Explore how values are spread across a dataset. Generate different distribution shapes, compare empirical data with theoretical models, and investigate probability, percentiles, tails and normality.
Interactive playground
Dataset controls
Change the distribution family and its parameters.
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Ready. Adjust the parameters and generate a dataset.
Parameter note: For continuous distributions, location and scale control the data generator. For binomial and Poisson, use p, trials and λ. The active dataset is reused throughout this module.
Active observations
A shared dataset powers every topic and chart on this page.
500 values
The histogram and empirical CDF describe the observed sample. The theoretical curves show the idealised model used to generate or approximate it.
Distribution at a glance
Quick numerical and visual summary of the active sample.
Topic 01
Empirical Distributions
Use the observed data to understand frequency, relative frequency and density.
Frequency histogram
Bar height represents the number of observations in each bin.
Relative frequency / density
Normalised histogram; total relative frequency sums to one.
Frequency distribution table
Class interval, midpoint, frequency, relative frequency and cumulative frequency.
Class interval
Midpoint
Frequency
Relative frequency
Cumulative frequency
Topic 02
Continuous Distributions
For a continuous random variable, probabilities are areas under a probability density function.
Normal distribution
Symmetric bell curve centred at the mean.
f(x) = [1 / (σ√2π)] e−(x−μ)² / 2σ²
Symmetric, bell-shaped and completely described by its mean μ and standard deviation σ.
Mean = median = mode.
Total area under the curve = 1.
68–95–99.7 rule is an approximation.
Uniform distribution
Constant density between the lower and upper bounds.
f(x) = 1 / (b − a), a ≤ x ≤ b
Every value in the interval has the same density. Probability is proportional to interval width.
Flat density across its support.
Mean = (a + b) / 2.
Variance = (b − a)² / 12.
Exponential distribution
High density near zero, followed by a long right tail.
f(x) = λe−λx, x ≥ 0
A right-skewed continuous model often used for waiting times between independent events at a constant rate.
Mean = 1 / λ.
Memoryless property.
Long right tail.
Probability density is not probability
For a continuous variable, the probability of one exact value is zero. Probabilities are found over intervals.
P(a ≤ X ≤ b) = ∫ab f(x) dx
In the chart, select an interval to estimate the proportion of observations inside it.
Continuous model explorer
Compare the observed sample histogram with a reference density curve.
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Topic 03
Discrete Distributions
Discrete random variables take countable values; probability is assigned to each possible outcome.
Binomial distribution
Discrete bars show probability for each possible success count.
P(X = k) = C(n,k) pk(1−p)n−k
Counts successes in a fixed number of independent trials, each with the same success probability.
Mean: np Variance: np(1−p)
Poisson distribution
Discrete event-count probabilities, often right-skewed for small λ.
P(X = k) = e−λ λk / k!
Models event counts in a fixed interval when events occur independently at a constant average rate.
Mean: λ Variance: λ
Probability mass function (PMF)
Adjust the parameters above to see how probability mass shifts across outcomes.
Topic 04
Cumulative Distribution Function & Percentiles
The CDF gives the probability that a random variable is less than or equal to a chosen value.
F(x) = P(X ≤ x)
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50th
Interpretation: If F(42) = 0.70, approximately 70% of observations are at or below 42. The inverse CDF (quantile function) answers the reverse question: what value marks a selected percentile?
Topic 05
Shape, Skewness & Tail Behaviour
Distribution shape influences which summaries and statistical methods are appropriate.
Symmetry
A symmetric distribution has similar shape on either side of its centre. For a symmetric unimodal distribution, mean, median and mode are close.
Positive (right) skew
A longer right tail pulls the mean toward larger values. The mean is often greater than the median.
Negative (left) skew
A longer left tail pulls the mean toward smaller values. The mean is often less than the median.
Bimodal distribution
A bimodal distribution has two prominent peaks. It can indicate that the dataset combines two different groups or processes.
Two peaks suggest two common value clusters.
Heavy and light tails
Heavy-tailed distributions place more probability in extreme values than a normal distribution. Tail behaviour matters in risk and anomaly analysis.
Compare the probability of extreme observations.
Current dataset: shape diagnostics
Use these indicators as descriptive evidence, not as a formal normality test.
Topic 06
Q–Q Plot & Normality
A quantile–quantile plot compares sample quantiles with theoretical normal quantiles.
Normal Q–Q plot
Points close to the reference line indicate approximate normality.
How to read the plot
Near a straight line: sample quantiles are broadly consistent with normal quantiles.
Curved pattern: possible skewness or a mismatch in tail behaviour.
Departures at the ends: possible heavy tails, light tails or extreme observations.
Isolated points: investigate potential outliers or data issues.
A Q–Q plot is a diagnostic, not proof that data follow a normal distribution. Consider sample size, context and other diagnostics.
Knowledge check
Test Your Understanding
Choose the best answer for each question.
1. What does the total area under a probability density curve represent?
2. Which distribution is commonly used to model the number of events in a fixed interval?
3. For a continuous random variable, what is P(X = 4)?
4. A long right tail usually indicates:
5. In a normal Q–Q plot, points following a straight line suggest:
Score: 0 / 5
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