Mean, Median, and Mode

How do we describe the "center" or "typical value" of a data set? These three measures help us understand data and make comparisons!

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What Are Measures of Central Tendency?

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

Three main measures:

• Mean: The average
• Median: The middle value
• Mode: The most frequent value

Each tells us something different about the data!

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What Is the Mean?

The mean is the arithmetic average of all values.

Formula: Mean = Sum of all values ÷ Number of values

Symbol: x̄ (read as "x bar")

Process:

1. Add all values
2. Divide by how many values there are

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

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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!

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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!

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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!

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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!

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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!

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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!

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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!

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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

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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.

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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.

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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.

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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!

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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!

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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!

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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.

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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!

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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.

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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!

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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!

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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!

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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

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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

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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!

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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!

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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

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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!

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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

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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

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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

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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

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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!