Descriptive Statistics Measures — Complete Comparison Guide
By CalcMulti Editorial Team··9 min read
Descriptive statistics summarise a dataset using two types of measures: central tendency (where is the centre of the data?) and spread (how dispersed is the data?). Choosing the right measure depends on your data type, distribution shape, and what question you are asking.
The most common mistake: using the mean and standard deviation for skewed data with outliers, when median and IQR would give a much more honest picture. This guide compares all key descriptive measures and tells you exactly when each is appropriate.
Side-by-Side Comparison
| Measure | Type | Formula/Method | Sensitive to Outliers? | Best For |
|---|---|---|---|---|
| Mean | Central tendency | Σx / n | Yes — strongly | Symmetric, approximately normal data |
| Median | Central tendency | Middle value when sorted | No — robust | Skewed data, income, prices, ordinal |
| Mode | Central tendency | Most frequent value(s) | No | Categorical data, multimodal distributions |
| Standard Deviation | Spread | √[Σ(x−x̄)²/(n−1)] | Yes | Symmetric, normal data alongside mean |
| IQR | Spread | Q3 − Q1 | No — robust | Skewed data or outliers alongside median |
| Range | Spread | max − min | Extremely sensitive | Quick overview; avoid for skewed data |
| Coefficient of Variation | Spread (relative) | (SD / mean) × 100% | Yes | Comparing spread across different scales |
| Variance | Spread | SD² | Yes | Mathematical analysis (ANOVA, regression) |
| Percentiles | Position/spread | Value at given rank | No | Growth charts, standardised tests, distributions |
Central Tendency — Mean vs Median vs Mode
The mean is the arithmetic average: sum of all values divided by n. It is the most commonly reported measure and has the best mathematical properties — it minimises the sum of squared deviations. But it is sensitive to extreme values: a single very large or very small value shifts the mean significantly.
The median is the middle value when data is sorted. For odd n, it is the exact middle; for even n, it is the average of the two middle values. The median is robust — extreme values do not affect it at all (as long as they stay on the same side). It is the preferred measure when data is skewed (e.g., income, house prices, reaction times).
The mode is the most frequently occurring value. For continuous data, it is often not useful (each value may appear only once). For categorical data (e.g., most popular product size, most common blood type), the mode is the only sensible central tendency measure.
| Scenario | Use | Reason |
|---|---|---|
| Exam scores in a class (normal-ish) | Mean | Symmetric; mean is most efficient |
| Household income in a city | Median | Right-skewed; a few billionaires inflate the mean |
| House prices in a neighbourhood | Median | Right-skewed; outliers distort mean |
| Most popular T-shirt size | Mode | Categorical — mean/median are meaningless for sizes |
| Age of employees (bimodal) | Both mean and mode | Report both; describe the two peaks |
| Response time on a website (skewed) | Median + percentiles | Skewed; p50, p95, p99 are most informative |
Spread Measures — SD, IQR, Range, and More
Standard deviation (SD) is the most commonly used spread measure for approximately normal data. It is in the same units as the data, directly interpretable, and the natural companion to the mean. SD is sensitive to outliers because it depends on the mean.
Interquartile Range (IQR = Q3 − Q1) covers the middle 50% of data. It is completely robust to outliers — extreme values outside the "whiskers" do not change Q1 or Q3. IQR is the natural companion to the median. Used in box plots.
Range (max − min) is the simplest spread measure but the least useful: one extreme outlier can make range enormous. Use it only as a rough orientation when the dataset is small and well-behaved.
Coefficient of Variation (CV = SD/mean × 100%) allows comparing spread across datasets with different units or different means. "Investment A has CV = 15%; Investment B has CV = 40%" — Investment B is more volatile relative to its mean return.
How Outliers Affect Each Measure — Example
Dataset: Monthly salaries of 7 employees (£000s): 30, 32, 33, 34, 35, 36, 200.
Mean: (30+32+33+34+35+36+200)/7 = 400/7 = £57,143. The CEO (£200k) inflates the mean to £57k — not representative of the typical employee.
Median: Sorted: 30, 32, 33, 34, 35, 36, 200. Middle value = 34 = £34,000. Not affected by the outlier at all. Much more representative of a typical employee.
Mode: No mode (all values appear once).
SD: Very large due to the outlier (≈ 58.7k). Mean ± 1 SD = £57k ± £59k — a range of [−£2k, £116k]. Meaningless.
IQR: Q1 = 32, Q3 = 35.5, IQR = 3.5 (£3,500). Not affected by the CEO's salary. Accurately describes that the middle 50% of regular employees earn within a £3,500 range.
Conclusion: For this skewed dataset with an outlier, report median (£34k) and IQR (£3.5k). The mean and SD give a misleading picture.
Matching Measures to Distribution Shape
The distribution shape is the most important factor in choosing which measures to report.
| Distribution Shape | Central Tendency | Spread | Visualisation |
|---|---|---|---|
| Symmetric, normal | Mean | Standard Deviation | Histogram, bell curve |
| Right-skewed (long right tail) | Median | IQR | Box plot, histogram |
| Left-skewed (long left tail) | Median | IQR | Box plot, histogram |
| Bimodal (two peaks) | Both modes + mean | SD of each peak if separable | Histogram showing both peaks |
| Heavily tailed | Median | IQR or percentiles | Box plot with outliers shown |
| Categorical (nominal) | Mode | Frequency table | Bar chart |
| Ordinal | Median | IQR, percentiles | Ordered bar chart, box plot |
The Five-Number Summary — A Complete Picture
The five-number summary provides a complete descriptive picture of a dataset without assuming any particular distribution: Minimum, Q1, Median (Q2), Q3, Maximum.
These five values are the basis of the box plot (box-and-whisker plot). Together they show: the range (min to max), the centre (median), the spread of the middle 50% (box from Q1 to Q3), and the presence of outliers (points beyond the whiskers at Q1 − 1.5×IQR and Q3 + 1.5×IQR).
When to use the five-number summary vs mean ± SD: always use the five-number summary when data is skewed, when outliers are present, or when you want to show the full distribution without assuming normality. Use mean ± SD for symmetric, approximately normal data where the empirical rule (68%/95%/99.7%) is meaningful.
Summary
The right descriptive statistics depend on your data's distribution shape and type. For symmetric normal data: report mean and SD. For skewed data or data with outliers: report median and IQR. For categorical data: report mode and frequencies.
- Mean and SD: use for symmetric, approximately normal, continuous data
- Median and IQR: use for skewed distributions, data with outliers, or ordinal data
- Mode: use for categorical data or to describe the most common value in multimodal distributions
- Always plot your data before choosing measures — a histogram reveals skewness and outliers
- Never report mean and SD for clearly skewed data — it gives a misleading picture of the typical value
Related Calculators
Arithmetic, weighted, geometric, harmonic mean
Median CalculatorMedian, quartiles, and IQR
Variance CalculatorPopulation and sample variance
Standard Deviation ExplainedComplete guide to SD
Five Number Summary CalculatorMin, Q1, median, Q3, max
When to Use Mean Median ModeDecision guide for central tendency
Statistics HubAll statistics calculators & guides
Frequently Asked Questions
Educational use only. Content is based on publicly documented mathematical formulas and reviewed for accuracy by the CalcMulti Editorial Team. Last updated: February 2026.