Mean, Median, and Mode | Open Textbooks

What Are Measures of Central Tendency?

When you have a set of numbers (data), you often want to describe it with a single value that represents the "typical" or "middle" value. We call these measures of central tendency. The three most common measures are:

📊

Mean

The average - add all values and divide by how many

🎯

Median

The middle value when data is in order

🔄

Mode

The value that appears most often

The Mean (Average)

Mean = Sum of All Values / Count of Values

The mean is what most people call the "average." It balances all the values in your data set.

Mean = (Sum of all values) / (Number of values)

Example: Finding the Mean

Test scores:

85 90 78 92 85

Step 1: Add all values: 85 + 90 + 78 + 92 + 85 = 430
Step 2: Divide by count: 430 / 5 = 86

The mean is 86

Pro Tip: Think of the mean as the "balance point" of your data. If you put all your values on a number line seesaw, the mean is where it would balance!

The Median (Middle Value)

Median = The Middle Value

Order your data from least to greatest, then find the value in the exact middle.

For odd count: Middle value | For even count: Average of two middle values

Example: Finding the Median (Odd Count)

Ages of 5 students:

14 12 15 11 13

Step 1: Order from least to greatest: 11, 12, 13, 14, 15
Step 2: Find the middle (position 3 of 5)

The median is 13

Example: Finding the Median (Even Count)

Prices of 6 items:

$5 $8 $12 $6 $9 $15

Step 1: Order from least to greatest: $5, $6, $8, $9, $12, $15
Step 2: Two middle values (positions 3 and 4): $8 and $9
Step 3: Average them: ($8 + $9) / 2 = $8.50

The median is $8.50

The Mode (Most Frequent)

Mode = Most Frequent Value

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

Count how many times each value appears - the highest count wins!

Example: Finding the Mode

Shoe sizes sold today:

7 8 9 8 10 8 7

Size 7: appears 2 times
Size 8: appears 3 times (most frequent!)
Size 9: appears 1 time
Size 10: appears 1 time

The mode is 8

Important! A data set can have:

No mode - if all values appear the same number of times
One mode - if one value appears most often
Multiple modes - if two or more values tie for most frequent

When to Use Each Measure

Measure	Best When...	Watch Out For...
Mean	Data is evenly distributed without extreme values	Outliers (extreme values) can skew the mean significantly
Median	Data has outliers or is skewed (like salaries or house prices)	Doesn't account for how spread out values are
Mode	You want to know the most common or popular value	May not exist or may have multiple modes

Real-World Applications

📝

Grades

Mean: Your GPA
Median: Middle performance
Mode: Most common grade

💰

Salaries

Median is preferred because a few high earners can skew the mean

🏀

Sports Stats

Mean: Season average
Mode: Most common score

📋

Surveys

Mode: Most popular answer
Median: Typical response

Try It: Mean Calculator

Enter your own numbers (separated by commas) and calculate the mean, median, and mode!

Statistics Calculator

Enter numbers separated by commas:

Your Data (sorted): -

Count: -

Sum: -

Mean: -

Median: -

Mode: -

Worked Examples

Let's solve some problems step by step.

Example 1: Finding All Three Measures

A basketball player scored these points in 7 games: 18, 22, 15, 22, 25, 22, 20. Find the mean, median, and mode.

1

Find the Mean:
Sum = 18 + 22 + 15 + 22 + 25 + 22 + 20 = 144
Mean = 144 / 7 = 20.57 (rounded to 2 decimal places)

2

Find the Median:
Order the data: 15, 18, 20, 22, 22, 22, 25
Middle value (position 4 of 7) = 22

3

Find the Mode:
15 appears 1 time, 18 appears 1 time, 20 appears 1 time
22 appears 3 times, 25 appears 1 time
Mode = 22

Mean = 20.57 | Median = 22 | Mode = 22

Example 2: The Impact of Outliers

Six employees earn these hourly wages: $12, $14, $15, $13, $14, $85. Find the mean and median. Which better represents the typical wage?

1

Find the Mean:
Sum = 12 + 14 + 15 + 13 + 14 + 85 = 153
Mean = 153 / 6 = $25.50

2

Find the Median:
Order: $12, $13, $14, $14, $15, $85
Two middle values: ($14 + $14) / 2 = $14

3

Compare:
The $85 outlier pulled the mean up to $25.50, but 5 out of 6 employees earn less than that!
The median of $14 better represents the "typical" wage.

Mean = $25.50 | Median = $14
The median ($14) better represents the typical wage because the $85 outlier skews the mean.

Example 3: Multiple Modes

Students were asked their favorite number from 1-10. Results: 3, 7, 3, 9, 7, 5, 3, 7, 2. Find the mode(s).

1

Count each value:
2: 1 time | 3: 3 times | 5: 1 time | 7: 3 times | 9: 1 time

2

Find the highest frequency:
Both 3 and 7 appear 3 times (the most)

This data set is bimodal (two modes): 3 and 7

Practice Problems

Try these problems on your own. Enter your answer and check if you're correct!

Problem 1: Finding the Mean

Find the mean of these quiz scores: 8, 9, 7, 10, 6

8 9 7 10 6

Mean:

Problem 2: Finding the Median

Find the median of: 23, 18, 31, 25, 19, 27, 22

Median:

Problem 3: Finding the Mode

Find the mode of: 4, 7, 4, 9, 4, 7, 2, 4

Mode:

Problem 4: Median with Even Count

Find the median of: 12, 8, 15, 10, 6, 14

Median:

Problem 5: Mean with Decimals

The heights of 4 plants are: 12.5 cm, 14.0 cm, 11.5 cm, 14.0 cm. What is the mean height?

Mean:

Check Your Understanding: Statistics Challenge

Test your knowledge of mean, median, and mode with this 6-question challenge!

Mean, Median, Mode Challenge

Score: 0 / 6

Question 1 of 6

Challenge Complete!

0/6

Next Steps

Key Takeaways:

Mean = Sum / Count (the average, affected by outliers)
Median = Middle value when ordered (not affected by outliers)
Mode = Most frequent value (can have 0, 1, or multiple modes)
Choose the right measure based on your data and what you want to show
When data has outliers, median often represents the "typical" value better

Practice finding all three measures for different data sets
Look for real-world examples of mean, median, and mode in the news
Move on to learn about data distribution in the next lesson