CF

Measures of Dispersion — Section 3.2 Notes

Range

  • Definition: The range measures the spread between the smallest and largest values in a data set.

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

  • Example: Data set 125, 175, 200, 225, 250

    • Largest = 250, Smallest = 125

    • Range = 250 − 125 = 125

  • Pros and cons:

    • Pros: Very simple and quick to compute

    • Cons: Only uses two data values (extremes)

    • Susceptible to extreme values; NOT resistant to outliers

Standard deviation

  • Concept:

    • Based on deviations from the mean

    • Deviations sum to zero by definition

    • Describes the "typical" deviation from the mean

    • Other options (MAD, variance) exist but are less intuitive for describing spread around the mean

  • Population standard deviation

    • Formula: \sigma = \sqrt{\frac{1}{N} \sum{i=1}^{N} (xi - \mu)^2}

    • Interpretation: Average squared deviation from the mean, then take the square root

  • Sample standard deviation

    • Formula: s = \sqrt{\frac{1}{n-1} \sum{i=1}^{n} (xi - \bar{x})^2}

    • Uses n−1 in the denominator to make the estimator unbiased for the population SD

  • Hand calculation illustration (population, N=5): data 1, 2, 3, 4, 5

    • Mean: \mu = \frac{1+2+3+4+5}{5} = 3

    • Deviations: (-2,-1,0,1,2)

    • Squared deviations: (4,1,0,1,4)

    • Sum: \sum (x_i-\mu)^2 = 10

    • Population variance: \sigma^2 = \frac{10}{5} = 2

    • Population SD: \sigma = \sqrt{2} \approx 1.4142

  • Example with a sample (same data, n=5):

    • Sum of squared deviations: 10

    • Sample variance: s^2 = \frac{10}{n-1} = \frac{10}{4} = 2.5

    • Sample SD: s = \sqrt{2.5} \approx 1.5811

  • Calculation tables: Conceptual tool to organize the steps (x, x−μ, (x−μ)²) for both population and sample calculations

A few notes about standard deviation

  • Rounding:

    • Be careful with rounding intermediate calculations; small changes can affect the final SD value

  • Interpretation:

    • Large deviations from the mean lead to a larger SD; small deviations lead to a smaller SD

  • Resistance:

    • Standard deviation is NOT resistant to outliers; a few extreme values can substantially increase SD

  • n−1 denominator (why):

    • The use of n−1 makes the estimator of the population variance unbiased when sampling

Comparing standard deviations

  • When to compare:

    • If two samples/populations share the same units and are on the same scale, SDs can be directly compared

  • When units differ:

    • Use the coefficient of variation (CV), a unitless measure of relative spread

  • Coefficient of variation (CV):

    • Formula: \text{CV} = \frac{\sigma}{|\mu|}

    • Interpretation: SD expressed as a fraction of the mean; enables comparison across different units/scales

Variance

  • Relationship to SD:

    • Variance is the square of the standard deviation

    • Population variance: \sigma^2 = \frac{1}{N} \sum{i=1}^{N} (xi - \mu)^2

    • Sample variance: s^2 = \frac{1}{n-1} \sum{i=1}^{n} (xi - \bar{x})^2

  • Interpretation:

    • In the hand-calculation context, variance is the quantity inside the square root

    • Units are squared, which can make interpretation less intuitive

  • Role in inferential statistics:

    • Variance is foundational for many inferential procedures, including hypothesis testing and confidence intervals

Notes about StatCrunch

  • Mean:

    • The mean formula is the same for samples and populations; StatCrunch treats the mean as the same quantity in either case

  • Standard deviation in StatCrunch:

    • The default “standard deviation” in StatCrunch is the sample SD

    • To obtain the population SD, choose “unadj. standard deviation”

  • Practical takeaway for exams:

    • Know how to compute by hand (sum of squared deviations, divide by N or n−1, take the square root)

The empirical rule (68-95-99.7 rule)

  • Typically described for bell-shaped (normal) distributions:

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

    • About 95% lie within 2 standard deviations

    • About 99.7% lie within 3 standard deviations

  • The empirical rule also aligns with the idea of a normal distribution where most data cluster near the mean

  • Visual shorthand:

    • 1 SD: 68%

    • 2 SD: 95%

    • 3 SD: 99.7%

  • Special tail details (for normal):

    • About 0.15% lie beyond 3 SD on each tail (total ~0.3% beyond |Z|>3)

    • About 2.35% lie beyond 2 SD on each tail (total ~4.7% beyond |Z|>2)

Example: Bell-shaped distribution with mean 150 and standard deviation 15

  • 68% of observations lie between which values?

    • Between 150 - 15 = 135 and 150 + 15 = 165

    • Interval: [135, 165]

  • 95% of observations lie between which values?

    • Between 150 - 2\times 15 = 120 and 150 + 2\times 15 = 180

    • Interval: [120, 180]

  • What percentage lie between 105 and 195?

    • This is within 3 standard deviations: 150 \pm 3\cdot 15 = 105, 195

    • Percentage: approximately 99.7%

Chebyshev’s inequality

  • Scope:

    • Applies to any distribution, not just bell-shaped

  • Statement:

    • For any distribution with mean (\mu) and standard deviation (\sigma), at least \left(1 - \frac{1}{k^2}\right) \times 100\% of observations lie within (k) standard deviations of the mean

  • Key takeaway:

    • Guarantees a minimum proportion of observations within (k\sigma) of the mean, regardless of distribution shape

Example: Chebyshev (mean = 150, SD = 15)

  • Question: At least what percentage lie between 120 and 180?

    • Here, 120 and 180 are within 2 standard deviations of the mean ((k=2))

    • Minimum percentage: \left(1 - \frac{1}{2^2}\right) \times 100\% = \left(1 - \frac{1}{4}\right) \times 100\% = 75\%

Empirical rule vs Chebyshev: a quick comparison

  • Distribution requirements:

    • Empirical rule assumes a bell-shaped (normal) distribution

    • Chebyshev’s inequality places no assumption on shape; it applies to any distribution

  • Nature of the statement:

    • Empirical rule gives approximate percentages for normal data

    • Chebyshev provides a guaranteed minimum ("at least"), not an exact proportion

  • Practical interpretation:

    • If you know the data are roughly normal, use the empirical rule for quick estimates

    • If the distribution shape is unknown or non-normal, use Chebyshev to obtain a conservative bound

Quick recap and study-ready takeaways

  • Range: simplest spread measure; sensitive to outliers

  • Standard deviation: core measure of spread around the mean; population SD uses N in the denominator, sample SD uses n−1

  • Variance: the squared SD; easier to manipulate in algebraic/statistical derivations; units are squared

  • Coefficient of variation: unitless way to compare spread across different means/units

  • The empirical rule: best for normal distributions; gives quick spread estimates

  • Chebyshev’s inequality: universal spread bound; useful when distribution is unknown

  • When using software (e.g., StatCrunch): know which SD you’re getting by default and how to obtain the population version if needed