Measures of Dispersion: Range, Standard Deviation, and Variance

Chapter 3.2: Measures of Dispersion

Introduction

Measures of dispersion are statistical tools used to describe the spread or variability of a data set. Understanding dispersion helps in interpreting how data values differ from the center and from each other. The main measures discussed here are the range, standard deviation, and variance.

Range

Definition and Calculation

The range is the simplest measure of dispersion. It is calculated as the difference between the largest and smallest values in a data set.

• Formula: $R = \text{Largest Data Value} - \text{Smallest Data Value}$

• Interpretation: The range provides a measure of the total spread of the data but does not account for how the data are distributed between the extremes.

Example: Given the exam scores: 82, 77, 90, 71, 62, 68, 74, 84, 94, 88

• Largest value = 94

• Smallest value = 62

• Range: $R = 94 - 62 = 32$

Standard Deviation

Definition and Motivation

The standard deviation measures the average distance of each data value from the mean. It is a widely used measure of spread because it considers all data points and their deviations from the mean.

• Population Standard Deviation (σ): The square root of the mean of the squared deviations from the population mean.

• Sample Standard Deviation (s): The square root of the mean of the squared deviations from the sample mean, divided by (n - 1), where n is the sample size.

Formulas

• Population Standard Deviation: $\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}$

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

Degrees of Freedom: In the sample standard deviation, (n - 1) is used in the denominator. This is called the degrees of freedom, reflecting that only n - 1 values are free to vary when calculating the sample mean.

Example: Comparing Variability

Consider the temperatures on April 14th for 21 years in two cities:

City	Mean	Median	Standard Deviation
Des Moines, Iowa	53.95	54.7	11.05
San Francisco, California	53.96	53.8	3.15

Interpretation: Although both cities have similar means, Des Moines has a much larger standard deviation, indicating greater variability in temperature compared to San Francisco.

Example: Calculating Sample Standard Deviation

Given the sample: 62, 88, 77, 68

• Sample mean ($\bar{x}$): $\frac{62 + 88 + 77 + 68}{4} = 73.75$

• Sum of squared deviations: $(62-73.75)^2 + (88-73.75)^2 + (77-73.75)^2 + (68-73.75)^2 = 138.06 + 201.56 + 10.56 + 33.06 = 383.24$

• Sample standard deviation: $s = \sqrt{\frac{383.24}{3}} \approx 11.3$

Additional info: Calculations are shown for clarity; in practice, statistical software can be used for larger data sets.

Interpreting Standard Deviation

• A larger standard deviation indicates more spread out data.

• A smaller standard deviation indicates data are closer to the mean.

• Standard deviation provides a rough estimate of the typical distance of a data value from the mean.

Variance

Definition and Calculation

The variance is the square of the standard deviation. It measures the average squared deviation from the mean and is expressed in squared units of the original data.

• Population Variance: $\sigma^2$

• Sample Variance: $s^2$

Formulas

• Population Variance: $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$

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

Example: Calculating Variance

Data Set	Standard Deviation (s)	Variance (s2)
62, 88, 77, 68	11.3	127.69
20, 13, 4, 8, 10	6	36

Calculation: Variance is simply the square of the standard deviation. For example, for the first data set: $s^2 = (11.3)^2 = 127.69$.