mean for all entries in a population.

• Population Standard Deviation (): The square root of the population variance.

• Sample Variance (): The average of the squared deviations from the mean for a sample, divided by .

• Sample Standard Deviation (): The square root of the sample variance.

Formulas:

• Population Variance:

• Population Standard Deviation:

• Sample Variance:

• Sample Standard Deviation:

Example: For the data set 8, 10, 4, 6, 7, 7, 9, 10, 7, 6, 5, 11 (n = 12):

• Mean is calculated.

• Sample variance

• Sample standard deviation

Interpreting Standard Deviation

The standard deviation measures the typical amount an entry deviates from the mean. A larger standard deviation indicates more spread out data.

• If all entries are the same, the standard deviation is 0.

• As entries become more spread out, the standard deviation increases.

The Empirical Rule (68–95–99.7 Rule)

For data with a symmetric, bell-shaped (normal) distribution:

• About 68% of data lie within one standard deviation of the mean.

• About 95% within two standard deviations.

• About 99.7% within three standard deviations.

Example: If the mean height of women is 64.1 inches with a standard deviation of 2.6 inches, then about 68% of women are between 61.5 and 66.7 inches tall.

Chebychev’s Theorem

Chebychev’s Theorem applies to any data set, regardless of distribution shape. It states that at least of the data values lie within standard deviations of the mean (for ).

• For : At least 75% of data lie within 2 standard deviations.

• For : At least 88.9% of data lie within 3 standard deviations.

Example: If the mean age in Georgia is 41.7 years with a standard deviation of 20.85 years, at least 75% of ages are between 0 and 83.4 years old.

Standard Deviation for Grouped Data

When data are grouped into classes, estimate the mean and standard deviation using class midpoints and frequencies.

• Sample mean: , where is frequency and is class midpoint.

• Sample standard deviation:

Example: For a frequency distribution of number of children in 50 households, the sample mean is about 1.8 children and the sample standard deviation is about 1.7 children.

Coefficient of Variation (CV)

The coefficient of variation expresses the standard deviation as a percentage of the mean, allowing for comparison between data sets with different units or means.

• Population:

• Sample:

Example: If a basketball team has a mean height of 75 inches (standard deviation 3.4 inches) and a mean weight of 210 pounds (standard deviation 19.7 pounds):

• CV for heights:

• CV for weights:

• Interpretation: Weights are more variable than heights.