ganak.appMean Calculator
Free mean calculator that computes arithmetic, geometric, harmonic, root mean square, and trimmed mean instantly. Enter your numbers and see step-by-step formulas, comparison charts, and the mean inequality relationship. Essential for statistics, data analysis, and math homework.
Supports commas, spaces, tabs, or new lines. Paste from spreadsheets works too.
Removes 10% from each end (0 values per side) for the trimmed mean.
All Mean Types
Five types of means computed for your dataset
Mean Type Comparison
Visual comparison of all mean types — H ≤ G ≤ A ≤ RMS
Mean Inequality: Harmonic (25.87) ≤ Geometric (31.49) ≤ Arithmetic (38.60) ≤ RMS (46.18)
Step-by-Step Calculations
How each mean type is computed from your data
Arithmetic Mean
Mean = (x₁ + x₂ + ... + xₙ) / n
Step 1: Add all values
12 + 25 + 30 + 40 + 86 = 193
Step 2: Divide by count (5)
193 / 5 = 38.6000
Geometric Mean
GM = (x₁ × x₂ × ... × xₙ)^(1/n)
Step 1: Take the natural log of each value and sum them
Σ ln(xᵢ) = 17.2482
Step 2: Divide by count and exponentiate
e^(17.2482 / 5) = 31.4891
Harmonic Mean
HM = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)
Step 1: Sum the reciprocals
Σ(1/xᵢ) = 0.1933
Step 2: Divide count by the reciprocal sum
5 / 0.1933 = 25.8673
Root Mean Square
RMS = √(Σxᵢ² / n)
Step 1: Square each value and sum
Σ(xᵢ²) = 10,665
Step 2: Divide by count and take square root
√(10,665 / 5) = 46.1844
Trimmed Mean (10%)
Remove 0 values from each end, then average
Step 1: Sort and remove 0 lowest + 0 highest
Remaining 5 values: 12, 25, 30, 40, 86
Step 2: Average the remaining values
Sum / Count = 193 / 5 = 38.6000
What Is a Mean?
Understanding the most common measure of central tendency
A mean is a single number that represents the center of a dataset. While "average" and "mean" are often used interchangeably, "mean" is the precise mathematical term. There are actually several types of means, each suited for different kinds of data — from test scores and temperatures to growth rates and speeds.
Most common formula
Arithmetic Mean = (x₁ + x₂ + ... + xₙ) / n
The arithmetic mean is what most people think of when they hear "average". But for multiplicative data like investment returns, the geometric mean is more accurate. For rates and ratios, the harmonic mean gives the correct answer. Choosing the right mean type matters — using the wrong one can lead to misleading conclusions.
Types of Means Explained
When to use arithmetic, geometric, harmonic, RMS, or trimmed mean
Arithmetic Mean
The sum of all values divided by the count. Best for additive data like test scores, temperatures, or heights. Sensitive to outliers — a single extreme value can shift the result significantly.
Use for: grades, measurements, survey results
Geometric Mean
The nth root of the product of n values. Best for multiplicative data like growth rates, compound interest returns, or population changes. Only defined for positive numbers.
Use for: investment returns, CAGR, bacterial growth, inflation rates
Harmonic Mean
The count divided by the sum of reciprocals. Best for averaging rates, speeds, or ratios where the denominator varies. For example, if you drive 60 mph for one trip and 40 mph for the return, the harmonic mean (48 mph) is the correct average speed, not the arithmetic mean (50 mph).
Use for: average speed, price-to-earnings ratios, fuel efficiency
Root Mean Square (Quadratic Mean)
The square root of the average of squared values. Used in physics, engineering, and signal processing where magnitude matters regardless of sign. RMS is always greater than or equal to the arithmetic mean.
Use for: AC voltage, error analysis, vibration magnitude
Trimmed Mean
The arithmetic mean after removing a percentage of values from each end of the sorted dataset. This reduces the impact of outliers while still using most of the data. A 10% trimmed mean removes the lowest 10% and highest 10% of values.
Use for: Olympic scoring, salary analysis, data with suspected outliers
The Mean Inequality
A fundamental relationship between mean types
For any set of positive numbers, the four classical means always follow this order:
Harmonic ≤ Geometric ≤ Arithmetic ≤ RMS
This is known as the QM-AM-GM-HM inequality. All four means are equal only when every value in the dataset is the same. The more spread out the data, the larger the gaps between mean types. This relationship is useful as a sanity check — if your geometric mean is larger than your arithmetic mean, something is wrong.
Example: {2, 8}
- Harmonic Mean: 2 / (1/2 + 1/8) = 3.2
- Geometric Mean: √(2 × 8) = 4.0
- Arithmetic Mean: (2 + 8) / 2 = 5.0
- RMS: √((4 + 64) / 2) = 5.83
3.2 ≤ 4.0 ≤ 5.0 ≤ 5.83 ✓
Calculating Means in Excel & Google Sheets
Built-in formulas for each mean type
Spreadsheet software has built-in functions for most mean types. Here are the formulas you need:
Excel
- =AVERAGE(A1:A10)
- =GEOMEAN(A1:A10)
- =HARMEAN(A1:A10)
- =TRIMMEAN(A1:A10, 0.2)
Google Sheets
- =AVERAGE(A1:A10)
- =GEOMEAN(A1:A10)
- =HARMEAN(A1:A10)
- =TRIMMEAN(A1:A10, 0.2)
For Root Mean Square (RMS), there is no built-in function. Use this array formula in Excel: =SQRT(SUMSQ(A1:A10)/COUNT(A1:A10)). In Google Sheets, you can use the same formula.
Frequently Asked Questions
Common questions about means, types of averages, and when to use each