Descriptive Statistics
Mean, Median & Mode Calculator
Enter any set of numbers to instantly compute all measures of central tendency, measures of spread, a sorted frequency table, step-by-step calculation details, and a ready-to-use APA 7th edition reporting statement.
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Dataset
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Measures of Central Tendency
Measures of Spread & Dispersion
Sorted Dataset & Frequency Table
APA 7th edition reporting statement
Copy-paste template
Step-by-step calculation details
Interpretation Report
Plain-language explanation of each measure
1
The Mean (Arithmetic Average)
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2
The Median (Middle Value)
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3
The Mode (Most Frequent Value)
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4
Spread & Variability
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5
Distribution Shape & Which Measure to Use
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Review all interpretations before use in academic or professional submissions.
Reference · Statistical Theory
How Mean, Median & Mode Work
Complete theoretical foundation, formulas, assumptions, when to use each measure, and primary academic references.
The Three Measures of Central Tendency
Measures of central tendency describe the "centre" or "typical value" of a distribution. There are three primary measures, each with distinct mathematical properties, appropriate use cases, and sensitivities to outliers.
1. The Mean (Arithmetic Mean)
x̄ = Σxᵢ / n
where:
Σxᵢ = sum of all values in the dataset
n = total number of values
Population mean: μ = Σxᵢ / N
Sample mean: x̄ = Σxᵢ / n
- Definition: The sum of all values divided by the count of values.
- Sensitivity: Sensitive to extreme values (outliers). A single very large or very small value can substantially shift the mean away from the "typical" value.
- Scale of measurement: Requires interval or ratio data (e.g., test scores, heights, temperatures).
- When to use: Appropriate when data is approximately symmetrically distributed with no extreme outliers.
2. The Median
Step 1: Sort all values in ascending order.
Step 2:
If n is odd: Median = value at position (n+1)/2
If n is even: Median = average of values at positions n/2 and (n/2)+1
Example (odd n=5): [2, 4, 6, 8, 10] → Median = 6
Example (even n=6): [2, 4, 6, 8, 10, 12] → Median = (6+8)/2 = 7
- Definition: The middle value of an ordered dataset. Exactly 50% of values fall below and 50% above the median.
- Sensitivity: Resistant (robust) to extreme outliers. Income and housing price data are almost always reported using the median for this reason.
- Scale of measurement: Requires at least ordinal data.
- When to use: Preferred over the mean when data is skewed or contains extreme outliers.
3. The Mode
Mode = the value(s) that appear with the highest frequency.
Types:
No mode: Each value appears exactly once.
Unimodal: One value has the highest frequency.
Bimodal: Two values share the highest frequency.
Multimodal: Three or more values share the highest frequency.
- Definition: The value (or values) that occur most frequently in the dataset.
- Uniqueness: The mode is the only measure of central tendency that can be used with nominal (categorical) data (e.g., the most common eye colour in a sample).
- Multiple modes: A dataset can have more than one mode if two or more values share the highest frequency (bimodal or multimodal).
- When to use: Most useful for categorical or discrete data, and to identify the most common value or score in a distribution.
Measures of Spread (Dispersion)
Range = Maximum − Minimum
Variance (Sample): s² = Σ(xᵢ − x̄)² / (n−1) [Bessel's correction]
Variance (Population):σ² = Σ(xᵢ − μ)² / N
Standard Deviation (Sample): s = √s²
Standard Deviation (Population):σ = √σ²
Coefficient of Variation (CV): CV = (s / x̄) × 100%
Interquartile Range (IQR): IQR = Q3 − Q1
Q1 = median of the lower half (below the overall median)
Q3 = median of the upper half (above the overall median)
- Why n−1 (Bessel's correction)? When estimating population variance from a sample, dividing by n−1 instead of n corrects for the tendency of a sample to underestimate population variance. This is an unbiased estimator of the population variance.
- Standard Deviation vs. Variance: Standard deviation is the square root of variance, expressed in the same units as the original data — making it more interpretable.
- IQR: The range of the middle 50% of the data. Resistant to outliers, often reported alongside the median.
Distribution Shape & Skewness
Key relationship: In a perfectly symmetric normal distribution, the mean = median = mode. When these three values diverge, the distribution is skewed. If mean > median, the distribution is positively (right) skewed. If mean < median, it is negatively (left) skewed.
APA 7th edition reporting format
APA Template: "Descriptive statistics were computed for [variable]. The mean was [x̄] (SD = [s]). The median was [Mdn = value] and the mode was [value]. These values suggest that the distribution was [approximately symmetric / positively skewed / negatively skewed]."
Primary references
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
Gravetter, F. J., & Wallnau, L. B. (2021). Statistics for the behavioral sciences (10th ed.). Cengage Learning.
Creswell, J. W., & Creswell, J. D. (2023). Research design: Qualitative, quantitative, and mixed methods approaches (6th ed.). SAGE Publications.
American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.).