What is the difference between mean, median, and mode?

The mean is the sum of all values divided by the count. The median is the middle value when sorted. The mode is the value that appears most frequently. Each summarizes data differently — mean uses every value, median uses only middle, mode uses frequency.

When should I use median instead of mean?

Use the median when your data has extreme outliers or is skewed, as the mean is sensitive to outliers. For example, median income is often more representative than mean income because a few very wealthy people pull the mean up while leaving median unchanged.

Can data have more than one mode?

Yes — bimodal (two modes), trimodal (three), or multimodal (more). Examples: heights of mixed adult population (men and women peaks); restaurant arrival times (lunch and dinner peaks); test scores with one cluster of strong and one of weak students.

What is a weighted average?

A weighted average gives some values more importance than others. For values x₁, x₂, ... with weights w₁, w₂, ...: weighted mean = Σ(wᵢ × xᵢ) / Σwᵢ. Used for GPAs (weighted by credit hours), portfolio returns (weighted by investment amounts), course grades (assignments weighted by points).

What is geometric mean?

GM = (x₁ × x₂ × ... × xₙ)^(1/n). Used for rates that multiply, like investment returns. Different from arithmetic mean. For 10%, 20%, 30% annual returns: arithmetic mean = 20%, but geometric mean (true compounded) = 19.7%. GM ≤ AM for positive numbers.

What is harmonic mean?

HM = n / Σ(1/xᵢ). Used for averaging rates over equal distances. For two trips of same distance at 60 km/h and 100 km/h: average speed = 75 km/h (harmonic mean), not 80 km/h. Different from arithmetic mean unless all values are equal.

Why is median income reported instead of mean?

Income distribution is heavily right-skewed — most people earn modest incomes but a few earn enormous amounts. Mean is pulled up by the wealthy. Median represents "typical" income better. Census Bureau and OECD use median for housing prices and household income for this reason.

How does sample size affect averages?

Larger samples give more stable estimates of population mean (Law of Large Numbers). Small samples (< 30) are sensitive to individual values. Median is more stable for small samples. Standard error decreases as 1/√n — quadruple the sample to halve the error.

Average Calculator

Enter a list of numbers to calculate the mean (average), median (middle value), and mode (most frequent value). Also shows the range, sum, and count of the data set.

When people say "average", they usually mean one of three different statistics: mean, median, or mode. Each summarizes a dataset in a different way, and choosing the right one matters. For symmetric, outlier-free data, mean works well. For skewed data with outliers (incomes, house prices), median better represents "typical". For categorical data (most popular shoe size), mode is the natural choice.

The **mean** (arithmetic average) is total ÷ count. It's the most familiar measure but is heavily influenced by extreme values. If 9 friends earn $50K and one earns $50M, their mean income is over $5M — misleading. The mean's power comes from its mathematical properties (foundation of variance, regression, ANOVA) and its conservation under summation.

The **median** is the middle value when sorted. For odd-count data, it's the central value; for even-count, it's the average of the two central values. Median is robust to outliers — that same group of 10 with one billionaire still has median income $50K. This is why "median household income" and "median home price" appear in news, not mean.

The **mode** is the most frequently occurring value. Datasets can have one mode (unimodal), two (bimodal), or more (multimodal). For continuous data, mode often gets binned. Categorical data — favorite color, T-shirt size — needs mode as the only sensible "average".

Common applications: grade calculations, sports statistics, scientific research, business analytics, demographic analysis, quality control, and any situation needing a single summary value for a list of data points.

Inputs

Value 1

Value 2

Value 3

Value 4

Value 5

Value 6 (0 to skip)

Value 7 (0 to skip)

Value 8 (0 to skip)

Results

Mean (Average)

Median

Mode

No mode (all unique)

Range

Sum

150

Count

Min

Max

Last updated: May 29, 2026

Formula

**Mean (arithmetic average):** mean = (sum of all values) / (count) For data {x₁, x₂, ..., xₙ}: μ = (x₁ + x₂ + ... + xₙ) / n **Median:** 1. Sort data in order. 2. If n is odd: median = middle value. 3. If n is even: median = average of two middle values. **Mode:** Most frequently occurring value(s). Datasets can be: - Unimodal: one mode. - Bimodal: two modes. - Multimodal: more than two. - No mode: all values appear with equal frequency. **Worked example:** Data: 5, 8, 12, 12, 15, 20, 25 Mean: (5+8+12+12+15+20+25) / 7 = 97 / 7 ≈ 13.86 Median: middle value = 12 (4th of 7 sorted) Mode: 12 (appears twice) **Effect of outliers:** Data: 5, 5, 5, 5, 5, 5, 5, 5, 5, 50 Mean: (8×5 + 50) / 10 = 90/10 = 9 (skewed by 50) Median: middle = 5 Mode: 5 The mean shifts dramatically; median and mode are robust. **Range, sum, count:** - Range = max - min - Sum = total of all values - Count = number of values - Mean = sum / count **Other types of averages:** **Weighted mean:** μ_weighted = Σ(wᵢ × xᵢ) / Σwᵢ Used when values have different importance. **Geometric mean:** GM = (x₁ × x₂ × ... × xₙ)^(1/n) Used for rates and ratios (e.g., investment returns). **Harmonic mean:** HM = n / Σ(1/xᵢ) Used for averaging rates over equal distances (e.g., average speeds). For symmetric data: HM ≤ GM ≤ AM (arithmetic mean). **Examples of harmonic mean:** For 100 km at 60 km/h, then 100 km at 100 km/h: Average speed = 2 × 60 × 100 / (60 + 100) = 75 km/h (NOT 80). **Percentile averaging:** Quartiles divide data into 4 equal parts: - Q1 (25th percentile) - Q2 (50th percentile = median) - Q3 (75th percentile) **Robust averages:** - **Trimmed mean**: discards top and bottom n% before averaging. - **Winsorized mean**: replaces extreme values with cap values. - **Median**: most robust to outliers. **Examples by data type:** | Data | Best measure | |---|---| | Heights | Mean (normally distributed) | | Income | Median (long right tail) | | House prices | Median | | Favorite color | Mode (categorical) | | Test scores | Mean (often normal) | | Reaction times | Median (skewed) | | Survey ratings (1-5) | Median or mode | | Stock returns | Geometric mean (compounding) | **When mean misleads:** Mean of household incomes: $80K. - Could be 10 people earning $80K each. - Or 9 earning $50K and 1 earning $350K. Same mean, very different distributions. Median ($50K vs $80K) clarifies. **Symmetric vs skewed:** For symmetric distributions: mean = median = mode (approximately). Right-skewed (long right tail): mean > median > mode. Left-skewed (long left tail): mean < median < mode. This is why "income distribution is right-skewed" → median < mean. **Standard deviation (also useful):** σ = √(Σ(xᵢ - μ)² / n) or √(Σ(xᵢ - μ)² / (n-1)) for samples Measures spread of data around the mean. **Software:** - **Excel**: AVERAGE(), MEDIAN(), MODE(). - **Python (pandas)**: .mean(), .median(), .mode(). - **R**: mean(), median(), names(sort(-table(x)))[1] for mode. - **MATLAB**: mean, median, mode. - **SQL**: AVG(), or use percentiles via window functions. **Common applications:** - **Education**: grade averaging, statistics for tests. - **Sports**: batting averages, ERA, scoring averages. - **Business**: average sales, customer ratings, prices. - **Finance**: investment returns (geometric for compounding). - **Manufacturing**: quality control (process averages). - **Healthcare**: vital sign averages, BMI distributions. - **Demographics**: income, household size, age.

How to use this calculator

Enter each value in the provided fields (use 0 for unused slots).
Calculator returns mean, median, mode, sum, count, and range.
For more than 8 values, use a spreadsheet or programming tool.
Mean: best for symmetric data without outliers.
Median: best for skewed data or data with outliers.
Mode: best for categorical or discrete data.

Worked examples

Test scores

**Scenario:** Student's test scores: 85, 90, 78, 92, 88. Class average? **Calculation:** Sum = 433. Mean = 433/5 = 86.6. Median = 88 (middle after sorting). No mode (all unique). **Result:** Mean and median both ~87, consistent. Indicates fairly symmetric distribution. Outliers would skew mean more than median. Final grade typically uses mean.

Home prices

**Scenario:** Recent home sales: $180K, $190K, $200K, $210K, $220K, $1,500K (mansion). **Calculation:** Mean = 2500/6 = $416,667. Median: sorted middle = ($200K + $210K)/2 = $205,000. **Result:** Mean ($417K) is misleadingly high due to mansion outlier. Median ($205K) better represents typical home in this neighborhood. Why real estate reports always use median home price.

T-shirt sizes

**Scenario:** Shirt sizes sold this week: S, S, M, M, L, L, L, L, XL, XL. **Calculation:** Mode = L (appears 4 times). Mean and median don't apply to non-numeric data. **Result:** L is the most common (modal) size. Restock more L; less of S and XL. Mode is the only meaningful "average" for categorical data like sizes, colors, models.

When to use this calculator

**Use averages for:**

- **Summary statistics**: representing a dataset with a single number. - **Sports analysis**: batting averages, scoring averages, ERAs. - **Academic grading**: course averages, GPA calculations. - **Business**: average sales, customer satisfaction, prices. - **Demographics**: income, age, household size. - **Quality control**: process means, deviations. - **Survey research**: rating averages. - **Financial returns**: geometric mean for compounding.

**Choosing the right average:**

- **Mean**: symmetric data, no outliers, mathematical operations needed. - **Median**: skewed data, outliers present, robust summary. - **Mode**: categorical data, identifying most common value.

**When mean misleads:**

Whenever data is skewed or has outliers, mean overstates "typical". - Income data (right-skewed). - House prices (right-skewed). - Wait times (right-skewed). - Sports salaries (extreme outliers). - File sizes on a system (some giant, most small).

Always check distribution shape before using mean alone.

**Symmetric vs skewed reminder:**

For normally distributed data: mean ≈ median ≈ mode. For right-skewed: mean > median > mode. For left-skewed: mean < median < mode.

If mean and median differ significantly, distribution is skewed.

**Common applications:**

- **GPA**: typically arithmetic mean of grades (weighted by credit hours). - **Batting average**: hits / at-bats (a proportion, technically not an average). - **Median home price**: real estate report standard. - **Salary surveys**: often report median to avoid CEO skew. - **Customer reviews**: usually mean (with caveats). - **Manufacturing yield**: mean production rate. - **Healthcare**: mean blood pressure, median life expectancy.

**Specialized averages:**

- **Weighted mean**: when items have different importance. - **Geometric mean**: for rates and ratios (investment returns). - **Harmonic mean**: for rates over equal distances (speeds). - **Trimmed mean**: drop extreme values (used in scoring).

**Weighted GPA example:**

3-credit class with A (4.0): contributes 4.0 × 3 = 12 points. 4-credit class with B (3.0): contributes 3.0 × 4 = 12 points. Total: 24 / 7 = 3.43 GPA.

Weighted by credit hours, not equal.

**Trimmed mean (Olympic scoring):**

Drop highest and lowest scores from judges, average the rest. Reduces effect of biased judges.

**Software for big datasets:**

- **Excel/Google Sheets**: built-in functions. - **Python (pandas)**: .mean(), .median(), .mode(). - **R**: statistical analysis standard. - **SQL**: AVG() for mean; percentile functions for median. - **Tableau, Power BI**: business intelligence.

**Pitfalls:**

- **Reporting mean for skewed data**: misleading. - **Forgetting outliers**: skew mean dramatically. - **Calculating average of averages**: usually wrong (need weighted). - **Treating ordinal as numeric**: rating 1-5 isn't really numeric. - **Mode for continuous data**: usually need to bin first. - **No mode reported as "no average"**: can have multiple modes. - **Confusing arithmetic, geometric, harmonic**: each has its use. - **Sample vs population**: sample mean is estimator of population mean.

Common mistakes to avoid

Using mean when median is more appropriate (outliers, skew).
Averaging averages without proper weighting.
Reporting mean for income/house prices (use median).
Forgetting mode for categorical data.
Confusing arithmetic, geometric, harmonic means.
Including outliers without comment.
Reporting mean without measure of spread (standard deviation).
Using percentage values' average instead of weighted percentages.

Frequently Asked Questions

Sources & further reading

SponsoredShop Top Deals on AmazonSupport CalcMountain — browse top-rated products at no extra cost to you.

Related Calculators

Percentage Calculator

Calculate percentages: find X% of Y, what percent X is of Y, or X is Y% of what.

Significant Figures Calculator

Count significant figures in a number and round to a specific number of sig figs.