Mean Median Mode Calculator — Free Statistics Tool

Enter a list of numbers separated by commas, spaces, or semicolons. Click Calculate to instantly see every descriptive statistic. Mean: Sum of all values ÷ count. The arithmetic average. Median: Middle value of the sorted list. For an even count, the average of the two middle values. Mode: Value(s) that appear most often. No mode if all values are unique. Range: Maximum − Minimum. Q1 (First Quartile): Median of the lower half of the data. Q3 (Third Quartile): Median of the upper half of the data. IQR (Interquartile Range): Q3 − Q1. Measures the spread of the middle 50%. Outliers: Values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR (Tukey's method). Std Dev: Square root of the average squared deviation from the mean (population formula).

Understanding mean, median, and mode is foundational to statistics, data science, and everyday decision-making. Here is a complete reference guide. **Mean (Arithmetic Average)** Formula: x̄ = (Σxᵢ) / n The mean uses every value, making it the most informative measure — but also the most sensitive to outliers. If nine employees earn $40,000 and one earns $1,000,000, the mean salary of ~$136,000 is misleading. In such cases, use the median. **Median (Middle Value)** Formula: sort data; median = middle value (or average of two middle values) The median is position-based, not value-based, so extreme values don't distort it. Real estate listings, income statistics, and wait times all use the median for this reason. **Mode (Most Frequent Value)** The mode is the only measure usable with categorical data ("what is the most common shirt size?"). A data set can be unimodal (one mode), bimodal (two modes), multimodal, or have no mode. **Range vs IQR vs Standard Deviation** Range (max − min) is quick but fragile — one outlier changes it dramatically. IQR (Q3 − Q1) covers the middle 50% of data and ignores extremes, making it robust. Standard deviation accounts for every data point and is the gold standard for measuring spread in symmetric, normal distributions. **Quartiles Explained** Q1 (25th percentile): 25% of values fall below this point. Q2 (50th percentile): This is the median. Q3 (75th percentile): 75% of values fall below this point. The box in a box plot spans Q1 to Q3 (the IQR), with whiskers extending to the fences. **Outlier Detection (Tukey's Fence)** Lower fence = Q1 − 1.5 × IQR Upper fence = Q3 + 1.5 × IQR Any value outside these fences is a mild outlier. Values beyond Q1 − 3×IQR or Q3 + 3×IQR are extreme outliers. This is the method used by most statistical software including SPSS and R. **When to Use Each Measure** Symmetric data, no outliers → Mean ≈ Median; use mean. Skewed data or outliers → Use median. Categorical data → Use mode. Measuring spread → Use standard deviation (normal) or IQR (skewed/outliers).