# Average Calculator Use our free average calculator to find the mean, weighted average, and GPA. Shows step-by-step work for any set of numbers. --- - **Canonical URL:** https://dothecalculation.com/calculators/average-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology - **Reviewed by:** Dr. Arthur Pendelton, PhD, PhD in Mathematical Statistics, Cambridge University (https://dothecalculation.com/about/team/arthur-pendelton) --- ## Compare mean, median, mode, and other averages Analyze datasets with arithmetic, geometric, and harmonic means alongside median, mode, quartiles, and outlier handling. - Multiple average types - Outlier-sensitive and robust summaries - Median and quartile interpretation ## Not all averages answer the same question “Average” is often used casually, but mathematics contains multiple different averages because different problems require different central summaries. The arithmetic mean measures equal additive contribution. The median measures the center after sorting and is resistant to outliers. The mode identifies the most frequent value. The geometric mean is suited for multiplicative growth rates, while the harmonic mean is useful for rates and ratios like speed or price per unit. A good analyst chooses the average that matches the data-generating process. If a dataset contains extreme outliers, the median may be more meaningful than the arithmetic mean. If values represent growth multipliers, the geometric mean is often the correct model. If values are rates on equal distances or equal units, the harmonic mean may be the correct aggregator. ## Core formulas for common averages **Arithmetic mean**: $$\bar{x} = \frac{x_1 + x_2 + ... + x_n}{n}$$ **Median**: the middle value after sorting, or the average of the two middle values when n is even. **Mode**: the most frequently occurring value(s). **Geometric mean**: $$G = (x_1 x_2 ... x_n)^{1/n}$$ **Harmonic mean**: $$H = \frac{n}{\sum_{i=1}^{n} 1/x_i}$$ These formulas produce different answers whenever the dataset is skewed, non-linear, or frequency-weighted in different ways. ## Outliers and Tukey’s fences A robust average workflow should identify outliers rather than blindly summarizing them. A common rule uses the Interquartile Range (IQR). Compute Q1 and Q3, then IQR = Q3 - Q1. Tukey’s lower and upper fences are: $$\text{Lower Fence} = Q1 - 1.5 \times IQR$$ $$\text{Upper Fence} = Q3 + 1.5 \times IQR$$ Values beyond these thresholds are flagged as outliers. Removing them often changes the arithmetic mean substantially while leaving the median comparatively stable. This is why the median is often preferred in skewed distributions. ## When to choose each average Use the arithmetic mean for balanced additive data. Use the median for salaries, housing prices, or skewed samples. Use the mode for category-heavy datasets where the most common value matters. Use the geometric mean for CAGR-style performance. Use the harmonic mean for rates like average speed over equal distances or average multiples in valuation work. ## Worked example: the same data, four different averages Take nine salaries, in thousands: 28, 30, 31, 33, 34, 36, 38, 41, and 240. The final figure is a founder's package, and it is the only value that matters to how this dataset behaves. **Arithmetic mean** = (28 + 30 + 31 + 33 + 34 + 36 + 38 + 41 + 240) ÷ 9 = 511 ÷ 9 ≈ **56.8**. Eight of the nine people earn less than the average, which is a good sign the mean is describing the outlier rather than the group. **Median** = the 5th value once sorted = **34**. This is a figure a typical employee would recognise as typical. **Mode** = none; every value appears once. Modes are informative for categorical or heavily rounded data and usually useless for continuous measurements like these. **Trimmed mean** (drop the highest and lowest, average the remaining seven) = (30 + 31 + 33 + 34 + 36 + 38 + 41) ÷ 7 = 243 ÷ 7 ≈ **34.7**, which lands close to the median and shows how little the bulk of the data disagrees once the outlier is removed. The mean is not wrong here — it correctly reports total pay divided by headcount, which is exactly the number a payroll budget needs. It is simply answering a different question from "what does someone here typically earn". Report both, and the 22-point gap between them becomes the finding rather than a discrepancy to hide. ## Weighted averages, and when an unweighted mean misleads A plain mean gives every observation equal say, which is wrong whenever the observations represent groups of different sizes. The weighted mean is Σ(wᵢxᵢ) ÷ Σwᵢ, where each weight is the count, credit value, or volume that observation stands for. Course grades are the standard case. Scoring 90% on a 2-credit module and 70% on a 4-credit module is not a 80% average: it is (2 × 90 + 4 × 70) ÷ 6 = 460 ÷ 6 ≈ **76.7%**. Averaging the percentages directly overstates the result by more than three points, and the same error inflates blended conversion rates, average review scores across products with different review counts, and portfolio returns across positions of different sizes. The reverse mistake is weighting when you should not. If each observation genuinely represents one equally important unit — nine people, nine days, nine trials — an unweighted mean is correct and adding weights invents structure that is not there. ## Averages of rates and percentages need care Averaging rates by adding them and dividing is almost always wrong, because the denominators differ. Driving 60 km at 30 km/h and back at 60 km/h does not average 45 km/h: the trip takes 2 hours out and 1 hour back, so it is 120 km in 3 hours = **40 km/h**. That is the harmonic mean, n ÷ Σ(1/xᵢ), which is the right tool whenever you are averaging rates over a fixed quantity of the thing in the denominator. Growth rates compound rather than add, so they call for the geometric mean. A portfolio returning +50% then −50% has an arithmetic mean of 0% and an actual outcome of −25% (1.5 × 0.5 = 0.75). The geometric mean, (Π(1 + rᵢ))^(1/n) − 1, returns roughly −13.4% per year, which is the constant rate that would produce the same final balance. Percentages of different bases cannot be averaged at all without their bases. A 10% conversion rate on 50 visitors and a 2% rate on 5,000 visitors blend to (5 + 100) ÷ 5,050 ≈ **2.1%**, not 6%. Whenever you are tempted to average percentages, reconstruct the numerators and denominators and divide the totals instead. ## Frequently asked questions ### What is the arithmetic mean? It is the sum of all values divided by the number of values. ### What is the median? It is the middle value of an ordered dataset, or the average of the two middle values when there is an even count. ### What is the mode? The mode is the value or values that occur most frequently in the dataset. ### What is the geometric mean used for? It is used for multiplicative growth processes such as investment returns, ratios, and repeated percentage changes. ### What is the harmonic mean used for? It is used for rates and ratios, especially when the denominator quantity is held constant. ### Why is the mean sensitive to outliers? Because every value contributes directly to the sum, so a very large or very small number can pull the mean strongly. ### Why is the median resistant to outliers? Because it depends only on order position, not the magnitude of extreme values. ### Can a dataset have more than one mode? Yes. Datasets can be bimodal, trimodal, or multimodal if multiple values share the highest frequency. ### When is there no mode? If all values occur exactly once, there is technically no repeated most-frequent value. ### Should I remove outliers before averaging? Only if there is a principled reason. Outliers may be valid observations, errors, or a different regime entirely. ## Related concepts - **Median** — The positional center of an ordered dataset. - **Interquartile Range** — The spread between the 25th and 75th percentiles. - **Geometric mean** — The multiplicative average used for rates of growth. ## Related guides - [Standard Deviation Guide: Measure Data Spread Clearly](https://dothecalculation.com/blog/math/standard-deviation-explained) — Understand variance, population vs sample formulas, and data spread with worked examples tied to the live DTC standard deviation calculator. ## Related calculators - [Binomial Distribution Probability Calculator](https://dothecalculation.com/calculators/binomial-distribution-calculator) — Compute binomial probability mass function, cumulative distribution, mean, and variance instantly with this free statistics calculator. - [Standard Deviation Calculator](https://dothecalculation.com/calculators/standard-deviation-calculator) — Calculate mean, variance, and standard deviation for any dataset instantly with this free step-by-step statistics calculator. - [Student's t-Test Calculator](https://dothecalculation.com/calculators/t-test-calculator) — Run a one-sample Student's t-test against a hypothesized mean and get the t-statistic, degrees of freedom, and two-tailed p-value instantly. - [Chi-Square Test Calculator](https://dothecalculation.com/calculators/chi-square-test-calculator) — Run a Chi-Square goodness-of-fit test on observed vs. expected category counts and get the test statistic, degrees of freedom, and p-value. - [Combination Calculator](https://dothecalculation.com/calculators/combination-calculator) — Calculate combinations (nCr) instantly with step-by-step work, with or without repetition. Perfect for lottery odds, poker hands, and team selection. - [Confidence Interval Calculator](https://dothecalculation.com/calculators/confidence-interval-calculator) — Calculate confidence intervals for population means and proportions, including margin of error, with this free statistics calculator. --- _This calculator is for educational and planning purposes. Verify calculations manually for critical academic, professional, or scientific applications._ --- _Source: [Do The Calculation](https://dothecalculation.com/calculators/average-calculator). Quote freely with attribution and a link to this page._