Mean, Median, and Mode

For Elementary Students

Three Ways to Find the "Middle"

When you have a group of numbers, you might want to find one number that represents the whole group. There are three different ways to do this!

Think about it like this: If you want to describe your test scores to a friend, which ONE number best represents all your scores?

Mean (The Average)

The mean is what most people call the "average."

How to find it: Add up all the numbers, then divide by how many numbers there are.

Example: Your test scores are: 80, 90, 70, 85, 95

Step 1: Add them all up

80 + 90 + 70 + 85 + 95 = 420

Step 2: Count how many scores (5 scores)

Step 3: Divide the total by the count

420 ÷ 5 = 84

Mean = 84

What it means: Your "average" score is 84!

Median (The Middle Number)

The median is the middle number when you line up all the numbers in order.

How to find it:

1. Put the numbers in order from smallest to biggest
2. Find the one in the middle

Example: Test scores: 70, 80, 85, 90, 95

70, 80, 85, 90, 95
        ↑
     middle!

Median = 85 (it's the third number out of five)

What If There's No Exact Middle?

Example: Numbers: 4, 8, 10, 14 (four numbers — no exact middle!)

4, 8, 10, 14
   ↑  ↑
  two middle numbers

Step 1: Find the two middle numbers: 8 and 10

Step 2: Find their average

(8 + 10) ÷ 2 = 18 ÷ 2 = 9

Median = 9

Mode (The Most Common)

The mode is the number that appears most often (the most popular number!).

Example: Numbers: 5, 3, 5, 8, 5, 9, 3

Count each number:

• 5 appears 3 times ← appears the most!
• 3 appears 2 times
• 8 appears 1 time
• 9 appears 1 time

Mode = 5 (because 5 shows up the most)

What If Two Numbers Tie?

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

• 2 appears twice
• 4 appears twice
• Everything else appears once

This set has TWO modes: 2 and 4

Memory trick: "Mode" sounds like "Most" — the MOST common number!

For Junior High Students

Three Measures of Center

When you have a set of numbers (a data set), you often want one number that represents the "middle" or "typical" value. There are three common ways to find it, each useful in different situations.

Vocabulary:

• Data set — a collection of numbers
• Measure of center — a single value that represents the center or typical value of the data

Mean (Average)

The mean is what most people call the "average." Add up all the values and divide by how many there are.

Formula: mean = sum of all values ÷ number of values

Symbol: Often written as x̄ (pronounced "x-bar")

Example: Test scores: 80, 90, 70, 85, 95

Step 1: Sum: 80 + 90 + 70 + 85 + 95 = 420

Step 2: Count: 5 scores

Step 3: Mean: 420 ÷ 5 = 84

Interpretation: The average test score is 84.

Properties of the mean:

• Uses ALL the data values
• Sensitive to extreme values (outliers)
• Most commonly used measure

Median (Middle Value)

The median is the middle number when the values are lined up in order.

Steps:

1. Sort the numbers from smallest to largest
2. Find the middle one

Example 1 (odd count): Numbers: 3, 7, 9, 12, 15

Already sorted! Count: 5 numbers (odd)

Middle position: (5 + 1) ÷ 2 = 3rd position

Median = 9 (the third of five numbers)

Example 2 (even count): Numbers: 4, 8, 10, 14

Count: 4 numbers (even)

Two middle numbers are 8 and 10 (positions 2 and 3)

Median = (8 + 10) ÷ 2 = 9

Properties of the median:

• Not affected by extreme values (outliers)
• Good for skewed data
• Represents the "typical" value better when there are outliers

The median divides the data in half: 50% of values are below it, 50% are above it.

Mode (Most Frequent)

The mode is the value that appears most often in the data set.

Example: Numbers: 5, 3, 5, 8, 5, 9, 3

Frequency count:

• 5 appears 3 times
• 3 appears 2 times
• 8 and 9 appear 1 time each

Mode = 5

Types of modes:

A data set can have:

• One mode (unimodal) — one value appears most often

  • Example: 1, 2, 2, 3 → mode is 2

• Two modes (bimodal) — two values tie for most frequent

  • Example: 1, 2, 2, 3, 4, 4 → modes are 2 and 4

• Three or more modes (multimodal)

  • Example: 1, 1, 2, 2, 3, 3 → modes are 1, 2, and 3

• No mode — all values appear the same number of times

  • Example: 1, 2, 3, 4 → no mode

Properties of the mode:

• Only measure that can be used for non-numeric data (like colors or categories)
• Not always unique
• Not affected by extreme values

When to Use Each Measure

Comparing Mean, Median, and Mode

Example: Salaries at a company: $30k, $35k, $40k, $42k, $500k

Mean: ($30k + $35k + $40k + $42k + $500k) ÷ 5 = $647k ÷ 5 = $129.4k

Median: Sort: $30k, $35k, $40k, $42k, $500k → middle is $40k

Mode: No mode (all values appear once)

Which is best? The median ($40k) better represents a "typical" salary because the mean ($129.4k) is pulled up by the one very high salary ($500k).

Effects of Outliers

An outlier is a value much higher or lower than the others.

Example: Test scores: 85, 88, 90, 92, 30 (the 30 is an outlier)

Mean: (85 + 88 + 90 + 92 + 30) ÷ 5 = 385 ÷ 5 = 77

Median: Sort: 30, 85, 88, 90, 92 → middle is 88

The outlier (30) pulled the mean down to 77, but the median (88) still represents the typical score.

Real-Life Applications

Weather: "The average temperature in July is 85°F" (mean)

Housing: "The median home price is $300,000" (median used because extreme prices skew the data)

Fashion: "Size 8 is the most common shoe size" (mode)

Income: "The median household income is $65,000" (median preferred over mean due to very high incomes)

Calculating with Frequency Tables

Example: Survey results

Mean: (1×2 + 2×5 + 3×3) ÷ (2+5+3) = (2+10+9) ÷ 10 = 21 ÷ 10 = 2.1

Median: 10 total values → middle values are 5th and 6th

• Values in order: 1,1,2,2,2,2,2,3,3,3
• Middle: 2 and 2 → median = 2

Mode: 2 appears most (5 times) → mode = 2

Practice