Average Calculator
Calculate the mean, median, mode, range, and more from a set of numbers. See also Percentage Calculator and Standard Deviation Calculator.
How to Calculate Average (Mean)
The arithmetic mean (average) is calculated by adding all numbers in a data set and dividing by the count of numbers. It is the most commonly used measure of central tendency. The median is the middle value when numbers are sorted, and the mode is the most frequently occurring value. Each measure provides different insights into the data distribution.
Average Formulas
Arithmetic Mean = Sum of all values / Count of values
Geometric Mean = (x1 × x2 × ... × xn)^(1/n)
Harmonic Mean = n / (1/x1 + 1/x2 + ... + 1/xn)
Example
Numbers: 10, 20, 30, 40, 50
Mean = (10 + 20 + 30 + 40 + 50) / 5 = 150 / 5 = 30
Median = 30 (middle value)
Mode = No mode (all values appear once)
Range = 50 − 10 = 40
Types of Averages
Arithmetic Mean is the standard average — best for evenly distributed data. Geometric Mean is used for growth rates, percentages, and ratios — it reduces the impact of extreme values. Harmonic Mean is used for rates and ratios like speed — it gives more weight to smaller values. Median is resistant to outliers and is preferred when data is skewed. Mode identifies the most common value and works with non-numeric data too.
Frequently Asked Questions
What is the difference between mean and median?
Mean is the sum divided by count — it is affected by outliers. Median is the middle value — it is not affected by extreme values. For example, in [1, 2, 3, 4, 100], the mean is 22 but the median is 3.
When should I use geometric mean?
Use geometric mean for growth rates, investment returns, and data that multiplies together. For example, if an investment grows 10%, 20%, and 30% over three years, the geometric mean gives the true average annual growth rate.
Can a data set have multiple modes?
Yes. A data set with two modes is bimodal, three modes is trimodal, and more is multimodal. If all values appear equally often, there is no mode.
Solved Examples — Averages
Example: A student scores 78, 85, 92, 88, and 67 on five exams. Find the mean, median, and range.
Solution:
Step 1: Mean = (78 + 85 + 92 + 88 + 67) / 5 = 410 / 5 = 82
Step 2: Sort values: 67, 78, 85, 88, 92. Median = 85 (middle value)
Step 3: Range = 92 − 67 = 25
Answer: Mean = 82, Median = 85, Range = 25
Example: An investment grows by 10%, 15%, and 20% over three years. Find the average annual growth rate (geometric mean).
Solution:
Step 1: Convert to multipliers: 1.10, 1.15, 1.20
Step 2: Geometric Mean = (1.10 × 1.15 × 1.20)^(1/3)
Step 3: = (1.518)^(1/3) ≈ 1.1495
Step 4: Average growth rate = 14.95% per year
Answer: ≈ 14.95% average annual growth
Example: A car travels 60 km at 40 km/h and 60 km at 60 km/h. What is the average speed for the whole trip?
Solution:
Step 1: Use harmonic mean for rates: HM = 2 / (1/40 + 1/60)
Step 2: = 2 / (0.025 + 0.01667) = 2 / 0.04167
Step 3: = 48 km/h
Note: The arithmetic mean (50 km/h) would be incorrect here!
Answer: 48 km/h (harmonic mean)
Example: The temperatures (°F) for a week are: 72, 68, 75, 79, 82, 71, 74. Find mean and median.
Solution:
Step 1: Sum = 72 + 68 + 75 + 79 + 82 + 71 + 74 = 521
Step 2: Mean = 521 / 7 ≈ 74.43°F
Step 3: Sorted: 68, 71, 72, 74, 75, 79, 82. Median = 74°F
Answer: Mean ≈ 74.43°F, Median = 74°F
Practice Questions
Try these on your own:
- Find the mean of: 14, 22, 18, 25, 31 (Answer: 22)
- Find the median of: 3, 7, 9, 12, 15, 18 (Answer: 10.5)
- Find the mode of: 4, 7, 4, 9, 7, 4, 11 (Answer: 4)
- A class of 30 students has a mean score of 72. A class of 20 students has a mean of 80. What is the combined mean? (Answer: 75.2)
- Find the geometric mean of 4, 8, and 16 (Answer: 8)
- The harmonic mean of two speeds, 30 mph and 70 mph, for equal distances is? (Answer: 42 mph)
Common Mistakes to Avoid
The most common mistake is using the arithmetic mean when data is skewed by outliers — in such cases the median is a better measure of central tendency. For example, if salaries in a company are $40K, $45K, $50K, $55K, and $500K, the mean ($138K) is misleading while the median ($50K) better represents a typical salary. Another frequent error is using the arithmetic mean for rates and ratios — always use the harmonic mean for averaging speeds or rates over equal distances. Students also confuse "no mode" with "mode is zero" — if no value repeats, the dataset has no mode. When combining averages from different groups, you cannot simply average the averages unless both groups are the same size; instead, use a weighted average. Finally, remember that the geometric mean only works with positive numbers.
Key Takeaways
- Arithmetic mean = sum of values ÷ count. Best for symmetric, evenly distributed data.
- Median is the middle value when sorted — it resists outliers and works best for skewed distributions.
- Mode is the most frequent value — useful for categorical data and identifying common occurrences.
- Geometric mean is ideal for growth rates and multiplicative data (e.g., investment returns).
- Harmonic mean is the correct average for rates (e.g., speed, price-to-earnings ratios).
- Always choose the type of average that matches your data and what you're trying to measure.