Median and Mode

Learn to find the median (middle value) and mode (most frequent value) of a data set.

Elementary20 minLesson
Printable worksheet

Definition

The median and mode are two measures of central tendency that help us describe the "center" of a data set.
Median: The middle value when data is arranged in order.
  • If there's an odd number of values, the median is the middle one
  • If there's an even number, the median is the average of the two middle values
Mode: The value that appears most often.
  • A data set can have one mode, multiple modes, or no mode at all
Data: 2,3,5,5,7→Median=5, Mode=5\text{Data: } 2, 3, 5, 5, 7 \quad \rightarrow \quad \text{Median} = 5, \text{ Mode} = 5

Try it now

What is the median of: 3,5,7,9,113, 5, 7, 9, 11?

Worked Examples

Find the median of: 12,5,8,3,912, 5, 8, 3, 9
1

Arrange in order from least to greatest

3,5,8,9,123, 5, 8, 9, 12 → 5 values in order
2

Count the values

There are 5 values (odd number) → Look for the middle value
3

Find the middle position

Position = (5+1)÷2=3(5 + 1) \div 2 = 3 → 3rd value is the median
4

Identify the median

3,5,8,9,123, 5, \boxed{8}, 9, 12 → Median = 88

Common Mistakes

Forgetting to order the data before finding the median

Why it's wrong: Without ordering, you might pick a value that's not actually in the middle.

Correct: ALWAYS sort from least to greatest first: 5,2,8,1→1,2,5,85, 2, 8, 1 \rightarrow 1, 2, 5, 8

Saying there's no mode when all values appear once

Why it's wrong: This is actually correct! A data set can have no mode.

Correct: If every value appears the same number of times, there is no mode.

Confusing median position with median value

Why it's wrong: In 1,3,5,7,91, 3, 5, 7, 9, the median is at position 3, but the median VALUE is 55.

Correct: First find the position, then identify the value at that position.

For even data sets, picking one of the middle values instead of their average

Why it's wrong: With even counts, the median falls between two values.

Correct: For 4,6,8,104, 6, 8, 10: median = (6+8)÷2=7(6 + 8) \div 2 = 7, not just 66 or 88.

Interactive Visual

Dot Plot

Points in 5 games

01020304050

Each dot is one value (5 values in total).

Mean = 17

Median = 10

Mode = 10

Tap a column to count its dots.

Dot Plot

3456789

Each dot is one value (7 values in total).

Median = 5

Mode = 3

Tap a column to count its dots.

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

15 problems
Problem 1 of 15
Easy
What is the median of: 3,5,7,9,113, 5, 7, 9, 11?

Why It Matters

Median and mode give us different insights than the mean:
  • Median for salaries: When a CEO earns millions and most employees earn modest wages, the median shows what a "typical" worker earns better than the mean
  • Mode for shoe sizes: A shoe store needs to know which size sells most, not the average size
  • Median for home prices: Real estate agents use median prices because a few mansions can skew the mean
  • Mode for survey results: Finding the most popular choice in a vote or survey
Choosing the right measure helps you understand your data correctly!

Real World Applications

Real Estate Prices

Home prices are often reported as median rather than mean because a few luxury homes can drastically increase the average.

Example:

In a neighborhood, home prices are: 200000, 220000, 250000, 280000, and 1500000 dollars. The mean is 490000 dollars, but the median is 250000 dollars - much more representative of typical homes.

1Try It Yourself

Five houses sold for: 180000, 195000, 210000, 225000, and 750000 dollars.

What is the median home price?

Step 1: Write the mathematical expression

Order the values and find the middle one:

Retail Inventory

Stores use mode to determine which products to stock more of.

Example:

T-shirt sales by size: S(12), M(28), L(28), XL(15). The modes are M and L - the store should order more medium and large shirts.

2Try It Yourself

A bookstore tracks daily sales: Monday 15, Tuesday 22, Wednesday 22, Thursday 18, Friday 22 books.

What is the mode of daily book sales?

Step 1: Write the mathematical expression

Find the value that appears most often:

Test Scores

Teachers use median to understand class performance without outliers affecting the result.

Example:

Class scores: 45, 72, 75, 78, 80, 82, 85, 88, 92. The median is 80, showing where the middle student performed.

3Try It Yourself

Six students scored: 65, 70, 78, 84, 90, 95 on a test.

What is the median score?

Step 1: Write the mathematical expression

Find the average of the two middle values:

Key Takeaways

  • 1The median is the middle value when data is ordered from least to greatest
  • 2For an odd number of values, the median is the middle one
  • 3For an even number, the median is the average of the two middle values
  • 4The mode is the value that appears most frequently
  • 5A data set can have no mode, one mode, or multiple modes
  • 6Use median when outliers might skew the data; use mode to find the most common value

Frequently Asked Questions

Yes! In 3,5,5,5,73, 5, 5, 5, 7, the number 55 is both the median (middle value) and the mode (appears most often).
Yes! In 3,5,5,5,73, 5, 5, 5, 7, the number 55 is both the median (middle value) and the mode (appears most often).
If every value appears exactly once, there is no mode. For example, 2,4,6,82, 4, 6, 8 has no mode.
The mean can be pulled by extreme values. If 4 students score 80 and one scores 5, the mean is 65, but the median (80) better represents typical performance.

Glossary

Median
The middle value of a data set when arranged in order
Mode
The value that appears most frequently in a data set
Bimodal
A data set with two modes (two values that appear equally often and more than others)
Central tendency
A measure that represents the center or typical value of a data set (mean, median, or mode)
Outlier
A value that is much higher or lower than most other values in a data set
Today's challengeOne problem, three tries

More in This Topic

Mean (Average)

Elementary20 min