Mean
The arithmetic average: add all observations and divide by the number of observations.
x̄ = (Σ xᵢ) / n
The mean uses every value and is sensitive to extreme observations.
Statistics Lab · Module 01
Descriptive Statistics Playground
Explore how data can be summarised using measures of central tendency, dispersion, position and distribution shape. Change the dataset and watch the statistics update.
Dataset controls
Generate a dataset or work with your own observations.
Generate a new sample each time.
The first numeric column will be detected automatically.
Current dataset
Preview of the observations being analysed.
Minimum — · Maximum —
Calculation convention
Quartiles use linear interpolation (R-7). Variance and standard deviation switch between population and sample formulas.
Quartiles use linear interpolation (R-7). Variance and standard deviation switch between population and sample formulas.
Statistical summary
All metrics are recalculated from the current observations.
Distribution visualisation
Explore the shape, centre and spread of the current dataset.
Histogram
Frequency of observations across value intervals.
FrequencyMeanMedian
Box plot
Five-number summary and IQR outliers.
Dot plot
Individual observations along a number line.
Frequency polygon
Connects class midpoints to show the distribution profile.
Cumulative frequency (ogive)
Shows how many observations fall at or below each boundary.
What does the data tell us?
Automatically generated interpretation of the current dataset.
Topic 01
Measures of Central Tendency
Identify the centre or typical value of a dataset.
Median
The middle value after sorting the observations. For an even number of values, average the two middle observations.
Median = middle ordered value
The median is resistant to extreme values and useful for skewed data.
Mode
The most frequently occurring observation. A dataset can have one mode, multiple modes or no repeated mode.
Mode = value with highest frequency
Useful for numerical and categorical data.
Compare the centre
When the mean and median differ, the distribution may be asymmetric or influenced by extreme values.
The comparison will update from your dataset.
Topic 02
Measures of Dispersion
Quantify how spread out the observations are.
Range
Difference between the maximum and minimum values.
Range = max(x) − min(x)
Variance
Average squared deviation from the mean. The sample version uses n − 1 in the denominator.
s² = Σ(xᵢ − x̄)² / (n − 1)
Standard deviation
Square root of variance. It expresses spread in the same units as the original data.
s = √s²
Coefficient of variation
Standard deviation relative to the mean, commonly expressed as a percentage. Most meaningful for ratio-scale data with a positive mean.
CV = (s / x̄) × 100%
Deviation from the mean
See how each observation contributes to the sum of squared deviations.
Topic 03
Position & Quartiles
Understand where an observation sits relative to the rest of the data.
Quartiles
Divide ordered data into four parts. Q1 is the 25th percentile, Q2 is the median and Q3 is the 75th percentile.
Q1 = P₂₅ · Q2 = P₅₀ · Q3 = P₇₅
Interquartile range
The width of the middle 50% of the observations.
IQR = Q3 − Q1
Percentile explorer
Choose a percentile to find its value in the current dataset.
75th
75th percentile: —
Topic 04
Distribution Shape
Describe symmetry, tail direction and tail weight.
Skewness
Measures asymmetry. Positive skew has a longer right tail; negative skew has a longer left tail.
g₁ = m₃ / m₂3/2
This playground reports adjusted Fisher–Pearson sample skewness when n is sufficient.
Kurtosis
Describes tail weight relative to a normal reference. Excess kurtosis is zero for a normal distribution under the population convention.
Excess kurtosis = m₄ / m₂² − 3
The playground reports bias-adjusted sample excess kurtosis when possible.
Shape interpretation
Calculated from the active observations.
Topic 05
Outliers & Data Exploration
Identify unusual observations and inspect their influence.
IQR outlier rule
An observation is flagged when it falls below the lower fence or above the upper fence.
Lower fence = Q1 − 1.5 × IQR
Upper fence = Q3 + 1.5 × IQR
Upper fence = Q3 + 1.5 × IQR
Interpret outliers carefully
An outlier is not automatically an error. Investigate the source, context and measurement process before removing it.
Outlier detection is a diagnostic step, not an automatic deletion rule.
Detected observations
Values outside the 1.5 × IQR fences.
Topic 06
Additional Descriptive Measures
Go beyond the standard summary: examine absolute deviation, relative position and alternative averages.
Mean absolute deviation (MAD)
The average absolute distance of observations from the mean. Unlike variance, it remains in the original data units.
MAD = Σ|xᵢ − x̄| / n
MAD will update with the active dataset.
Geometric mean
A multiplicative average, useful for growth rates and ratios. It is defined here only when every observation is positive.
GM = (Π xᵢ)1/n
Geometric mean will update with the active dataset.
Harmonic mean
A reciprocal-based average, often useful for averaging rates when the numerator is fixed. It requires positive observations.
HM = n / Σ(1 / xᵢ)
Harmonic mean will update with the active dataset.
Z-scores
Express how far a value lies from the mean in standard-deviation units. A positive score is above the mean; a negative score is below it.
z = (x − x̄) / s
Z-score: —
Five-number summary
Minimum, first quartile, median, third quartile and maximum.
Sorted observations & rank
Inspect the ordered data and the empirical position of each observation.
| Rank | Value | Percentile rank (≤ value) |
|---|
Knowledge check
Test your understanding
Answer the questions and check your score.
