Calculate the arithmetic mean (average) of a set of numbers with step-by-step solutions and detailed explanations
Separate numbers with commas or spaces
Numbers: 85, 92, 78, 90, 88
Numbers: 120, 150, 135, 180, 160
Numbers: 72, 68, 75, 70, 73
Numbers: 3.5, 4.2, 3.8, 4.0, 3.9
Numbers: 1000, 1500, 1200, 1800, 1400
The arithmetic mean (average) of n numbers is calculated by summing all values and dividing by the count:
Where are the numbers and is the count.
Example:
Mean of 10, 20, 30 = (10 + 20 + 30) / 3 = 60 / 3 = 20
• The mean is the most common measure of central tendency
• It represents the "average" or "typical" value in a dataset
• Every value contributes equally to the calculation
• The mean is sensitive to outliers (extreme values)
• Used extensively in statistics, finance, and data analysis
📊 Academic Performance
Calculate average test scores, GPA, or class performance metrics
💰 Financial Analysis
Track average daily sales, monthly expenses, or investment returns
⚕️ Health Metrics
Monitor average blood pressure, heart rate, or weight measurements
🌡️ Environmental Data
Analyze average temperature, rainfall, or pollution levels
❌ Ignoring Outliers
One extreme value can significantly skew the mean. Consider using median for skewed data.
❌ Mixing Units
Ensure all numbers use the same units (e.g., don't mix feet and inches).
❌ Division by Zero
You need at least one number to calculate a mean.
✅ Best Practice
Check for outliers and consider whether mean or median is more appropriate for your data.
| Measure | Definition | Best Used When | Example |
|---|---|---|---|
| Mean | Sum divided by count | Data is symmetric with no outliers | Test scores: 80, 85, 90 → Mean = 85 |
| Median | Middle value when sorted | Data has outliers or is skewed | Salaries: 40K, 45K, 200K → Median = 45K |
| Mode | Most frequent value | Categorical or discrete data | Shoe sizes: 8, 8, 8, 9, 10 → Mode = 8 |
Free lessons on mean, median, mode, and summarizing quantitative data.
Open-source college-level textbook covering measures of central tendency.
Simple explanations with interactive examples for understanding averages.
A recruiter once told me the "average" salary at their company was $145,000. I asked for the median. $89,000. Guess which number they put in the job listing.
The CEO made $1.2 million. Two VPs made $400k each. That pulled the mean up by $56,000 over the median — a number that better represents what most employees actually earn. Same dataset, two "averages," two completely different stories.
This isn't a math trick. It's the most common way data gets used to mislead, and it happens in salary reports, real estate listings, and government statistics every single day.
Five software engineers at a startup earn: $75k, $82k, $89k, $95k, and $380k. That last one is the CTO who also codes.
$144,200. Not a single person on that team actually earns that amount. The CTO's salary yanked the average up by $55,000 over what the typical engineer makes. The mean is honest — it's mathematically correct — but it's misleading when outliers exist.
This is exactly why the Bureau of Labor Statistics reports median household income, not mean. They know the billionaires would wreck the number.
Line up the same five salaries in order: $75k, $82k, $89k, $95k, $380k. The middle value is $89k. That's the median.
The CTO's $380k doesn't budge it. Replace $380k with $3.8 million and the median is still $89k. That's the superpower of the median: it's immune to extremes.
Same Data, Different "Average"
One outlier pulls the mean $55k above the median
When someone says "average home price in this neighborhood is $850,000," ask: mean or median? If one mansion sold for $4 million last month, the mean could be wildly misleading. Real estate agents know this. They pick whichever number makes the listing look better.
Mode is the most frequently occurring value. In the salary dataset above, there is no mode — every value appears once. Mode is useless for continuous data like salaries or temperatures.
Where mode shines: categorical data. What's the most popular shoe size at a store? The most common response on a survey? The most frequent error code in a server log? Mean and median can't answer these questions. Mode can.
A dataset can have no mode, one mode, or multiple modes (bimodal, multimodal). If your class test scores cluster around both 65 and 90, that bimodal distribution tells you something important — the class has two distinct groups, and the mean of 77 represents neither of them.
Use Mean When...
Data is roughly symmetric with no extreme outliers. Test scores in a normal class, daily temperatures over a month, manufacturing measurements. The mean uses every data point, which is its strength and weakness.
Use Median When...
Data is skewed or has outliers. Income, home prices, hospital wait times, wealth distribution. Anything where a few extreme values would distort the "typical" picture.
Your GPA is a weighted mean — and it hides the same kind of detail. A 3.5 GPA could mean all B+'s, or it could mean half A's and half C's. The average tells you the center but not the shape.
The p-value faces a similar problem — a single number that gets treated as the whole story when it's really just one piece of the puzzle.
Mean adds all values and divides by count — it's sensitive to outliers. Median is the middle value when data is sorted — it's resistant to outliers. For symmetric data they're nearly identical. For skewed data (income, home prices), median better represents the "typical" value.
Use mode for categorical data where mean and median don't make sense — most popular product, most common survey response, most frequent error type. Mode is also useful for identifying clusters in bimodal distributions where the mean falls between two peaks and represents neither group.
Because executive compensation skews the mean upward dramatically. A company where most people earn $70-90k but the CEO earns $5M would show a mean salary far above what any typical employee earns. The median ignores the extremes and shows what the middle person actually makes.
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