Measures of Center

Introduction

Measures of center describe the "typical" or "middle" value in a dataset. They help us answer: "What is a representative value?" The three main measures — mean, median, and mode — each have different properties and appropriate uses.

The Mean

Definition

Mean ($\bar{x}$): The arithmetic average

Formula: $\bar{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + ... + x_n}{n}$

Where:

• $\sum x_i$ = sum of all values
• $n$ = number of observations

Calculating the Mean

Example 1: Test scores: 85, 90, 78, 92, 88

$\bar{x} = \frac{85 + 90 + 78 + 92 + 88}{5} = \frac{433}{5} = 86.6$

Mean test score = 86.6 points

Example 2: Heights (in inches): 64, 67, 65, 70, 64

$\bar{x} = \frac{64 + 67 + 65 + 70 + 64}{5} = \frac{330}{5} = 66$

Mean height = 66 inches

Properties of the Mean

Uses all data:

• Every value contributes
• Change any value, mean changes
• Adding up all deviations from mean = 0

Balance point:

• If data were on a number line with equal weights, mean is where it would balance
• Sum of distances below mean = sum of distances above mean

Sensitive to outliers:

• Extreme values pull mean toward them
• One very high/low value can change mean substantially

Example showing outlier effect:

Without outlier: 10, 12, 11, 13, 12
$\bar{x} = \frac{58}{5} = 11.6$

With outlier: 10, 12, 11, 13, 12, 100
$\bar{x} = \frac{158}{6} = 26.3$

The outlier (100) dramatically increased the mean from 11.6 to 26.3!

When to Use the Mean

Appropriate when: ✓ Distribution is roughly symmetric
✓ No extreme outliers
✓ Need to use all data values
✓ Want mathematical properties (use in further calculations)

Not appropriate when: ❌ Distribution is heavily skewed
❌ Outliers present
❌ Want resistant measure
❌ Data is ordinal (ranked) only

The Median

Definition

Median: The middle value when data is ordered

• 50th percentile
• Splits data in half
• Half values below, half above

Finding the Median

Step 1: Order data from smallest to largest

Step 2: Find middle position

If $n$ is odd: Median = middle value
Position = $\frac{n+1}{2}$

If $n$ is even: Median = average of two middle values
Positions = $\frac{n}{2}$ and $\frac{n}{2} + 1$

Examples

Example 1 (odd n): Scores: 78, 85, 90, 82, 88

Step 1: Order: 78, 82, 85, 88, 90
Step 2: $n = 5$ (odd), position = $\frac{5+1}{2} = 3$
Median = 85 (the 3rd value)

Example 2 (even n): Scores: 78, 85, 90, 82, 88, 92

Step 1: Order: 78, 82, 85, 88, 90, 92
Step 2: $n = 6$ (even), positions = 3 and 4
Step 3: Values are 85 and 88
Median = $\frac{85 + 88}{2} = 86.5$

Properties of the Median

Resistant to outliers:

• Position-based, not value-based
• Extreme values don't affect it much
• More stable measure for skewed data

Example: Data: 10, 12, 11, 13, 12 → Median = 12
With outlier: 10, 12, 11, 13, 12, 100 → Median = 12

The outlier didn't change the median!

50-50 split:

• Half the data ≤ median
• Half the data ≥ median
• Useful for understanding data distribution

Not affected by exact values:

• Only needs order and middle position
• Works well for ordinal data (rankings)

When to Use the Median

Appropriate when: ✓ Distribution is skewed
✓ Outliers are present
✓ Want resistant measure
✓ Data is ordinal (ordered categories)
✓ Interested in "typical" individual

Examples where median is better:

• Income (right-skewed, few very high earners)
• Home prices (right-skewed, few very expensive homes)
• Reaction times (right-skewed, occasional very slow responses)

The Mode

Definition

Mode: The most frequently occurring value

• Can have one mode (unimodal)
• Can have multiple modes (bimodal, multimodal)
• Can have no mode (all values occur once)

Finding the Mode

Count frequency of each value, identify most common

Example 1: Scores: 85, 90, 85, 92, 88, 85

• 85 appears 3 times
• 90, 92, 88 each appear once
• Mode = 85

Example 2: Scores: 85, 90, 85, 92, 90, 88

• 85 appears twice
• 90 appears twice
• Modes = 85 and 90 (bimodal)

Example 3: Scores: 85, 90, 92, 88, 82

• All values appear once
• No mode

When to Use the Mode

Appropriate when: ✓ Categorical data
✓ Want most common value
✓ Describing bimodal distributions

Examples:

• "The most common car color is white" (mode of categorical data)
• "The distribution is bimodal with peaks at 65 and 72" (describing shape)

Not very useful for: ❌ Continuous numerical data (values rarely repeat)
❌ Summarizing center of distribution

Comparing Mean and Median

Relationship to Distribution Shape

Symmetric distribution: $Mean \approx Median$

Both measures give similar values, either can be used

Right-skewed distribution: $Mean > Median$

Mean pulled right by high values in tail
Median more representative of "typical" value

Left-skewed distribution: $Mean < Median$

Mean pulled left by low values in tail
Median more representative of "typical" value

Visual Representation

Symmetric: Mean and median at same location (center of distribution)

Right-skewed: Mean to the right of median (toward tail)

Left-skewed: Mean to the left of median (toward tail)

Choosing Between Mean and Median

Use Mean when:

• Distribution is symmetric
• No outliers or extreme skewness
• Want to use all data
• Need for further calculations (variance, hypothesis tests)

Use Median when:

• Distribution is skewed
• Outliers are present
• Want resistant measure
• Ordinal data
• Interested in "typical" individual rather than arithmetic average

Real-world example: Income

Town income data:

• Median income: 45,000 dollars
• Mean income: 75,000 dollars

Mean is much higher because a few very wealthy residents pull it up. The median of 45,000 dollars better represents the "typical" resident's income.

Weighted Mean

Definition

Weighted Mean: When values have different importance or frequency

Formula: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

Where:

• $w_i$ = weight for each value
• $x_i$ = data value

Example: Course Grade

Your course grade is calculated as:

• Tests: 60% of grade (weight = 0.60)
• Homework: 25% of grade (weight = 0.25)
• Final: 15% of grade (weight = 0.15)

Scores:

• Test average: 85
• Homework average: 92
• Final exam: 78

Weighted mean: $\bar{x}_w = 0.60(85) + 0.25(92) + 0.15(78)$ $= 51 + 23 + 11.7 = 85.7$

Course grade = 85.7%

Note: Cannot just average 85, 92, and 78 because they have different weights!

Trimmed Mean

Definition

Trimmed Mean: Mean calculated after removing extreme values

Common: 5% trimmed mean (remove lowest 5% and highest 5%)

Purpose

• More resistant than regular mean
• Still uses most of data
• Compromise between mean and median

Example

Data (ordered): 10, 12, 13, 14, 15, 16, 17, 18, 19, 100

Regular mean: $\frac{234}{10} = 23.4$ (affected by outlier 100)

10% trimmed mean: Remove lowest 10% (10) and highest 10% (100)
$\frac{12+13+14+15+16+17+18+19}{8} = 15.5$

Trimmed mean (15.5) more representative than regular mean (23.4)

Common Mistakes

❌ Using mean with skewed data
Use median instead!

❌ Forgetting to order data for median
Always sort first!

❌ Reporting mode for continuous data
Usually not meaningful when values don't repeat

❌ Not specifying units
Always include units (inches, dollars, points, etc.)

❌ Confusing which measure to use
Consider shape and outliers

❌ Calculating mean of percentages
May need weighted mean if groups are different sizes

Quick Reference

Mean:

• Formula: $\bar{x} = \frac{\sum x_i}{n}$
• When: Symmetric, no outliers
• Property: Uses all data, sensitive to extremes
• Symbol: $\bar{x}$ (sample), $\mu$ (population)

Median:

• Method: Middle value when ordered
• When: Skewed, outliers present
• Property: Resistant, 50-50 split
• Symbol: M or $\tilde{x}$

Mode:

• Method: Most frequent value
• When: Categorical data, describe shape
• Property: Can have multiple or none

Relationship to shape:

• Symmetric: Mean ≈ Median
• Right-skewed: Mean > Median
• Left-skewed: Mean < Median

Remember: The best measure of center depends on the distribution's shape and the presence of outliers. When in doubt, report both mean and median!