30-35 minutes
Lesson 2: Measures of Spread (Variability)
Learning Objectives
By the end of this lesson, you will be able to:
- Explain why spread (variability) is just as important as center
- Calculate and interpret the range of a dataset
- Calculate quartiles and the interquartile range (IQR)
- Understand and calculate variance and standard deviation
- Apply the Empirical Rule (68-95-99.7 Rule) for bell-shaped distributions
- Choose appropriate measures of spread based on data characteristics
Why Does Spread Matter?
In Lesson 1, we learned that the mean tells us the "center" of the data. But the mean alone doesn't tell the whole story!
The Problem: Same Mean, Very Different Data!
Class A Test Scores: 70, 72, 75, 78, 80 Mean = 75
Class B Test Scores: 30, 50, 75, 100, 100 Mean = 75
Both classes have the same mean (75), but are they really similar?
- Class A: Scores are clustered tightly around 75 (everyone scored 70-80)
- Class B: Scores are all over the place! (30 to 100)
Conclusion: To truly understand data, we need to know both the center (mean) AND the spread (how scattered the values are)!
Spread (Variability): A measure of how spread out or scattered the data values are. Also called "dispersion" or "variability."
Low spread: Values are clustered close together
High spread: Values are scattered far apart
The Range (Simplest Measure)
Range: The difference between the maximum and minimum values in a dataset.
Formula:
Range = Maximum - Minimum
Example 1: Calculating Range
Dataset: Daily high temperatures (°F): 68, 72, 75, 81, 64, 77, 70
Step-by-step:
- Find the maximum: 81°F (hottest day)
- Find the minimum: 64°F (coolest day)
- Calculate the range: 81 - 64 = 17°F
Interpretation: Temperatures varied by 17 degrees over the week.
Pros and Cons of Range
Pros:
- Easy to calculate and understand
- Shows the full extent of the data
Cons:
- Uses only 2 values (max and min) – ignores everything else!
- Extremely sensitive to outliers
- Doesn't tell you how data is distributed in between
The Range Problem: Outlier Sensitivity
Commute times (minutes): 20, 22, 21, 23, 19, 120
Range = 120 - 19 = 101 minutes
That one day you got stuck in traffic (120 min) makes the range huge, even though most days you commute ~20 minutes. The range doesn't represent typical variability well when there are outliers!
Interquartile Range (IQR): Better Than Range!
The IQR solves the outlier problem by looking at the middle 50% of the data, ignoring the extreme tails.
Quartiles: Values that divide ordered data into four equal parts (quarters).
- Q1 (First Quartile): 25% of data is below Q1
- Q2 (Second Quartile): The median (50% of data below)
- Q3 (Third Quartile): 75% of data is below Q3
Interquartile Range (IQR) Formula:
IQR = Q3 - Q1
(The range of the middle 50% of the data)
Example 2: Calculating IQR
Dataset: Quiz scores: 12, 14, 15, 16, 18, 19, 20, 21, 24 (already ordered!)
Step 1: Find Q2 (the median)
9 values (odd), so median is the 5th value: Q2 = 18
Step 2: Find Q1 (median of lower half)
Lower half: 12, 14, 15, 16 (values below Q2)
Q1 = average of 14 and 15 = 14.5
Step 3: Find Q3 (median of upper half)
Upper half: 19, 20, 21, 24 (values above Q2)
Q3 = average of 20 and 21 = 20.5
Step 4: Calculate IQR
IQR = Q3 - Q1 = 20.5 - 14.5 = 6
Interpretation: The middle 50% of quiz scores span 6 points.
Why IQR is Better Than Range
- Resistant to outliers - Only uses middle 50% of data
- More representative - Shows typical variability, not extreme spread
- Used in boxplots - Essential for identifying outliers (Lesson 4!)
Variance and Standard Deviation: The Most Important Measures
Range and IQR give us a rough sense of spread, but variance and standard deviation are the most widely used measures because they:
- Use every data value (not just min/max or quartiles)
- Measure spread from the mean
- Have important mathematical properties for advanced statistics
Understanding Variance
Variance (s²): The average of the squared deviations from the mean. Measures how far values typically are from the mean.
Variance Formula (Sample):
s² = Σ(x - x̄)² / (n - 1)
Where:
• x = each data value
• x̄ = mean
• n = number of values
• Σ = "sum of"
Example 3: Calculating Variance
Dataset: Number of pets: 1, 2, 3, 4, 5
Step 1: Calculate the mean
x̄ = (1+2+3+4+5) / 5 = 15/5 = 3
Step 2: Calculate deviations from mean (x - x̄)
| x | x - x̄ | (x - x̄)² |
|---|---|---|
| 1 | 1 - 3 = -2 | (-2)² = 4 |
| 2 | 2 - 3 = -1 | (-1)² = 1 |
| 3 | 3 - 3 = 0 | 0² = 0 |
| 4 | 4 - 3 = 1 | 1² = 1 |
| 5 | 5 - 3 = 2 | 2² = 4 |
Step 3: Sum the squared deviations
Σ(x - x̄)² = 4 + 1 + 0 + 1 + 4 = 10
Step 4: Divide by (n-1)
s² = 10 / (5-1) = 10/4 = 2.5
Result: Variance = 2.5 pets²
Why Square the Deviations?
Notice in the table that some deviations are negative (below mean) and some are positive (above mean). If we just added them up, they'd cancel out to zero!
Squaring makes all deviations positive, so they add up to show total variability. It also gives more weight to values far from the mean.
Problem: Variance is in squared units (pets²? ). That's why we use standard deviation!
Standard Deviation: Back to Original Units!
Standard Deviation (s): The square root of the variance. Measures typical distance from the mean in the original units.
Standard Deviation Formula:
s = √(s²) = √[Σ(x - x̄)² / (n - 1)]
Example 3 (continued): Standard Deviation
From above, we calculated variance = 2.5
Standard Deviation: s = √2.5 ≈ 1.58 pets
Interpretation: The number of pets typically varies by about 1.58 from the mean of 3 pets. This is back in "pets" units, not "pets²"!
Interpreting Standard Deviation
- Small SD: Data points are clustered closely around the mean (low variability)
- Large SD: Data points are spread far from the mean (high variability)
- SD = 0: All values are exactly the same (no variability)
- Units: SD has the same units as the original data
Comparing Two Datasets with Same Mean
Dataset A: 10, 11, 12, 13, 14
- Mean = 12
- Standard Deviation ≈ 1.58
- Values are tightly clustered
Dataset B: 2, 6, 12, 18, 22
- Mean = 12
- Standard Deviation ≈ 7.91
- Values are widely spread
Conclusion: Dataset B has 5 times more variability than Dataset A, even though they have the same mean!
The Empirical Rule (68-95-99.7 Rule)
For bell-shaped (approximately normal) distributions, standard deviation has a special interpretation!
The Empirical Rule
For bell-shaped distributions:
- ~68% of data falls within 1 standard deviation of the mean (x̄ ± 1s)
- ~95% of data falls within 2 standard deviations of the mean (x̄ ± 2s)
- ~99.7% of data falls within 3 standard deviations of the mean (x̄ ± 3s)
Example 4: Using the Empirical Rule
SAT scores are approximately bell-shaped with:
- Mean = 1050
- Standard Deviation = 200
What percentage of students score between 850 and 1250?
Step 1: Calculate how many SDs from mean
850 = 1050 - 200 = Mean - 1 SD
1250 = 1050 + 200 = Mean + 1 SD
Step 2: Apply Empirical Rule
Within ±1 SD of the mean → ~68% of students
Answer: About 68% of students score between 850 and 1250.
What about 650 to 1450?
650 = 1050 - 400 = Mean - 2 SD
1450 = 1050 + 400 = Mean + 2 SD
Answer: About 95% of students score in this range!
Important: Empirical Rule Only for Bell-Shaped Distributions!
The 68-95-99.7 rule ONLY works for approximately normal (bell-shaped) distributions. It doesn't apply to skewed data!
When to use: Heights, test scores, measurement errors, many natural phenomena
When NOT to use: Income (right-skewed), house prices (right-skewed), strongly skewed data
Comparing Measures of Spread
| Measure | Formula | Uses All Data? | Resistant to Outliers? | When to Use |
|---|---|---|---|---|
| Range | Max - Min | No (only 2 values) | No | Quick, rough estimate |
| IQR | Q3 - Q1 | No (middle 50%) | Yes | Skewed data, with outliers |
| Variance | Σ(x-x̄)²/(n-1) | Yes | No | Math/stats theory (but units are squared) |
| Std Dev | √Variance | Yes | No | Most common! Original units, interpretable |
Check Your Understanding
Test your knowledge of measures of spread!
Question 1:
Calculate the range of: 15, 22, 18, 30, 12, 25
Question 2:
Two datasets both have a mean of 50. Dataset A has SD = 2, Dataset B has SD = 10. Which is true?
Question 3:
IQ scores are bell-shaped with mean = 100 and SD = 15. Using the Empirical Rule, about what percentage of people have IQs between 85 and 115?
Question 4:
Why do we square the deviations when calculating variance?
Question 5:
Which measure of spread is MOST resistant to extreme outliers?
Question 6:
A dataset has variance = 16. What is the standard deviation?
Key Takeaways
- Spread matters! Two datasets can have the same mean but very different variability.
- Range = Max - Min. Simple but sensitive to outliers.
- IQR = Q3 - Q1. Middle 50% spread, resistant to outliers. Great for skewed data!
- Variance = Average squared deviation from mean. In squared units.
- Standard Deviation = √Variance. Most common measure! Original units, interpretable.
- Empirical Rule: For bell-shaped data, ~68% within ±1 SD, ~95% within ±2 SD, ~99.7% within ±3 SD.
- Choosing measures: Use SD for symmetric data, IQR for skewed data or outliers.