Statistics - Fifth Grade

Complete Notes & Formulas

Understanding Statistics

What is Statistics?

Statistics is the study of collecting, organizing, analyzing, and interpreting data. It helps us understand information and make decisions based on numbers.

Measures of Central Tendency

These are numbers that describe the center or typical value of a data set.

Measure	What It Shows
Mean	The average of all numbers
Median	The middle number when data is ordered
Mode	The most frequently occurring number
Range	The spread of data (highest − lowest)

1. Find the Mode

What is Mode?

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

Formula

Mode = The number that appears most frequently

Steps to Find Mode

Step 1: Look at all the numbers in the data set

Step 2: Count how many times each number appears

Step 3: The number that appears most often is the mode

Important Facts

No Mode: If all numbers appear the same number of times

Example: 1, 2, 3, 4, 5 (no mode)

One Mode (Unimodal): One number appears most often

Example: 2, 3, 3, 4, 5 (mode = 3)

Two Modes (Bimodal): Two numbers appear equally most often

Example: 1, 2, 2, 3, 4, 4 (modes = 2 and 4)

Multiple Modes (Multimodal): More than two numbers appear equally most often

Example: 1, 1, 2, 2, 3, 3 (modes = 1, 2, and 3)

Examples

Example 1: Find the mode: 7, 5, 9, 5, 8, 5, 3

Count:

• 7 appears 1 time

• 5 appears 3 times

• 9 appears 1 time

• 8 appears 1 time

• 3 appears 1 time

Mode = 5 (appears most often)

Example 2: Find the mode: 12, 15, 18, 15, 20, 18

• 15 appears 2 times

• 18 appears 2 times

Modes = 15 and 18 (bimodal)

2. Find the Mean (Average)

What is Mean?

The mean (or average) is the sum of all values divided by the number of values.

Formula

Mean = Sum of All Values ÷ Number of Values

Mean = (x₁ + x₂ + x₃ + ... + xₙ) ÷ n

Steps to Find Mean

Step 1: Add all the numbers together

Step 2: Count how many numbers there are

Step 3: Divide the sum by the count

Examples

Example 1: Find the mean: 8, 12, 10, 14, 6

Step 1: Add: 8 + 12 + 10 + 14 + 6 = 50

Step 2: Count: 5 numbers

Step 3: Divide: 50 ÷ 5 = 10

Mean = 10

Example 2: Find the mean: 15, 20, 25, 30

Sum: 15 + 20 + 25 + 30 = 90

Count: 4 numbers

Mean: 90 ÷ 4 = 22.5

Mean = 22.5

3. Find the Median

What is Median?

The median is the middle number when data is arranged in order from smallest to largest.

Formulas

For ODD number of values:

Median = Middle number

For EVEN number of values:

Median = (Two middle numbers) ÷ 2

Steps to Find Median

Step 1: Put the numbers in order from smallest to largest

Step 2: Count how many numbers there are

Step 3: If odd count: Find the middle number

Step 4: If even count: Find the two middle numbers, add them, divide by 2

Examples

Example 1 (Odd): Find the median: 7, 3, 9, 5, 11

Step 1: Order: 3, 5, 7, 9, 11

Step 2: Count: 5 numbers (odd)

Step 3: Middle number = 7

Median = 7

Example 2 (Even): Find the median: 4, 8, 2, 10, 6, 12

Step 1: Order: 2, 4, 6, 8, 10, 12

Step 2: Count: 6 numbers (even)

Step 3: Two middle numbers: 6 and 8

Step 4: (6 + 8) ÷ 2 = 14 ÷ 2 = 7

Median = 7

4. Find the Range

What is Range?

The range shows how spread out the data is. It's the difference between the highest and lowest values.

Formula

Range = Highest Value − Lowest Value

Range = Maximum − Minimum

Steps to Find Range

Step 1: Find the largest number in the data set

Step 2: Find the smallest number in the data set

Step 3: Subtract: Largest − Smallest

Examples

Example 1: Find the range: 5, 12, 3, 18, 9, 7

Highest: 18

Lowest: 3

Range: 18 − 3 = 15

Range = 15

Example 2: Find the range: 25, 30, 22, 35, 28

Highest: 35, Lowest: 22

35 − 22 = 13

Range = 13

5. Mean: Find the Missing Number

Working Backwards

When you know the mean and some values, you can find a missing number by working backwards.

Steps to Find Missing Number

Step 1: Multiply the mean by the total count of numbers

Step 2: Add all the known numbers

Step 3: Subtract the sum of known numbers from the product

Formula

Missing Number = (Mean × Count) − Sum of Known Numbers

Examples

Example 1: The mean of 5 numbers is 12. Four numbers are 10, 15, 8, and 14. Find the missing number.

Step 1: Mean × Count = 12 × 5 = 60

Step 2: Sum of known: 10 + 15 + 8 + 14 = 47

Step 3: Missing number: 60 − 47 = 13

Missing number = 13

Check: (10 + 15 + 8 + 14 + 13) ÷ 5 = 60 ÷ 5 = 12 ✓

Example 2: Mean = 20, Data: 18, 22, x, 24. Find x.

Total needed: 20 × 4 = 80

Known sum: 18 + 22 + 24 = 64

x = 80 − 64 = 16

x = 16

6. Median: Find the Missing Number

Strategy

To find a missing number when you know the median, consider where the number must fall in the ordered list.

Steps

Step 1: Arrange known numbers in order

Step 2: Identify the position of the median

Step 3: Determine what value the missing number must be

Step 4: Use logic to find the missing number

Examples

Example 1: The median is 15. Data: 10, 12, x, 18, 20. Find x.

Step 1: 5 numbers (odd), so median is the 3rd number

Step 2: The 3rd number must be 15

Step 3: Since x is in position 3: x = 15

x = 15

Ordered: 10, 12, 15, 18, 20 (median = 15 ✓)

Example 2: Median = 8. Data: 5, 7, x, 10 (4 numbers). Find x.

4 numbers (even), median = (2nd + 3rd) ÷ 2

8 = (7 + x) ÷ 2

16 = 7 + x

x = 9

7. Range: Find the Missing Number

Strategy

When you know the range, you can find a missing number by using the formula: Range = Highest − Lowest

Two Scenarios

Scenario 1: Missing number is the highest

Missing number = Lowest + Range

Scenario 2: Missing number is the lowest

Missing number = Highest − Range

Examples

Example 1: Range = 12. Data: 5, 8, 11, x. If x is the largest, find x.

Lowest number: 5

Range: 12

Formula: x = Lowest + Range

x = 5 + 12 = 17

x = 17

Check: 17 − 5 = 12 ✓

Example 2: Range = 15. Data: x, 8, 12, 18. If x is the smallest, find x.

Highest: 18

x = Highest − Range

x = 18 − 15 = 3

x = 3

8-11. Interpret Charts and Graphs to Find Statistics

Reading Data from Graphs

You can find mean, median, mode, and range from various types of graphs by reading the data values first.

Types of Graphs

Bar Graphs: Read the height/length of each bar

Line Plots: Count the X's or dots above each value

Line Graphs: Read values from the points on the line

Tables: Read values directly from the table

Steps to Interpret

Step 1: Read the graph title and labels

Step 2: Extract all data values from the graph

Step 3: Apply the appropriate formula (mean, median, mode, or range)

Step 4: Calculate and check your answer

Example: Bar Graph

Favorite Ice Cream Flavors:

Flavor	Votes
Chocolate	12
Vanilla	8
Strawberry	12
Mint	6

Find: Mode, Mean, Median, Range

Mode: 12 (appears twice - Chocolate and Strawberry)

Mean:

Sum: 12 + 8 + 12 + 6 = 38

Count: 4 flavors

Mean: 38 ÷ 4 = 9.5

Median:

Order: 6, 8, 12, 12

Middle: (8 + 12) ÷ 2 = 10

Range:

12 − 6 = 6

Example: Line Plot

Quiz Scores (out of 10):

X
X    X    X
X    X    X    X
--|--|--|--|
7  8  9  10

Data: 7, 7, 8, 8, 9, 9, 10 (7 scores total)

Mode: 7, 8, and 9 (all appear 2 times - trimodal)

Mean: (7+7+8+8+9+9+10) ÷ 7 = 58 ÷ 7 = 8.29

Median: 8 (middle of 7 numbers)

Range: 10 − 7 = 3

Quick Reference: Statistics Formulas

Measure	Formula
Mean	Sum of all values ÷ Number of values
Median (Odd)	Middle number
Median (Even)	(Two middle numbers) ÷ 2
Mode	Most frequently occurring value
Range	Highest value − Lowest value