Mean, Median, Mode & Range
Four different ways to describe data — each tells a different story. Know which one to use and why.
The Four Measures
When someone says "average," they usually mean the mean — but there are actually four common ways to summarize data:
Mean (Average)
Formula: Add all values, divide by the count.
Mean = Sum of all values ÷ Number of values
Example 1
Test scores: 85, 92, 78, 96, 89
Mean = (85 + 92 + 78 + 96 + 89) ÷ 5 = 440 ÷ 5 = 88
Example 2
Daily temperatures: 72, 75, 68, 80, 75
Mean = (72 + 75 + 68 + 80 + 75) ÷ 5 = 370 ÷ 5 = 74°F
Median (Middle Value)
Steps: Sort the values in order, then find the middle one.
- Odd count: The middle value
- Even count: Average of the two middle values
Example 3 (Odd Count)
Data: 12, 5, 8, 3, 15
Sorted: 3, 5, 8, 12, 15
Median = 8
Example 4 (Even Count)
Data: 4, 7, 2, 9, 1, 6
Sorted: 1, 2, 4, 6, 7, 9
Median = (4 + 6) ÷ 2 = 5
Mode (Most Frequent)
The value that appears most often. A data set can have:
- One mode (unimodal): 2, 3, 3, 4, 5 → mode is 3
- Two modes (bimodal): 1, 2, 2, 3, 4, 4, 5 → modes are 2 and 4
- No mode: 1, 2, 3, 4, 5 → each appears once
Example 5
Shoe sizes sold today: 8, 9, 10, 9, 11, 8, 9, 10, 9
9 appears 4 times (most). Mode = 9
Range (Spread)
Range = Maximum value - Minimum value
Example 6
Temperatures: 62, 71, 58, 85, 73
Range = 85 - 58 = 27 degrees
When to Use Each
- Mean: Best for symmetric data without extreme outliers. Good for: test scores, temperatures, measurements.
- Median: Best when there are outliers or skewed data. Good for: income, home prices, wealth data.
- Mode: Best for categorical data. Good for: most popular shoe size, favorite color, most common grade.
- Range: Shows data variability. A large range means data is spread out; small range means clustered.
How Outliers Affect the Mean
Example 7: The Billionaire Problem
Five people in a room earn: $40K, $45K, $50K, $55K, $60K
Mean = $50K. Median = $50K. Both tell the same story.
Now Jeff Bezos walks in: $40K, $45K, $50K, $55K, $60K, $100,000,000,000
Mean = ~$16.7 billion 🤯. Median = $52.5K.
The mean is wildly misleading. The median barely changed and still represents the typical person. This is why median income is more meaningful than mean income.
A Complete Example
Daily coffee sales at a shop: 45, 52, 48, 51, 47, 52, 49
Mean: (45+52+48+51+47+52+49) ÷ 7 = 344 ÷ 7 ≈ 49.1
Median: Sorted: 45, 47, 48, 49, 51, 52, 52 → 49
Mode: 52 appears twice → 52
Range: 52 - 45 = 7
🎯 Try It Yourself
Test your understanding with these practice problems.
1. Find the mean of: 4, 8, 6, 10, 7
💡 Hint: Sum = 35, divide by 5
2. Find the median of: 12, 3, 7, 9, 15, 5, 8
💡 Hint: Sort first: 3,5,7,8,9,12,15. Middle value?
3. Find the mode of: 5, 3, 7, 3, 9, 5, 3, 8
💡 Hint: Which number appears most often?
4. Find the range of: 23, 45, 12, 67, 34
💡 Hint: Max - Min = 67 - 12
5. Salaries: $30K, $35K, $32K, $38K, $250K. Is mean or median more representative?
💡 Hint: The $250K is an outlier that pulls the mean way up
⚠️ Common Mistakes
- ✗Calculating the mean and calling it 'the average' when the median would be more meaningful (especially with outliers).
- ✗Forgetting to SORT the data before finding the median.
- ✗When there's an even number of values, forgetting to average the two middle values for the median.
- ✗Saying 'no mode' when all values appear once. That's correct — but some confuse this with 0.
- ✗Confusing range (a single number) with the data range (min to max values).
🌍 Real Life Example
Misleading House Prices
A neighborhood has 9 homes valued at: $200K, $210K, $220K, $225K, $230K, $235K, $240K, $250K, and $2,000,000. Mean price: $312K. Median price: $230K. The mean makes the neighborhood look expensive due to one mansion. A real estate agent showing you 'average home price of $312K' is technically correct but misleading. The median ($230K) better represents what most homes cost.
💡 Key Takeaway
Mean (average) uses all values and is affected by outliers. Median (middle value) resists outliers and represents the 'typical' value. Mode (most frequent) shows what's most common. Range shows spread. Use median for skewed data, mean for symmetric data, and mode for categorical data.