Chapter 3: Measures of Variation in Statistics – Range, Standard Deviation, and Variance

Describing, Exploring, and Comparing Data

Introduction

This chapter focuses on the essential statistical concepts used to describe, explore, and compare data sets. The primary measures of variation discussed are range, standard deviation, and variance. Understanding these measures is crucial for interpreting the spread and consistency of data.

Measures of Variation

Round-off Rule for Measures of Variation

• When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data.

Range

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

• Definition: The range of a set of data values is the difference between the maximum and minimum data values.

• Formula: $\text{Range} = \text{maximum data value} - \text{minimum data value}$

• Properties:

  • Uses only the maximum and minimum values, making it sensitive to extreme values (not resistant).

  • Does not account for all data values, so it may not accurately reflect overall variation.

• Example: For wait times: 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20 $\text{Range} = 75 - 20 = 55.0$ minutes

Standard Deviation

The standard deviation measures how much data values deviate from the mean, providing a more comprehensive view of variation than the range.

• Notation:

  • $s$ = sample standard deviation

  • $\sigma$ = population standard deviation

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

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

• Properties:

  • Measures deviation from the mean.

  • Never negative; zero only if all values are identical.

  • Larger values indicate greater variation.

  • Units are the same as the original data.

  • Can be affected by outliers (not resistant).

  • Sample standard deviation $s$ is a biased estimator of population standard deviation $\sigma$.

• Example Calculation:

  1. Compute mean: $\overline{x} = \dfrac{430}{11} = 39.1$ min

  2. Subtract mean from each value: $x - \overline{x}$

  3. Square each deviation: $(x - \overline{x})^2$

  4. Sum all squared deviations: $\sum (x - \overline{x})^2 = 2740.91$

  5. Divide by $n-1$: $\dfrac{2740.91}{10} = 274.091$

  6. Take square root: $s = \sqrt{274.091} = 16.556$

  7. Final answer: $s = 16.6$ minutes

Shortcut Formula for Sample Standard Deviation

• Used by calculators and software for efficiency.

• Example: $s = \sqrt{\dfrac{30,150}{110}} = 16.6$ minutes

• Additional info: The shortcut formula typically uses sums of squares and sums of data values to simplify calculations.

Range Rule of Thumb

• Understanding Standard Deviation: Most (about 95%) sample values lie within 2 standard deviations of the mean.

• Identifying Significant Values:

  • Significantly low: $\mu - 2\sigma$ or lower

  • Significantly high: $\mu + 2\sigma$ or higher

  • Not significant: between $\mu - 2\sigma$ and $\mu + 2\sigma$

• Estimating Standard Deviation: $s \approx \dfrac{\text{range}}{4}$

Variance

Variance quantifies the spread of data by averaging the squared deviations from the mean. It is the square of the standard deviation.

• Sample Variance: $s^2 = (s)^2$

• Population Variance: $\sigma^2 = (\sigma)^2$

• Properties:

  • Units are the squares of the original units.

  • Can increase dramatically with outliers (not resistant).

  • Never negative; zero only if all values are identical.

  • Sample variance $s^2$ is an unbiased estimator of population variance $\sigma^2$.

Why Divide by (n - 1)?

• Only $n-1$ values can be freely assigned; the last value is determined by the mean.

• Dividing by $n-1$ ensures sample variances $s^2$ center around the population variance $\sigma^2$.

• Dividing by $n$ would underestimate the population variance.

Empirical Rule for Bell-Shaped Distributions

The Empirical Rule applies to data sets with approximately normal (bell-shaped) distributions.

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

• About 95% of values fall within 2 standard deviations of the mean.

• About 99.7% of values fall within 3 standard deviations of the mean.

Example: IQ scores with mean 100 and standard deviation 15: 2 standard deviations: $100 - 30 = 70$, $100 + 30 = 130$ About 95% of IQ scores are between 70 and 130.

Chebyshev’s Theorem

Chebyshev’s Theorem provides minimum proportions of data within $K$ standard deviations of the mean, applicable to any data distribution.

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

• For $K = 2$: At least 75% of values within 2 standard deviations.

• For $K = 3$: At least 89% of values within 3 standard deviations.

• Example: IQ scores, mean 100, standard deviation 15: - At least 75% between 70 and 130 - At least 89% between 55 and 145

Comparing Variation in Different Samples or Populations

The coefficient of variation (CV) expresses standard deviation relative to the mean, allowing comparison across different data sets.

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

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

• Round CV to one decimal place (e.g., 25.3%).

Biased and Unbiased Estimators

• Sample standard deviation $s$ is a biased estimator of population standard deviation $\sigma$.

• Sample variance $s^2$ is an unbiased estimator of population variance $\sigma^2$.

Notation Summary

Symbol	Meaning
s	Sample standard deviation
s2	Sample variance
σ	Population standard deviation
σ2	Population variance