Class Notes on Variances, Standard Deviations, and Measures of Variation

Class Schedule Adjustments

• The instructor has moved the class schedule forward due to snow day delays.

• Homework Assignment #3 is now due on the 12th.

• Class will review material on Thursday instead of holding an exam.

• Communication about adjustments has been sent via email to all students.

Attendance Roll Call

• Students present: Owen, Brent, Destiny, Jacob, Eliana, Katie, Ronnie, Avon, Payton, Penn.

• Noted students are now back home due to various activities.

Hallmarks Information

• Hallmarks #1 and #2 are due today by midnight.

• The Pearson platform will close submissions at midnight.

Measures of Variation - Overview

• Section 32 covers measures of variation.

Range

• Definition: The range is a basic measure of variation calculated by subtracting the minimum value from the maximum value in a dataset.

• Formula: $ext{Range} = ext{Max value} - ext{Min value}$

• Characteristics:

  • Reflects only the extremes (maximum and minimum values).

  • Not resistant to outliers (a single extreme value can dramatically affect the range).

• Example Calculation:

  • For a data set: 20, …, 75

  • Max = 75, Min = 20

  • $ext{Range} = 75 - 20 = 55$

  • The dataset has 11 values, suggesting some variance.

Standard Deviation

• Definition: A more comprehensive measure of variation that indicates how much individual data values deviate from the mean.

• Notation:

  • Sample Standard Deviation: denoted by lowercase 's'

  • Population Standard Deviation: denoted by lowercase 'σ' (sigma).

• Importance: Indicates the spread of values in relation to the mean; essential for understanding the dispersion in a dataset.

• Mean: The central value around which the dataset is analyzed.

Calculation of Standard Deviation

1. Calculate the Mean: $\bar{x} = rac{ ext{Sum of all } x}{n}$

2. Determining Deviations: For each data value, subtract the mean and square the result to eliminate negatives:

   • Example:

     • If mean height is 66 inches, an individual height of 74 inches yields:

     • Deviations: 74 - 66 = 8

     • Squared deviation: $8^2 = 64$

3. Sum of Squared Deviations

4. Divide by n-1 (for sample variance):

   • Formula for Sample Variance: $s^2 = rac{ ext{Sum of Squared Deviations}}{n - 1}$

5. Square Root:

   • $s = ext{sqrt}(s^2)$

• Characteristics:

  • Mean of deviations can be zero; thus squaring removes this.

  • Standard deviation gives us a numerical value indicating the average distance of each data point from the mean.

  • Units are the same as the data set's original units.

Differences between Sample and Population Statistics

• Sample standard deviation tends to underestimate the population standard deviation.

• Using $n - 1$ in samples compensates for the bias.

• Population standard deviation uses just $n$ because all data points are included and no bias from sample selection occurs.

Variance

• Definition: Variance is the square of the standard deviation.

• Sample Variance: $(s^2)$

• Population Variance: $(\sigma^2)$

• Units of variance are squared units of the original dataset (e.g., if data in inches, variance in square inches).

Outliers and Variation

• Outliers can dramatically affect both range and standard deviation.

• Comparison between sample variance and population variance can guide estimations for datasets lacking full information.

Empirical Rule and Chebyshev's Theorem

Empirical Rule (Normal Distribution)

• States that:

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

  • Approximately 95% of the data falls within 2 standard deviations.

  • Approximately 99.7% of the data falls within 3 standard deviations.

Chebyshev's Theorem (Non-Normal Distributions)

• For any dataset, at least:

  • $(1 - rac{1}{k^2})$ of the data is contained within k standard deviations from the mean.

  • For example:

  • For k=2: $(1 - rac{1}{4}) = 0.75$ (at least 75% within 2 standard deviations)

  • For k=3: $(1 - rac{1}{9}) = 0.89$ (at least 89% within 3 standard deviations)

Coefficient of Variation (CV)

• Formula: $CV = rac{s}{\bar{x}} imes 100$, where s is the standard deviation and $\bar{x}$ is the mean.

• Purpose: It allows for the comparison of the degree of variation from one dataset to another, expressed as a percentage.

• A CV greater than 1% indicates significant differences between sample variations.

Practical Example - Analysis of Celebrity Net Worths

• Tasks involved:

  • Find the range, variance, and standard deviation for a dataset of celebrity net worths (in billions).

  • Understand limitations of findings (such as typicality) based on the exclusive nature of the sample data.

• Round off considerations: Round results to the appropriate units based on the nature of the data.

Blood Platelet Count Example Using Empirical Rule

• Given:

  • Mean = 255.3

  • Standard Deviation = 65.9

• Using the empirical rule:

  • 95% of women will have platelet counts between:

  • 123.5 (2 SDs below) and 387.1 (2 SDs above).

• Applying Chebyshev’s Theorem for different non-normal distributions.

Summary

• Today's focus was on measures of variation, calculation of range, standard deviation, variance, and understanding practical applications and implications of these statistics within various contexts.

• Next class will inquire about measures of relative standing and introduction to box plots.