Mean Calculator
Calculate the arithmetic, geometric, harmonic, and weighted mean of a data set. See also Median Calculator and Standard Deviation Calculator.
How to Calculate the Mean
The mean (average) is the most common measure of central tendency. To calculate the arithmetic mean, add all values in the data set and divide by the number of values. The geometric mean multiplies all values and takes the nth root, while the harmonic mean divides the count by the sum of reciprocals. Each type of mean is suited for different types of data and analysis.
Mean Formulas
Arithmetic Mean = (x₁ + x₂ + ... + xₙ) / n
Geometric Mean = (x₁ × x₂ × ... × xₙ)^(1/n)
Harmonic Mean = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)
Weighted Mean = Σ(wᵢ × xᵢ) / Σwᵢ
Example Calculation
Data: 10, 20, 30, 40, 50
Sum = 10 + 20 + 30 + 40 + 50 = 150
Count = 5
Arithmetic Mean = 150 / 5 = 30
Geometric Mean = (10 × 20 × 30 × 40 × 50)^(1/5) ≈ 26.051711
Harmonic Mean = 5 / (1/10 + 1/20 + 1/30 + 1/40 + 1/50) ≈ 21.897811
Comparison of Mean Types
| Type | Best For | Sensitive to Outliers? |
|---|---|---|
| Arithmetic Mean | General-purpose average | Yes |
| Geometric Mean | Growth rates, ratios | Less so |
| Harmonic Mean | Rates, speeds | Less so |
| Weighted Mean | Unequal importance values | Depends on weights |
Solved Examples
Example 1: Exam Scores
A class of 8 students scored the following on a math exam: 72, 85, 90, 68, 77, 93, 81, 88. Find all three types of mean.
Sum = 72 + 85 + 90 + 68 + 77 + 93 + 81 + 88 = 654
Arithmetic Mean = 654 / 8 = 81.75
Geometric Mean = (72 x 85 x 90 x 68 x 77 x 93 x 81 x 88)^(1/8) = 81.27
Harmonic Mean = 8 / (1/72 + 1/85 + 1/90 + 1/68 + 1/77 + 1/93 + 1/81 + 1/88) = 80.77
Answer: The arithmetic mean is 81.75, geometric mean is approximately 81.27, and harmonic mean is approximately 80.77. For positive data, the inequality HM ≤ GM ≤ AM always holds.
Example 2: Weighted GPA Calculation
A student earns grades in four courses: Biology (3 credits, grade 3.7), Calculus (4 credits, grade 3.3), English (3 credits, grade 4.0), and History (2 credits, grade 3.0). Find the weighted mean GPA.
Weighted Sum = (3)(3.7) + (4)(3.3) + (3)(4.0) + (2)(3.0)
= 11.1 + 13.2 + 12.0 + 6.0 = 42.3
Total Credits = 3 + 4 + 3 + 2 = 12
Weighted Mean = 42.3 / 12 = 3.525
Answer: The weighted GPA is 3.525. Without weighting, the simple average would be (3.7 + 3.3 + 4.0 + 3.0)/4 = 3.5, which underweights the 4-credit calculus course.
Example 3: Average Speed Using Harmonic Mean
A driver travels 60 km at 40 km/h and then 60 km at 60 km/h. What is the average speed for the entire trip?
Time for first leg = 60/40 = 1.5 hours
Time for second leg = 60/60 = 1.0 hours
Total distance = 120 km, Total time = 2.5 hours
Average speed = 120/2.5 = 48 km/h
Harmonic Mean = 2/(1/40 + 1/60) = 2/0.04167 = 48 km/h
Answer: The average speed is 48 km/h, which equals the harmonic mean of the two speeds. The arithmetic mean (50 km/h) would be incorrect because equal distances, not equal times, were traveled at each speed.
Practice Questions
Question 1
The daily temperatures (in Celsius) recorded over a week were: 22, 25, 19, 28, 24, 21, 23. Calculate the arithmetic mean temperature.
Answer: Sum = 22 + 25 + 19 + 28 + 24 + 21 + 23 = 162. Mean = 162/7 = 23.14 degrees Celsius.
Question 2
An investment returns 10%, 15%, -5%, and 20% over four consecutive years. What is the geometric mean annual return?
Answer: Convert to multipliers: 1.10, 1.15, 0.95, 1.20. Geometric Mean = (1.10 x 1.15 x 0.95 x 1.20)^(1/4) = (1.4421)^(0.25) = 1.0961. The geometric mean annual return is approximately 9.61%.
Question 3
A factory produces batches with defect rates of 2%, 3%, 1.5%, 4%, and 2.5%. Each batch has equal size. What is the mean defect rate?
Answer: Since batches are equal size, use arithmetic mean: (2 + 3 + 1.5 + 4 + 2.5)/5 = 13/5 = 2.6%. The mean defect rate is 2.6%.
Common Mistakes
Using arithmetic mean for rates or ratios
When averaging speeds over equal distances or rates over equal quantities, the harmonic mean is correct. Driving 100 km at 60 km/h and 100 km at 40 km/h gives an average of 48 km/h (harmonic), not 50 km/h (arithmetic).
Ignoring the effect of outliers
The arithmetic mean is highly sensitive to extreme values. A dataset of salaries (35k, 40k, 38k, 42k, 500k) has a mean of 131k, which misrepresents the typical salary. The median (40k) is more appropriate when outliers are present.
Averaging percentages without weighting
If Store A sells 100 items with a 20% return rate and Store B sells 900 items with a 5% return rate, the overall return rate is not (20+5)/2 = 12.5%. It is (20+45)/1000 = 6.5%. Always weight by sample size when combining rates from groups of different sizes.
Applying geometric mean to negative values
The geometric mean requires all positive values. If your data includes zeros or negatives, the geometric mean is undefined. For investment returns that include losses, convert to multipliers first (a loss of 10% becomes 0.90).
Key Takeaways
- The arithmetic mean is the sum divided by the count and works best for symmetric data without outliers.
- The geometric mean is appropriate for multiplicative processes like growth rates, compound interest, and index numbers.
- The harmonic mean is correct for averaging rates when the denominator varies (e.g., speed over equal distances).
- The weighted mean accounts for unequal importance and is essential for GPA, portfolio returns, and survey data.
- For any positive dataset, the inequality HM ≤ GM ≤ AM always holds (AM-GM-HM inequality).
- Always consider whether your data has outliers, is skewed, or involves rates before choosing which mean to report.
Frequently Asked Questions
What is the difference between mean and average?
In everyday language, "average" and "mean" are used interchangeably. Technically, "average" can refer to mean, median, or mode, but "mean" specifically refers to the arithmetic mean — the sum divided by the count.
When should I use geometric mean instead of arithmetic mean?
Use geometric mean when dealing with percentages, growth rates, or ratios. For example, average investment returns over multiple years should use geometric mean because returns compound multiplicatively.
What is weighted mean used for?
Weighted mean is used when some values contribute more than others. For example, calculating a course grade where exams count more than homework — each score is multiplied by its weight before averaging.
Can the mean be a value not in the data set?
Yes. The mean is often a value that does not appear in the original data. For example, the mean of 1 and 2 is 1.5, which is not in the set.