Descriptive Statistics: Measures of Variation (Chapter 2, Section 2.4)

Measures of Variation

Introduction

Measures of variation are essential in statistics for understanding how data values are spread or dispersed around the mean. This section covers the range, variance, standard deviation, and related concepts, providing tools to interpret and compare variability in data sets.

Range

The range is the simplest measure of variation, representing the difference between the maximum and minimum values in a data set.

• Definition: The range is the difference between the largest and smallest data entries.

• Formula: $\text{Range} = \text{Max. data entry} - \text{Min. data entry}$

• Requirement: Data must be quantitative.

• Example: For Corporation A, with salaries (in thousands): 41, 38, 39, 45, 47, 41, 41, 42, 44, 37, the range is $47 - 37 = 10$ ($10,000).

Variation in Data Sets

Variation describes how much the data values differ from each other. Even if two data sets have the same mean, median, and mode, their spread can be very different.

• Key Point: Greater variation means data values are more spread out from the mean.

• Example: Corporation B's salaries are more spread out than Corporation A's, as shown in their frequency distributions.

Deviation, Variance, and Standard Deviation

Deviation

• Definition: The deviation of a data entry $x$ is its difference from the mean.

• Population: $x - \mu$

• Sample: $x - \overline{x}$

Population Variance and Standard Deviation

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

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

• Properties:

  • Measures spread about the mean.

  • Always $\geq 0$; $\sigma = 0$ only if all entries are identical.

  • Units are the same as the data.

Sample Variance and Standard Deviation

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

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

Steps to Calculate Variance and Standard Deviation

1. Find the mean ($\mu$ for population, $\overline{x}$ for sample).

2. Find the deviation for each entry.

3. Square each deviation.

4. Sum the squared deviations.

5. Divide by $N$ (population) or $n-1$ (sample) for variance.

6. Take the square root for standard deviation.

Example Calculation (Population)

• Given $N = 10$, $\mu = 41.5$

• Sum of squared deviations: $SS_x = 88.5$

• Population variance: $\sigma^2 = \frac{88.5}{10} = 8.9$

• Population standard deviation: $\sigma = \sqrt{8.9} \approx 3.0$

Example Calculation (Sample)

• Given $n = 12$, $\overline{x} = 7.5$, $SS_x = 39$

• Sample variance: $s^2 = \frac{39}{11} \approx 3.5$

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

Using Technology

Statistical software and calculators can quickly compute mean and standard deviation from data tables.

• Example: Office rental rates in Los Angeles: mean $= 36.9$, sample standard deviation $= 17.4$.

Interpreting Standard Deviation

• Definition: Standard deviation quantifies the typical deviation from the mean.

• Greater spread in data results in a larger standard deviation.

• Example: Data sets with identical means can have different standard deviations depending on their spread.

Estimating Standard Deviation

• If all entries are the same, $\sigma = 0$.

• If entries deviate by $\pm 1$, $\sigma = 1$.

• If entries deviate by $\pm 1$ or $\pm 3$, $\sigma \approx 2.2$.

Empirical Rule (68–95–99.7 Rule)

For bell-shaped (normal) distributions:

• About 68% of data within 1 standard deviation of the mean.

• About 95% within 2 standard deviations.

• About 99.7% within 3 standard deviations.

Example:

• Mean height of women (20–29): $64.2$ inches, $s = 2.9$ inches.

• Heights between $58.4$ and $64.2$ inches (2 standard deviations below mean): about $47.72 ext{%}$ of women.

Chebychev’s Theorem

Applies to any data set, regardless of distribution shape.

• At least $1 - \frac{1}{k^2}$ of data lies within $k$ standard deviations of the mean ($k > 1$).

• For $k = 2$: at least $75 ext{%}$ within 2 standard deviations.

• For $k = 3$: at least $88.9 ext{%}$ within 3 standard deviations.

• Example: For Georgia age data, at least $75 ext{%}$ of residents are between $0$and $81.9$ years old.

Standard Deviation for Grouped Data

• For frequency distributions, use class midpoints to estimate mean and standard deviation.

• Formula: $s = \sqrt{\frac{\sum f(x - \overline{x})^2}{n-1}}$ where $f$ is frequency, $x$ is class midpoint, $n$ is total entries.

• Example: Number of children per household: sample mean $\approx 1.8$, sample standard deviation $\approx 1.7$.

Coefficient of Variation (CV)

• Definition: CV expresses standard deviation as a percentage of the mean, allowing comparison of variability between different data sets.

• Population: $CV = \frac{\sigma}{\mu} \times 100\%$

• Sample: $CV = \frac{s}{\overline{x}} \times 100\%$

• Example: Basketball team heights: mean $= 72.8$ in, $\sigma = 3.3$ in, $CV = 4.5\%$; weights: mean $= 187.8$ lb, $\sigma = 17.7$ lb, $CV = 9.4\%$. Weights are more variable than heights.

Summary Table: Key Formulas

Measure	Population Formula	Sample Formula
Variance	$\sigma^2 = \frac{\sum (x - \mu)^2}{N}$	$s^2 = \frac{\sum (x - \overline{x})^2}{n-1}$
Standard Deviation	$\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$	$s = \sqrt{\frac{\sum (x - \overline{x})^2}{n-1}}$
Coefficient of Variation	$CV = \frac{\sigma}{\mu} \times 100\%$	$CV = \frac{s}{\overline{x}} \times 100\%$

Additional info: These notes are based on textbook slides and cover all major objectives for Section 2.4, including examples, formulas, and interpretation strategies for measures of variation in statistics.