Measures of Variation in Statistics: Range, Variance, and Standard Deviation

Chapter Outline

• 2.1 Frequency Distributions and Their Graphs

• 2.2 More Graphs and Displays

• 2.3 Measures of Central Tendency

• 2.4 Measures of Variation

• 2.5 Measures of Position

Measures of Variation

Section 2.4 Objectives

• Find the range of a data set

• Find the variance and standard deviation of a population and a sample

• Use the Empirical Rule and Chebyshev's Theorem to interpret standard deviation

• Approximate the sample standard deviation for grouped data

• Use the coefficient of variation to compare variation in different data sets

Range

The range is the simplest measure of variation in a data set. It is defined as the difference between the maximum and minimum data entries.

• Formula: $\text{Range} = \text{Maximum data entry} - \text{Minimum data entry}$

• The range is a single value and is sensitive to outliers.

Example: If the starting salaries for Corporation A are 37, 39, 41, 41, 42, 44, 44, 45, 47, 52 (in thousands), then:

• $\text{Range} = 52 - 37 = 15$ (thousand dollars)

Variation

Variation describes how much the data values differ from each other. Two data sets can have the same mean but different levels of variation.

• Data sets with greater spread have higher variation.

• Graphs can help visualize variation between data sets.

Deviation, Variance, and Standard Deviation

These are more sophisticated measures of variation that consider how each data value differs from the mean.

• Deviation: The difference between a data entry $x$ and the mean $\mu$ (population) or $\bar{x}$ (sample).

• Variance: The average of the squared deviations from the mean.

• Standard Deviation: The square root of the variance; measures the typical amount an entry deviates from the mean.

Population Formulas

• Population Variance: $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$

• Population Standard Deviation: $\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$

Sample Formulas

• Sample Variance: $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$

• Sample Standard Deviation: $s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$

Steps to Calculate Population Variance and Standard Deviation

1. Find the mean $\mu$ of the population data set.

2. Find the deviation of each entry: $x - \mu$.

3. Square each deviation.

4. Add up the squared deviations: $\sum (x - \mu)^2$.

5. Divide by $N$ for variance; take the square root for standard deviation.

Steps to Calculate Sample Variance and Standard Deviation

1. Find the mean $\bar{x}$ of the sample data set.

2. Find the deviation of each entry: $x - \bar{x}$.

3. Square each deviation.

4. Add up the squared deviations: $\sum (x - \bar{x})^2$.

5. Divide by $n-1$ for variance; take the square root for standard deviation.

Example: Calculating Population Standard Deviation

• Given data: 37, 39, 41, 41, 42, 44, 44, 45, 47, 52

• Mean $\mu = 43.1$

• Sum of squared deviations $\sum (x - \mu)^2 = 85.9$

• Population variance: $\sigma^2 = \frac{85.9}{10} = 8.59$

• Population standard deviation: $\sigma = \sqrt{8.59} \approx 2.9$

Example: Calculating Sample Standard Deviation

• Given sample data: 4, 7, 6, 9, 5, 9, 8, 7, 9

• Mean $\bar{x} = 7.1$

• Sum of squared deviations $\sum (x - \bar{x})^2 = 28$

• Sample variance: $s^2 = \frac{28}{8} = 3.5$

• Sample standard deviation: $s = \sqrt{3.5} \approx 1.9$

Interpreting Standard Deviation

• Standard deviation measures the typical amount an entry deviates from the mean.

• The more spread out the data, the greater the standard deviation.

Empirical Rule (68-95-99.7 Rule)

For data sets with approximately normal (bell-shaped) distributions:

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

• About 95% falls within two standard deviations.

• About 99.7% falls within three standard deviations.

Example: If the mean height of women is 64.2 inches and the standard deviation is 2.8 inches, then about 68% of heights are between 61.4 and 67.0 inches.

Chebyshev's Theorem

Chebyshev's Theorem applies to any data set, regardless of distribution shape.

• At least $1 - \frac{1}{k^2}$ of the data values lie within $k$ standard deviations of the mean, for $k > 1$.

• For $k = 2$, at least 75% of data within 2 standard deviations.

• For $k = 3$, at least 88.9% of data within 3 standard deviations.

Standard Deviation for Grouped Data

When data is presented as a frequency distribution, use the following formula to estimate the sample standard deviation:

• $s = \sqrt{\frac{\sum f(x - \bar{x})^2}{n - 1}}$ where $f$ is the frequency of each class and $x$ is the class midpoint.

Coefficient of Variation

The coefficient of variation (CV) is a standardized measure of dispersion, useful for comparing variation between data sets with different units or means.

• Formula: $\text{CV} = \frac{s}{\bar{x}} \times 100\%$ (for sample), $\text{CV} = \frac{\sigma}{\mu} \times 100\%$ (for population)

• Higher CV indicates greater relative variability.

Summary Table: Measures of Variation

Measure	Formula	Interpretation
Range	$\text{Max} - \text{Min}$	Difference between largest and smallest values
Population Variance	$\sigma^2 = \frac{\sum (x - \mu)^2}{N}$	Average squared deviation from mean (population)
Sample Variance	$s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$	Average squared deviation from mean (sample)
Population Standard Deviation	$\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$	Typical deviation from mean (population)
Sample Standard Deviation	$s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$	Typical deviation from mean (sample)
Coefficient of Variation	$\frac{s}{\bar{x}} \times 100\%$ or $\frac{\sigma}{\mu} \times 100\%$	Relative variation (unitless, %)

Additional info:

• Standard deviation is always non-negative.

• Variance is in squared units; standard deviation is in the same units as the data.

• Empirical Rule applies only to approximately normal distributions.

• Chebyshev's Theorem is more general and applies to all distributions.