Basic Distributions

Why this matters

Distributions describe the shape of your data. As a Data Analyst, this helps you:

• Pick the right summary stats (e.g., median instead of mean for skewed data).
• Estimate probabilities (e.g., how often a queue will be empty or overflow).
• Choose appropriate visualizations (histogram vs. bar chart) and transformations (log for right-skew).
• Validate assumptions for A/B tests and forecasting.

Concept explained simply

A distribution tells you how likely different values are. Think of it as the "shape" a random process tends to produce over many repeats.

• Discrete vs. continuous: Counts (0,1,2,...) vs. measurements (any real value).
• Parameters: Small set of numbers that summarize the shape (e.g., mean, variance, rate).
• Skew and tails: Right-skewed means long tail to the right; heavy tails mean more extreme values than Normal.

Common distributions at a glance

• Bernoulli(p): Single yes/no outcome (clicked vs. not).
• Binomial(n, p): Number of successes out of n independent trials.
• Poisson(lambda): Count of events in a fixed interval when events happen independently at a constant average rate.
• Uniform(a, b): All values in [a, b] equally likely.
• Normal(mu, sigma): Bell curve; many natural aggregates; CLT makes sums/means tend toward Normal.
• Log-normal(mu_log, sigma_log): Positive, right-skewed amounts; log of data is roughly Normal.
• Exponential(rate): Time between Poisson events; memoryless.
• t-distribution(df): Like Normal but with heavier tails; used when estimating means with small samples and unknown variance.

Mental model: Generative stories

• Bernoulli: Flip a biased coin once. Heads probability = p.
• Binomial: Flip the same coin n times; count heads.
• Poisson: Events pop up randomly at average rate lambda per interval; count how many occur in an interval.
• Exponential: Wait time until the next random event from a Poisson process.
• Normal: Many small independent effects add up (measurement error, heights of people, average of many samples).
• Log-normal: Multiply small independent effects (e.g., price growth factors) — taking logs turns products into sums.

Worked examples

Example 1: Support emails per hour

Scenario: Over many hours, you record counts of support emails: 0,1,2,... You suspect a Poisson process.

1. Estimate lambda: the average count per hour (mean of the data). Suppose the mean is 3.2 emails/hour.
2. Quick check: For Poisson, variance is approximately equal to the mean. If sample variance is near 3.2, it supports Poisson.
3. Probability of zero emails in an hour: P(X=0) = exp(-lambda) = exp(-3.2) ≈ 0.040.

Example 2: Transaction amounts (skewed)

Scenario: Purchase amounts are positive and right-skewed; a few very large orders exist.

1. Hypothesis: Log-normal. Take log(amount) and plot a histogram.
2. If log(amount) looks roughly bell-shaped, use Normal summaries on the log scale.
3. Median on the original scale equals exp(mean of log(amount)). If mean log = 2.1, median ≈ exp(2.1) ≈ 8.17.

Example 3: Email campaign conversion

Scenario: Each of 1,000 recipients independently converts with probability p.

1. Model: Binomial(n=1000, p).
2. Expected conversions: n * p. If p = 0.03, expected = 30.
3. Standard deviation: sqrt(n p (1-p)) ≈ sqrt(1000 * 0.03 * 0.97) ≈ 5.4.
4. Rough 95% range: expected ± 2*sd → 30 ± 10.8 → about 19 to 41.

Exercises

Complete the tasks below. Solutions are available in toggles. Your progress is saved if you are logged in; otherwise, you can still practice for free.

Exercise 1: Match scenarios to distributions

For each scenario, select the most suitable distribution and estimate its key parameter(s):

1. A. Number of app crashes per day (rare, independent events).
2. B. Whether a user clicks a button in a single app session.
3. C. Number of clicks out of 200 ad impressions with stable probability.
4. D. Time between consecutive signups on a landing page.
5. E. Daily revenue values that are positive and heavily right-skewed.

Estimate parameters using these summaries (hypothetical):

• Average crashes/day: 1.4
• Click probability per impression: 0.05
• Impressions in C: n = 200
• Median time between signups: about 2 minutes
• Mean of log(daily revenue): 3.2; SD of log(daily revenue): 0.6

• I picked one distribution per scenario.
• I estimated parameters using the given numbers.
• My choices match the data type (count, yes/no, time, positive skew).

Exercise 2: Compute simple probabilities

Use the distribution formulas or standard approximations:

1. Poisson with lambda = 3 per hour: probability of zero events in the next hour?
2. Normal with mean = 48 and SD = 12 hours: probability a ticket resolves in over 72 hours?
3. Log-normal where log(amount) ~ Normal(mean = 2.0, SD = 0.5): what is the median amount?
4. Small sample mean: n = 25, sample mean = 68, sample SD = 10. 95% CI for the population mean?

• I wrote the formula used for each calculation.
• I showed a rounded numeric answer.
• For (4), I used the t-multiplier (not Normal) since SD is estimated.

Common mistakes and self-check

• Using Normal on raw, right-skewed amounts. Self-check: Plot histogram and try log; if log looks bell-shaped, prefer log-normal summaries.
• Treating counts with mean near 0 as Normal. Self-check: If mean is small and variance ≈ mean, Poisson often fits better.
• Forgetting independence. Self-check: If events cluster (dependence), Poisson may understate variance (overdispersion).
• Using z-interval with small n and unknown sigma. Self-check: Use t with df = n-1 when sigma is unknown and n is small.
• Confusing median and mean under log-normal. Self-check: Median = exp(mean of log data), not exp(mean of raw).

Practical projects

• Helpdesk volume model: Collect hourly ticket counts for two weeks. Fit Poisson (estimate lambda). Compare observed variance to lambda; note any hours with overdispersion.
• Revenue shape check: Take 3 months of order amounts. Plot raw and log histograms. Report median, IQR on raw; and mean, SD on log. Present one slide with recommendation.
• CTR stability: For daily impressions and clicks, model clicks ~ Binomial(n, p). Compute daily p-hats and a 95% interval for p. Flag days outside the interval and explain practical reasons.

Mini challenge

You observe: (i) 60% of minutes have 0 signups, 30% have 1, 10% have 2, almost none have 3+. Suggest a distribution and estimate its main parameter. Then, propose a quick check to validate your choice.

Hint

Rates per minute with many zeros often suggest a Poisson with lambda around the average count per minute; check mean vs variance.

Who this is for

• Entry-level and aspiring Data Analysts who want to interpret data distributions clearly.
• Professionals switching from reporting to analytical modeling.

Prerequisites

• Basic arithmetic and percentages.
• Comfort with averages, variance, and standard deviation.
• Ability to read histograms and bar charts.

Learning path

• Before: Descriptive statistics (mean, median, variance), data types (categorical vs numeric).
• This subskill: Recognize and use basic distributions in EDA.
• After: Sampling and the Central Limit Theorem, hypothesis testing, regression assumptions.

Next steps

• Re-check your last project: Which distributions did you assume implicitly? Were they appropriate?
• Build a small notebook/template to test Poisson vs. overdispersion and to visualize log-normal candidates.
• Take the Quick Test below. You can take it for free; sign in to save your progress.