Mean, Median, and Mode

Measures of Central Tendency

Measures of central tendency describe the "center" or "typical value" of a data set.

Three main measures:

• Mean (average)
• Median (middle value)
• Mode (most frequent)

Each tells us something different about the data!

The Mean (Average)

The mean is the sum of all values divided by the number of values.

Formula: Mean = (sum of all values) / (number of values)

Or: x̄ = (x₁ + x₂ + x₃ + ... + xₙ) / n

Example 1: Find the mean of 5, 8, 12, 15, 20

Sum: 5 + 8 + 12 + 15 + 20 = 60 Count: 5 values Mean: 60 ÷ 5 = 12

Example 2: Find the mean of 3, 7, 9, 10, 11

Sum: 3 + 7 + 9 + 10 + 11 = 40 Count: 5 values Mean: 40 ÷ 5 = 8

Example 3: Test scores: 85, 90, 78, 92, 88, 95

Sum: 528 Count: 6 Mean: 528 ÷ 6 = 88

The average test score is 88.

Properties of the Mean

The mean:

• Uses ALL data values
• Can be affected by extreme values (outliers)
• May not be an actual data value
• Is used in further statistical calculations
• Represents "balance point" of data

Example with outlier: Data: 10, 12, 14, 15, 16, 100

Mean = 167 ÷ 6 = 27.83

The 100 pulls the mean up significantly!

The Median

The median is the middle value when data is arranged in order.

Steps to find median:

1. Arrange data in order (least to greatest)
2. If odd number of values: median is the middle value
3. If even number of values: median is average of two middle values

Example 1 (odd): Find median of 7, 3, 9, 5, 11

Step 1: Order: 3, 5, 7, 9, 11 Step 2: 5 values (odd), so middle is 3rd value Median: 7

Example 2 (even): Find median of 4, 8, 12, 15, 20, 25

Step 1: Already ordered Step 2: 6 values (even), so average 3rd and 4th values Middle values: 12 and 15 Median: (12 + 15) ÷ 2 = 13.5

Example 3: Data: 85, 90, 78, 92, 88, 95

Ordered: 78, 85, 88, 90, 92, 95 Middle values: 88 and 90 Median: (88 + 90) ÷ 2 = 89

Properties of the Median

The median:

• Not affected by extreme values (resistant to outliers)
• Always a value that divides data in half
• May or may not be an actual data value
• Better than mean when data has outliers
• Represents the 50th percentile

Example with outlier: Data: 10, 12, 14, 15, 16, 100

Median: (14 + 15) ÷ 2 = 14.5

The outlier (100) doesn't affect the median!

The Mode

The mode is the value that appears most frequently.

Example 1: Find mode of 3, 5, 7, 5, 9, 5, 11

5 appears three times (most frequent) Mode: 5

Example 2: Test scores: 85, 90, 85, 92, 88, 85, 95

85 appears three times Mode: 85

Example 3: 2, 4, 6, 8, 10

All values appear once No mode (or all values are modes)

Example 4: 1, 2, 2, 3, 3, 4

Both 2 and 3 appear twice Bimodal: modes are 2 and 3

Properties of the Mode

The mode:

• Always an actual data value
• Can have no mode, one mode, or multiple modes
• Useful for categorical data (colors, types, etc.)
• Not affected by extreme values
• Easy to identify in small data sets

Terms:

• Unimodal: one mode
• Bimodal: two modes
• Multimodal: more than two modes
• No mode: all values appear same number of times

Comparing Mean, Median, and Mode

Example: 2, 3, 4, 4, 5, 6, 7, 20

Mean: (2+3+4+4+5+6+7+20) ÷ 8 = 51 ÷ 8 = 6.375 Median: (4+5) ÷ 2 = 4.5 Mode: 4

Observations:

• Mean is pulled up by outlier (20)
• Median better represents "typical" value here
• Mode is most frequent value

When to Use Each Measure

Use MEAN when:

• Data has no extreme outliers
• You want to use all data values
• Data is numerical and roughly symmetric
• You need it for further calculations

Use MEDIAN when:

• Data has outliers or is skewed
• You want the "middle" value
• Income, home prices (often skewed data)
• You want a resistant measure

Use MODE when:

• Data is categorical
• You want the most popular/common value
• Multiple modes are meaningful
• Shoe sizes, favorite colors, etc.

Weighted Mean

A weighted mean gives different values different importance.

Formula: Weighted mean = Σ(value × weight) / Σ(weights)

Example: Test scores with different weights

Homework (10%): 90 Quiz (20%): 85 Test (70%): 92

Weighted mean = (90×10 + 85×20 + 92×70) / (10+20+70) = (900 + 1700 + 6440) / 100 = 9040 / 100 = 90.4

Overall grade: 90.4%

Example 2: GPA calculation

Course A (3 credits): 4.0 Course B (3 credits): 3.5 Course C (4 credits): 3.0

GPA = (4.0×3 + 3.5×3 + 3.0×4) / (3+3+4) = (12 + 10.5 + 12) / 10 = 34.5 / 10 = 3.45

Finding a Missing Value

Example: Five test scores have mean 85. Four scores are 80, 85, 90, 88. Find the fifth score.

Let x = fifth score Mean = (80 + 85 + 90 + 88 + x) / 5 = 85

Solve: 343 + x = 425 x = 82

The fifth score is 82.

Example 2: Three numbers have mean 12 and median 10. Two numbers are 8 and 10. Find the third.

Since median is 10, and we have 8 and 10, the third number must be ≥ 10.

Order will be: 8, 10, x (where x ≥ 10)

Mean: (8 + 10 + x) / 3 = 12 18 + x = 36 x = 18

Range

The range measures spread (not central tendency).

Formula: Range = maximum value - minimum value

Example: 3, 7, 12, 15, 20

Range = 20 - 3 = 17

The data spans 17 units.

Effect of Outliers

Example: Compare with and without outlier

Original data: 5, 6, 7, 8, 9 Mean: 7 Median: 7 Mode: none

Add outlier: 5, 6, 7, 8, 9, 50 Mean: 85 ÷ 6 = 14.17 (changed significantly!) Median: (7 + 8) ÷ 2 = 7.5 (barely changed) Mode: none

Conclusion: Median is more resistant to outliers.

Symmetric vs. Skewed Data

Symmetric distribution: Mean ≈ Median ≈ Mode Example: 2, 4, 6, 8, 10 Mean = Median = 6

Right-skewed (positive skew): Mode < Median < Mean Few large values pull mean up Example: 1, 2, 3, 4, 20

Left-skewed (negative skew): Mean < Median < Mode Few small values pull mean down Example: 1, 17, 18, 19, 20

Real-World Applications

Example 1: Salaries Company salaries: $30k,$32k, $35k,$38k, $40k,$250k (CEO)

Mean: $70.83k (misleading due to CEO) Median:$36.5k (better represents typical salary)

Example 2: Test Scores Class scores: 65, 70, 75, 80, 85, 90, 95

Mean = 80 (class average) Median = 80 (middle score) No mode (good distribution)

Example 3: Shoe Sizes Sizes sold: 7, 8, 8, 9, 9, 9, 9, 10, 10, 11

Mode = 9 (stock more size 9s!) Mean = 9 (but can't have size 9.0 shoe)

Data with Frequency Tables

Example: Frequency table of quiz scores

| Score | Frequency | |-------|-----------| | 7 | 2 | | 8 | 5 | | 9 | 8 | | 10 | 3 |

Mean: Sum = (7×2) + (8×5) + (9×8) + (10×3) = 14 + 40 + 72 + 30 = 156 Count = 2 + 5 + 8 + 3 = 18 Mean = 156 ÷ 18 = 8.67

Median: 18 values, so median is average of 9th and 10th values Looking at cumulative: 2, 7, 15, 18 9th and 10th values are both 9 Median = 9

Mode: 9 appears 8 times (most frequent) Mode = 9

Quartiles (Preview)

Quartiles divide data into four parts:

• Q1: 25th percentile (lower quartile)
• Q2: 50th percentile (median)
• Q3: 75th percentile (upper quartile)

Example: 2, 4, 6, 8, 10, 12, 14, 16, 18

Median (Q2) = 10 Lower half: 2, 4, 6, 8 → Q1 = 5 Upper half: 12, 14, 16, 18 → Q3 = 15

Common Mistakes to Avoid

1. Forgetting to order data for median Always sort first!

2. Not averaging two middle values for even count With 6 values, median = average of 3rd and 4th

3. Confusing mean with median Mean = average, Median = middle

4. Saying "no mode" when values appear once Either say "no mode" or "all values are modes"

5. Not using all data values for mean Include every value in the sum

6. Arithmetic errors Double-check calculations!

Choosing the Best Measure

Scenario 1: Average income in a neighborhood with one billionaire → Use MEDIAN (outliers present)

Scenario 2: Average test score for class → Use MEAN (symmetric data, no outliers)

Scenario 3: Most popular ice cream flavor → Use MODE (categorical data)

Scenario 4: Typical home price → Use MEDIAN (often skewed data)

Effect of Transformations

Adding a constant to all values: Mean, median, and mode all increase by that constant

Example: 2, 4, 6, 8 (mean = 5) Add 3 to each: 5, 7, 9, 11 (mean = 8)

Multiplying all values by constant: Mean, median, and mode all multiply by that constant

Example: 2, 4, 6, 8 (mean = 5) Multiply by 2: 4, 8, 12, 16 (mean = 10)

Box Plots Connection

Box plot (box-and-whisker plot) shows:

• Minimum
• Q1 (lower quartile)
• Median (Q2)
• Q3 (upper quartile)
• Maximum

The median is the line inside the box!

Calculator Tips

Most calculators have statistical functions:

• Enter data into lists
• Calculate 1-variable statistics
• Get mean (x̄), median (Med), etc.

Verify by hand for small datasets!

Quick Reference

Mean: Sum ÷ Count

Median: Middle value (ordered data)

• Odd count: middle value
• Even count: average of two middle values

Mode: Most frequent value

Best measure:

• Symmetric data → Mean
• Skewed/outliers → Median
• Categorical → Mode

Practice Strategy

Level 1: Small datasets (5-7 values) Find mean, median, mode

Level 2: Even vs. odd counts Practice both median cases

Level 3: Data with outliers Compare mean and median

Level 4: Find missing values Given mean or median, find unknown

Level 5: Real-world applications Decide which measure to use

Tips for Success

• Always organize data first
• Check if count is odd or even for median
• Remember mean uses ALL values
• Median is resistant to outliers
• Mode can be "none" or multiple values
• Double-check arithmetic
• Think about which measure makes sense for the context
• Practice with real-world examples
• Understand when each measure is most useful
• Use estimation to check if answers are reasonable