The mean is what people usually mean by "average"!

Calculating the Mean

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

Step 1: Add all values 85 + 90 + 78 + 92 + 85 = 430

Step 2: Count how many values 5 scores

Step 3: Divide Mean = 430 ÷ 5 = 86

The mean score is 86!

Mean with Different Numbers

Example: Ages: 12, 15, 13, 11, 14, 15

Sum = 12 + 15 + 13 + 11 + 14 + 15 = 80 Count = 6 Mean = 80 ÷ 6 ≈ 13.33 years

Mean can be a decimal even if all data are whole numbers!

What Is the Median?

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

Process:

1. Arrange data from smallest to largest
2. Find the middle value
3. If two middle values, take their average

The median divides the data in half!

Finding the Median (Odd Number of Values)

Example: 7, 3, 9, 5, 1

Step 1: Arrange in order 1, 3, 5, 7, 9

Step 2: Find the middle 1, 3, 5, 7, 9

Median = 5 (middle value)

With odd number of values, there's one middle value!

Finding the Median (Even Number of Values)

Example: 8, 4, 6, 2

Step 1: Arrange in order 2, 4, 6, 8

Step 2: Find the two middle values 2, 4, 6, 8

Step 3: Average the two middle values Median = (4 + 6) ÷ 2 = 10 ÷ 2 = 5

Median = 5

With even number of values, average the two middle values!

What Is the Mode?

The mode is the value that appears most frequently.

Example: 5, 7, 3, 7, 9, 7, 4

7 appears three times (more than any other)

Mode = 7

The mode is the most common value!

No Mode

Some data sets have no mode:

Example: 5, 8, 11, 14, 17

Each value appears only once.

No mode (or all values are modes)

This is perfectly fine!

Multiple Modes (Bimodal or Multimodal)

Data can have more than one mode:

Example: 3, 5, 5, 7, 9, 9, 11

Both 5 and 9 appear twice.

Modes = 5 and 9 (bimodal)

Bimodal = two modes Multimodal = more than two modes

Comparing Mean, Median, and Mode

Data: 2, 4, 4, 5, 7, 9

Mean: (2 + 4 + 4 + 5 + 7 + 9) ÷ 6 = 31 ÷ 6 ≈ 5.17

Median: 2, 4, 4, 5, 7, 9 → (4 + 5) ÷ 2 = 4.5

Mode: 4 (appears twice)

All different! Each measures "center" differently.

When to Use Each Measure

Mean:

• Use when data is fairly symmetric
• All values contribute
• Sensitive to extreme values

Median:

• Use with skewed data or outliers
• Not affected by extreme values
• Better for income, home prices

Mode:

• Use for categorical data
• Most common or popular item
• Favorite color, shoe size, etc.

Effect of Outliers on Mean

Outlier: A value much larger or smaller than others

Example: Salaries: $30,000,$32,000, $35,000,$33,000, $200,000

Mean: ($30,000 +$32,000 + $35,000 +$33,000 + $200,000) ÷ 5 =$330,000 ÷ 5 = $66,000

Median: $30,000,$32,000, **$33,000**,$35,000, $200,000 =$33,000

The outlier ($200,000) pulls the mean way up!** **The median ($33,000) better represents typical salary.

Symmetric Data

In symmetric data, mean ≈ median:

Example: 10, 12, 14, 16, 18

Mean = 70 ÷ 5 = 14 Median = 14 (middle value)

When distribution is balanced, mean and median are similar!

Skewed Data

Right-skewed (positive skew): Few large values

• Mean > Median
• Large values pull mean higher

Left-skewed (negative skew): Few small values

• Mean < Median
• Small values pull mean lower

Median is more resistant to skewness!

Real-World Example: Test Scores

Class test scores: 65, 70, 75, 75, 80, 80, 80, 85, 90, 95

Mean: 795 ÷ 10 = 79.5

Median: 65, 70, 75, 75, 80, 80, 80, 85, 90, 95 = (80 + 80) ÷ 2 = 80

Mode: 80 (appears three times)

Most students scored around 80!

Real-World Example: Home Prices

Home prices in neighborhood: $150K,$160K, $170K,$175K, $180K,$900K

Mean: $1,735K ÷ 6 ≈$289K

Median: $150K,$160K, $170K,$175K, $180K,$900K = ($170K +$175K) ÷ 2 = $172.5K

Median ($172.5K) better represents typical home!** **Mean ($289K) inflated by the $900K mansion.

Finding Missing Value Given Mean

Problem: Four numbers have mean of 15. Three numbers are 12, 14, and 18. Find the fourth.

Solution: Mean = 15, Count = 4 Sum of all four = 15 × 4 = 60

Known sum: 12 + 14 + 18 = 44 Missing value: 60 - 44 = 16

The fourth number is 16!

Mode for Categorical Data

Mode works great for non-numerical data:

Favorite colors: Blue, Red, Blue, Green, Blue, Yellow, Red, Blue

Mode = Blue (appears 4 times)

Can't calculate mean or median for colors!

Mode is the only measure that works for categorical data.

Data Set with All Three Equal

Example: 10, 10, 10, 10, 10

Mean: 50 ÷ 5 = 10 Median: 10 (middle value) Mode: 10 (appears five times)

When all values are the same, all three measures equal that value!

Adding a Value to Data Set

Original: 5, 7, 9 (Mean = 7, Median = 7)

Add 11: New data: 5, 7, 9, 11

New mean: 32 ÷ 4 = 8 New median: (7 + 9) ÷ 2 = 8

Adding values changes mean and median!

Range (Quick Introduction)

Range measures spread, not center:

Range = Largest value - Smallest value

Example: 3, 7, 9, 12, 14

Range = 14 - 3 = 11

Range tells how spread out the data is!

Step-by-Step: Finding All Three

Data: 15, 12, 18, 12, 20, 15, 12

Mean: Sum = 15 + 12 + 18 + 12 + 20 + 15 + 12 = 104 Count = 7 Mean = 104 ÷ 7 ≈ 14.86

Median: Order: 12, 12, 12, 15, 15, 18, 20 Middle (4th value) = 15

Mode: 12 appears three times (most frequent) Mode = 12

Mean of Grouped Data

When data is already grouped:

Example:

• Score 70: 2 students
• Score 80: 5 students
• Score 90: 3 students

Calculate: (70×2 + 80×5 + 90×3) ÷ (2 + 5 + 3) = (140 + 400 + 270) ÷ 10 = 810 ÷ 10 = 81

Mean score = 81

Using Mean to Find Total

If mean = 12 and there are 8 values:

Sum of all values = Mean × Count = 12 × 8 = 96

Useful for reverse problems!

Weighted Mean

Different values have different importance:

Example: Final grade

• Tests (worth 3): 80, 85, 90
• Homework (worth 1): 95

Weighted mean: (80×3 + 85×3 + 90×3 + 95×1) ÷ (3 + 3 + 3 + 1) = (240 + 255 + 270 + 95) ÷ 10 = 860 ÷ 10 = 86

Some values count more!

Common Mistakes to Avoid

❌ Mistake 1: Not ordering data for median

• Must arrange in order first!

❌ Mistake 2: Forgetting to average two middle values

• When even count, take mean of two middle values

❌ Mistake 3: Confusing mode with median

• Mode = most frequent
• Median = middle value

❌ Mistake 4: Thinking there must be a mode

• Data can have no mode (all different)
• Or multiple modes

❌ Mistake 5: Rounding too early

• Keep decimals until final answer
• Then round appropriately

Which Measure to Report?

Use mean when:

• Data is symmetric
• No extreme outliers
• All values matter equally

Use median when:

• Data has outliers
• Data is skewed
• Want "typical" value (salaries, home prices)

Use mode when:

• Data is categorical
• Want most common value
• Finding most popular item

Context determines the best choice!

Real-World Applications

Education:

• Average test scores
• Median GPA
• Most common grade

Business:

• Mean sales per day
• Median customer spend
• Most popular product (mode)

Sports:

• Average points per game
• Median batting average
• Most common score

Economics:

• Median household income
• Average price
• Most common purchase

Problem-Solving Strategy

To find mean:

1. Add all values
2. Divide by count
3. Round if appropriate

To find median:

1. Order from smallest to largest
2. Find middle value(s)
3. If even count, average the two middle

To find mode:

1. Look for most frequent value
2. Can be none, one, or multiple
3. Count carefully!

To choose which to use:

1. Look at data distribution
2. Consider outliers
3. Think about context

Quick Reference

Mean (Average):

• Sum ÷ Count
• Affected by all values
• Sensitive to outliers

Median (Middle):

• Middle value when ordered
• Resistant to outliers
• Better for skewed data

Mode (Most Common):

• Most frequent value
• Can be none or multiple
• Only measure for categorical data

Remember:

• Mean uses all values
• Median uses position
• Mode uses frequency

Practice Tips

Tip 1: Always organize data first

• Makes finding median easier
• Helps spot mode
• Reduces errors

Tip 2: Check your work

• Does mean make sense?
• Is median actually in the middle?
• Did you count mode frequencies?

Tip 3: Consider outliers

• Look at data before choosing measure
• Think about what outliers mean

Tip 4: Practice with real data

• Sports statistics
• Grade averages
• Allowance or spending

Tip 5: Understand what each measures

• Mean = balance point
• Median = middle
• Mode = most popular

Summary

Three measures describe the center of data:

Mean:

• Arithmetic average: sum ÷ count
• Uses all values
• Sensitive to outliers
• Best for symmetric data

Median:

• Middle value when ordered
• Position-based measure
• Resistant to outliers
• Best for skewed data or outliers

Mode:

• Most frequent value
• Can be none, one, or multiple
• Only measure for categorical data
• Shows most common value

Key concepts:

• Each measures "center" differently
• Context determines best choice
• Outliers strongly affect mean
• Median more resistant to extremes

Applications:

• Comparing data sets
• Understanding typical values
• Making decisions based on data
• Describing distributions

Problem-solving:

• Calculate all three when possible
• Compare to understand data better
• Choose appropriate measure for context
• Consider outliers and distribution

Understanding mean, median, and mode is fundamental for data analysis and statistics!