Median and Mode
Learn to find the median (middle value) and mode (most frequent value) of a data set.
Definition
- 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
- A data set can have one mode, multiple modes, or no mode at all
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Worked Examples
Find the median of:
Arrange in order from least to greatest
→ 5 values in order
Count the values
There are 5 values (odd number) → Look for the middle value
Find the middle position
Position = → 3rd value is the median
Identify the median
→ Median =
Answer: The median is
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:
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 , the median is at position 3, but the median VALUE is .
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 : median = , not just or .
Interactive Visual
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Practice Problems
15 problemsWhat is the median of: ?
Why It Matters
- 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
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.
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.
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.
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
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