Lesson 4: Measures of Variability

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Class Data

Lesson 3 Review

Measures of Center

Last lesson we learned three ways to measure the “center” of a distribution:

Measure Formula Resistant? Best For
Mean \(\bar{x} = \frac{\sum x_i}{n}\) No Symmetric data
Median Middle value Yes Skewed data/outliers
Mode Most frequent Yes Categorical data

We also learned about percentiles, quartiles, and the five-number summary.

TipKey Insight from Lesson 3

Two datasets can have the same mean and median but look very different! Today we learn how to quantify that difference.

Lesson 4 Content

Objectives

  • Calculate and interpret range and inter-quartile range (IQR)
  • Calculate and interpret variance and standard deviation

Required Reading

Devore, Section 1.4: Measures of Variability

Why Measure Variability?

The Problem with Center Alone

Consider two archery targets:

Both archers might have similar average positions, but Archer A is clearly more consistent. We need measures of spread or variability to capture this difference.

The Range

The range is the simplest measure of spread:

\[ \text{Range} = \text{Maximum} - \text{Minimum} \]

Quiz scores: 72, 75, 78, 80, 82, 85, 88 
Maximum: 88 
Minimum: 72 
Range: 16 points

Problems with the range:

  • Uses only 2 data points (ignores everything in between)
  • Very sensitive to outliers
  • Tends to increase with sample size

The Interquartile Range (IQR)

The interquartile range (IQR) measures the spread of the middle 50% of the data:

\[ \text{IQR} = Q_3 - Q_1 \]

Quiz scores: 72, 75, 78, 80, 82, 85, 88 
Q1 (25th percentile): 76.5 
Q3 (75th percentile): 83.5 
IQR: 7 points

Advantages of IQR:

  • Resistant to outliers (uses quartiles, not extremes)
  • Focuses on the “typical” spread of the data
  • Useful for identifying outliers (1.5 × IQR rule)

The 1.5 × IQR Rule for Outliers

An observation is a potential outlier if it falls:

  • Below \(Q_1 - 1.5 \times \text{IQR}\), or
  • Above \(Q_3 + 1.5 \times \text{IQR}\)
Salaries (thousands): 45, 48, 52, 55, 58, 62, 68, 72, 85, 250 
Q1: 52.75 
Q3: 71 
IQR: 18.25 
Lower fence: 25.375 
Upper fence: 98.375 
Outliers: 250

NoteNote

The boxplot automatically shows values beyond 1.5 × IQR as individual points.

Variance and Standard Deviation

To measure spread, we want to know how far observations are from the center. The deviation of observation \(x_i\) from the mean is:

\[ x_i - \bar{x} \]

Problem: If we sum all deviations, they always equal zero!

Data: 2, 4, 6, 8, 10 
Mean: 6 
Deviations: -4, -2, 0, 2, 4 
Sum of deviations: 0

Solution: Square the deviations before summing!

The Sample Variance

The sample variance (\(s^2\)) is the average of the squared deviations (with a small adjustment):

\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]

WarningWhy n-1?

We divide by \(n-1\) instead of \(n\) because we lose one “degree of freedom” when we estimate the mean from the same data. This gives us an unbiased estimate of the population variance. We’ll discuss this more when we cover inference.

Computing the Sample Variance

Step-by-step method:

  1. Calculate the mean \(\bar{x}\)
  2. Subtract the mean from each value: \((x_i - \bar{x})\)
  3. Square each deviation: \((x_i - \bar{x})^2\)
  4. Sum the squared deviations: \(\sum(x_i - \bar{x})^2\)
  5. Divide by \(n-1\)

Example: Calculating Variance

Data: 4, 7, 5, 9, 10 
Step 1 - Mean: 7 
Step 2-3 - Deviations and Squared Deviations:
x_i      (x_i - x̄)  (x_i - x̄)²                     
--------------------------------------------------
4        -3.0         9.00           
7        0.0          0.00           
5        -2.0         4.00           
9        2.0          4.00           
10       3.0          9.00           
--------------------------------------------------

Step 4 - Sum of squared deviations: 26 
Step 5 - Variance (s²): 26 / 4 = 6.5

The Sample Standard Deviation

The sample standard deviation (\(s\)) is the square root of the variance:

\[ s = \sqrt{s^2} = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]

Data: 4, 7, 5, 9, 10 
Variance (s²): 6.5 
Standard Deviation (s): 2.55

Why take the square root?

  • The variance is in squared units (e.g., dollars², inches²)
  • The standard deviation is in the original units (e.g., dollars, inches)
  • This makes interpretation much easier!

Interpreting Standard Deviation

The standard deviation tells us roughly how far a typical observation is from the mean.

Mean: 70.92 
Standard Deviation: 4.56 
Most scores fall between 66.4 and 75.5

Properties of Standard Deviation

Property Description
Always non-negative \(s \geq 0\), equals 0 only when all values are identical
Same units as data Unlike variance, which is in squared units
Sensitive to outliers Like the mean, outliers inflate \(s\)
Useful benchmarks For bell-shaped data: ~68% within 1 SD, ~95% within 2 SD

Comparing Measures of Spread

Summary Table

Measure Formula Resistant? Uses All Data?
Range Max - Min No No (only 2 points)
IQR \(Q_3 - Q_1\) Yes No (only quartiles)
Variance \(s^2 = \frac{\sum(x_i-\bar{x})^2}{n-1}\) No Yes
Std Dev \(s = \sqrt{s^2}\) No Yes

Which Measure to Use?

Situation Recommended Measure
Quick summary Range (but note its limitations)
Data with outliers IQR
Identifying outliers 1.5 × IQR rule
General purpose Standard deviation
Statistical inference Variance/Standard deviation
TipRule of Thumb

Report IQR when you report the median; report standard deviation when you report the mean.

Practice Problems

Problem 1

The following data represents the number of hours cadets slept last night:

5, 6, 6, 7, 7, 7, 8, 8, 9

  1. Calculate the range
  2. Calculate the IQR
  3. Calculate the variance and standard deviation
  4. Are there any outliers using the 1.5 × IQR rule?

  1. Range = 9 - 5 = 4 hours

  2. IQR calculation:

    • Ordered: 5, 6, 6, 7, 7, 7, 8, 8, 9
    • Q1 = 6, Q3 = 8
    • IQR = 8 - 6 = 2 hours

  3. Variance and SD:

    • Mean = (5+6+6+7+7+7+8+8+9)/9 = 63/9 = 7
    • Deviations: -2, -1, -1, 0, 0, 0, 1, 1, 2
    • Squared: 4, 1, 1, 0, 0, 0, 1, 1, 4
    • Sum = 12
    • Variance = 12/8 = 1.5 hours²
    • Standard deviation = √1.5 = 1.22 hours

  4. Outlier check:

    • Lower fence = Q1 - 1.5(IQR) = 6 - 1.5(2) = 3
    • Upper fence = Q3 + 1.5(IQR) = 8 + 1.5(2) = 11
    • All values between 3 and 11, so no outliers

Problem 2

Two PT instructors recorded 2-mile run times (in minutes) for their sections:

Section A: 14, 15, 15, 16, 16, 16, 17, 17, 18 Section B: 12, 13, 16, 16, 16, 16, 19, 20, 20

  1. Calculate the mean for each section
  2. Calculate the standard deviation for each section
  3. Which section is more consistent? How do you know?
Section A:
  Mean: 16 minutes
  SD: 1.22 minutes
Section B:
  Mean: 16.44 minutes
  SD: 2.83 minutes

  1. Both sections have a mean of approximately 16 minutes

  2. Section A: SD ≈ 1.22 minutes Section B: SD ≈ 2.92 minutes

  3. Section A is more consistent because it has a smaller standard deviation. Even though both sections have the same average run time, Section A’s times are clustered more tightly around the mean.

Problem 3

A dataset has the following five-number summary: - Min = 10 - Q1 = 25 - Median = 35 - Q3 = 45 - Max = 100

  1. Calculate the IQR
  2. Determine the outlier fences
  3. Is the maximum value (100) an outlier?
  4. Sketch what the boxplot would look like

  1. IQR = Q3 - Q1 = 45 - 25 = 20

  2. Outlier fences:

    • Lower fence = Q1 - 1.5(IQR) = 25 - 1.5(20) = 25 - 30 = -5
    • Upper fence = Q3 + 1.5(IQR) = 45 + 1.5(20) = 45 + 30 = 75

  3. Yes, 100 is an outlier because it exceeds the upper fence of 75

  4. Boxplot sketch:

Summary

Key Takeaways

  1. Range: Max - Min; simple but sensitive to outliers
  2. IQR: Q3 - Q1; resistant to outliers, measures middle 50%
  3. 1.5 × IQR Rule: Values beyond \(Q_1 - 1.5 \times \text{IQR}\) or \(Q_3 + 1.5 \times \text{IQR}\) are potential outliers
  4. Variance (\(s^2\)): Average squared deviation from the mean
  5. Standard Deviation (\(s\)): Square root of variance; in original units
ImportantRemember
  • Report IQR with the median (for skewed data or outliers)
  • Report standard deviation with the mean (for symmetric data)
  • Standard deviation ≈ “typical distance from the mean”

Before You Leave

Today

  • Range and IQR as simple measures of spread
  • The 1.5 × IQR rule for identifying outliers
  • Variance and standard deviation as comprehensive measures
  • When to use each measure of variability

Any questions?

Next Lesson

Lesson 5: Exploratory Data Analysis Lab

  • Execute EDA using appropriate graphs and summaries
  • Justify choices of displays for variable types
  • Communicate findings with clear, concise annotations

Upcoming Graded Events

  • WebAssign 1.4 - Due before Lesson 5
  • Exploratory Data Analysis - Due Lesson 9
  • WPR I - Lesson 16