Module 04

Distributions

The shapes that data takes

Key Distribution Concepts

  • 01

    Normal Distribution

    The bell curve foundation

  • 02

    Skewness

    Asymmetry in data

  • 03

    Kurtosis

    Tail weight and peakedness

  • 04

    Other Distributions

    Uniform, binomial, and more

01

Normal Distribution

Interactive Normal Curve Explorer

Normal Distribution Explorer
0
1
P(X < 0): 0.5000 P(X > 0): 0.5000

Adjust the mean (μ) and standard deviation (σ) sliders to see how the normal distribution changes! Watch how μ shifts the curve left or right, while σ makes it wider or narrower. The shaded area shows the 68-95-99.7 rule in action—see what percentage of data falls within 1, 2, or 3 standard deviations.

Parameters
μ (mean), σ (std dev)
Key Rule
68-95-99.7

The Bell Curve

μ (mean)

The normal distribution (Gaussian) is symmetric and bell-shaped. The peak is at the mean, and it tapers equally on both sides. It's defined by just two parameters: mean (μ) and standard deviation (σ).

Parameters
μ (mean), σ (std dev)
Key Rule
68-95-99.7

Key Properties

Symmetric
Mean = Median = Mode
Asymptotic
Tails approach but never touch x-axis
Area = 1
Total probability under curve is 100%
Defined by μ, σ
Two parameters fully specify it

The normal distribution is symmetric (no skew), with tails extending infinitely but getting ever closer to zero. The total area under the curve equals 1 (100% probability).

Parameters
μ (mean), σ (std dev)
Key Rule
68-95-99.7

The Empirical Rule

-3σ-2σ-1σμ+1σ+2σ+3σ
68%95%99.7%

The 68-95-99.7 rule: ~68% of data falls within 1σ of the mean, ~95% within 2σ, and ~99.7% within 3σ. Memorize this—it's fundamental to statistical reasoning.

Parameters
μ (mean), σ (std dev)
Key Rule
68-95-99.7

Ubiquity of Normal

Where Normal Distributions Appear
Human heightsTest scoresBlood pressureMeasurement errorsIQ scoresBirth weights
Central Limit Theorem: Sample means are normally distributed regardless of population shape (with large n).

Many natural phenomena follow normal distributions. More importantly, the Central Limit Theorem says sample means tend toward normal, which is why most statistical tests assume normality.

Parameters
μ (mean), σ (std dev)
Key Rule
68-95-99.7
02

Skewness

Asymmetry in Distributions

Symmetric
Skew = 0
Right Skew
Skew > 0
Left Skew
Skew < 0

Skewness measures asymmetry. Positive (right) skew has a tail extending right—most values cluster left with a few high outliers. Negative (left) skew is the opposite.

Right Skew
Tail points right (positive)
Left Skew
Tail points left (negative)

Positive Skewness

▼ Mode▼ Median▼ Mean
Examples: Income, house prices, hospital stays, insurance claims

Right-skewed data has: Mode < Median < Mean. The long tail of high values pulls the mean up. Most people earn below the 'average' income because a few billionaires skew the mean.

Right Skew
Tail points right (positive)
Left Skew
Tail points left (negative)

Negative Skewness

▼ Mean▼ Median▼ Mode
Examples: Age at death (developed countries), exam scores (easy test)

Left-skewed data has: Mean < Median < Mode. The long tail of low values pulls the mean down. When most students ace an easy test, the few low scores create left skew.

Right Skew
Tail points right (positive)
Left Skew
Tail points left (negative)

How to Identify Skewness

📊 Visual Methods
• Histogram shape
• Box plot whisker lengths
• Q-Q plot deviation
📐 Numeric Methods
• Compare mean vs median
• Skewness coefficient
• Pearson's mode skewness
Quick check: If mean ≠ median, suspect skewness. Direction: mean is pulled toward the tail.

Multiple ways to detect skewness: visually (histogram, boxplot), or numerically (compare mean to median, calculate skewness coefficient). The mean-median comparison is quick and intuitive.

Right Skew
Tail points right (positive)
Left Skew
Tail points left (negative)
03

Kurtosis

Tail Weight & Peakedness

Leptokurtic
Heavy tails, sharp peak
Mesokurtic
Normal (baseline)
Platykurtic
Light tails, flat peak

Kurtosis measures tail heaviness. Leptokurtic (positive excess kurtosis) has heavier tails and more outliers. Platykurtic (negative) has lighter tails. Mesokurtic is normal.

Normal Kurtosis
3 (or 0 excess)
Focus On
Tail weight, not peak

What Kurtosis Really Means

Common misconception: Kurtosis is NOT about 'peakedness.' It's about tail weight—likelihood of extreme values.
High Kurtosis (Lepto)
• More outliers expected
• Extreme values more likely
• 'Fat tails'
Low Kurtosis (Platy)
• Fewer outliers expected
• Extreme values rare
• 'Thin tails'

Kurtosis tells you about outlier probability. High kurtosis means more data in tails = more outliers. This matters for risk assessment, financial modeling, and choosing statistical tests.

Normal Kurtosis
3 (or 0 excess)
Focus On
Tail weight, not peak

Kurtosis Values

0 (Normal)
←
Platykurtic
→
Leptokurtic
Excess Kurtosis: Measured relative to normal distribution (which has kurtosis = 3). Excess kurtosis = kurtosis - 3. Normal has excess kurtosis of 0.

Normal distribution has kurtosis of 3. We often report 'excess kurtosis' (kurtosis - 3) so normal = 0. Positive excess = heavier tails than normal. Negative = lighter tails.

Normal Kurtosis
3 (or 0 excess)
Focus On
Tail weight, not peak

Real-World Examples

Leptokurtic Examples
• Stock returns (fat tails)
• Income distributions
• Insurance claims
• Earthquake magnitudes
Platykurtic Examples
• Uniform distribution
• Quality control (bounded)
• Some survey responses
• Dice rolls
Warning: Financial models assuming normal tails underestimate risk. Stock market crashes are more common than normal predicts.

Stock returns have high kurtosis—extreme movements (crashes, rallies) happen more often than a normal distribution predicts. This is why 'black swan' events catch models off guard.

Normal Kurtosis
3 (or 0 excess)
Focus On
Tail weight, not peak
04

Other Distributions

Uniform Distribution

a (min)b (max)
All values equally likely between a and b

In a uniform distribution, all values within the range are equally likely. Rolling a fair die is discrete uniform (1-6). Choosing a random point on a line segment is continuous uniform.

Discrete
Binomial, Poisson
Continuous
Uniform, Exponential, Normal

Binomial Distribution

n=10 trials, p=0.5 (coin flips)
X-axis: # of successes (0 to 10)

Binomial counts successes in n independent trials, each with probability p. Flip a coin 10 times: how many heads? Parameters: n (trials), p (success probability).

Discrete
Binomial, Poisson
Continuous
Uniform, Exponential, Normal

Poisson Distribution

λ = 3 events per interval
X-axis: # of events (0, 1, 2, ...)

Poisson models rare events over time/space: emails per hour, typos per page, accidents per month. Parameter λ is the average rate. Events are independent and random.

Discrete
Binomial, Poisson
Continuous
Uniform, Exponential, Normal

Exponential Distribution

Time until next event (memoryless)
Examples: Wait times, lifetimes, decay

Exponential models waiting time until an event: time until next bus, time until equipment failure. It's 'memoryless'—the past doesn't affect future waiting time.

Discrete
Binomial, Poisson
Continuous
Uniform, Exponential, Normal

Inference Distributions

t-Distribution
Like normal but heavier tails (small n)
Chi-Square
Right-skewed, used for variance tests

t-distribution: like normal but with heavier tails, used when sample size is small. Chi-square: always positive, right-skewed, used for variance tests and categorical data.

Discrete
Binomial, Poisson
Continuous
Uniform, Exponential, Normal
★

Real-World Examples

Apply your knowledge to practical scenarios

01

Predicting Heights in a Population

Adult male heights in the US follow a normal distribution with mean μ = 70 inches (5'10") and standard deviation σ = 3 inches. A clothing company wants to stock sizes appropriately.

Sample Data
{
  "mean": 70,
  "sd": 3,
  "unit": "inches"
}
Q What percentage of men are between 67 and 73 inches tall (within 1 SD)? +
A About 68%. The 68-95-99.7 rule tells us 68% of data falls within ±1 SD of the mean in a normal distribution.
Q If the company only stocks sizes for men between 64 and 76 inches, what percentage of customers can they serve? +
A About 95%. This range is ±2 SD from the mean (70 ± 6), which covers approximately 95% of the population.
Q How tall is someone in the 97.5th percentile? +
A About 76 inches (6'4"). The 97.5th percentile is approximately mean + 2 SD = 70 + 6 = 76 inches.
Q If a man is 79 inches tall (6'7"), how unusual is this? +
A Very unusual—about 3 SDs above mean. Only ~0.15% of men are this tall or taller (beyond +3 SD in one tail).
Key Insights
  • Normal distribution allows precise probability calculations from mean and SD alone
  • The 68-95-99.7 rule is a powerful tool for quick estimates
  • Heights cluster predictably—extremes are mathematically rare
02

Why 'Average' Income is Misleading

A news headline claims: 'Average household income rises to $100,000!' But you earn $60,000 and feel like you're below average. Let's investigate the skewed reality of income distributions.

Sample Data
{
  "mean": 100000,
  "median": 70000,
  "mode": 50000
}
Q Given mean > median > mode, what type of skew does income have? +
A Right (positive) skew. The long tail of high earners pulls the mean above the median. Most people earn less than the 'average.'
Q If the median is $70,000, what does this tell us compared to the $100,000 mean? +
A Half of households earn less than $70K, half earn more. The mean is inflated by high earners—billionaires drastically pull up the average.
Q Which measure should journalists report for a fair picture of 'typical' income? +
A Median. It represents the true middle—the point where 50% earn less and 50% earn more. Mean can be misleading for skewed data.
Q You earn $60,000. Are you really 'below average'? +
A You're below the mean, but close to the mode (most common income). You're probably in the lower-middle of the distribution—quite normal!
Key Insights
  • Income, wealth, and house prices are almost always right-skewed
  • Median is the honest measure for skewed distributions
  • A few extreme values can drastically inflate the mean
03

Call Center Staffing with Poisson

A call center receives an average of λ = 4 calls per minute. Management needs to staff appropriately. Calls arrive randomly and independently (a Poisson process).

Sample Data
{
  "lambda": 4,
  "unit": "calls per minute"
}
Q What's the probability of receiving exactly 4 calls in a given minute? +
A About 19.5%. Using Poisson formula: P(X=4) = (e^-4 × 4^4) / 4! ≈ 0.195. Even at the average rate, you only see exactly that many 1 in 5 minutes.
Q What's the probability of receiving 0 calls in a minute (a quiet moment)? +
A About 1.8%. P(X=0) = e^-4 ≈ 0.018. Even with average of 4 calls/min, there's still a small chance of zero calls.
Q If they can handle at most 8 calls per minute, how often will they be overwhelmed? +
A About 5% of the time. P(X > 8) ≈ 0.05. You need to calculate P(X ≤ 8) and subtract from 1.
Q For a 5-minute interval, what distribution applies and what's the average? +
A Still Poisson, with λ = 20 calls (4 × 5 minutes). Poisson is additive over independent time intervals.
Key Insights
  • Poisson models count of rare events over fixed time/space
  • λ is both the mean AND variance of a Poisson distribution
  • Real call patterns often deviate—test your assumptions!
04

Fat Tails and Financial Crashes

Stock market returns are often modeled as normal distributions, but they actually have 'fat tails' (high kurtosis). This miscalculation has crashed banks and economies.

Q If returns were truly normal, a -20% daily drop (like Black Monday 1987) should occur once in how many years? +
A Once in billions of years. A -20% move would be about 20+ standard deviations from the mean—statistically 'impossible' under normal assumptions.
Q What does 'fat tails' or high kurtosis mean for market risk? +
A Extreme events (crashes and rallies) happen FAR more often than normal distribution predicts. The tails contain more probability than models assume.
Q Why did models fail in the 2008 financial crisis? +
A Risk models assumed normal distributions. They said certain loss events were '25-sigma events' (once per universe lifetime) when they actually happen every decade or so.
Q What's the practical lesson for risk management? +
A Never trust models that assume normality for financial data. Use stress testing, scenario analysis, and expect 'impossible' events to happen.
Key Insights
  • Financial returns have high kurtosis (fat tails)—more outliers than normal predicts
  • Black swan events are not as rare as Gaussian models suggest
  • The 2008 crisis was partly caused by blind faith in normal distribution assumptions
?

Distributions Quiz

Test your understanding • Passing score: 70%

0 of 14 answered
Q1 multiple-choice

Which three measures are equal in a perfectly normal distribution?

Explanation: In a perfectly symmetric normal distribution, mean = median = mode. They all sit at the center of the bell curve.
Q2 multiple-choice

According to the 68-95-99.7 rule, what percentage of data falls within 3 standard deviations of the mean?

Explanation: The empirical rule states that 99.7% of data in a normal distribution falls within ±3 standard deviations of the mean.
Q3 multiple-choice

Income data typically shows: Mean = $100K, Median = $70K. This distribution is:

Explanation: When mean > median, the distribution is right-skewed (positively skewed). High-income outliers pull the mean to the right.
Q4 multiple-choice

In a right-skewed distribution, what's the order of mean, median, and mode?

Explanation: In right-skewed distributions: Mode < Median < Mean. The tail extends right, pulling the mean in that direction.
Q5 multiple-choice

What does kurtosis primarily measure?

Explanation: Kurtosis measures tail weight—how likely extreme values (outliers) are. High kurtosis means 'fat tails' with more probability in the extremes.
Q6 multiple-choice

A distribution with heavy tails and more outliers than normal is called:

Explanation: Leptokurtic distributions have positive excess kurtosis—heavier tails than normal, meaning more outliers and extreme values.
Q7 multiple-choice

You flip a fair coin 10 times and count heads. Which distribution describes this?

Explanation: Binomial distribution models the count of successes in n independent trials, each with probability p. Coin flips are the classic example.
Q8 multiple-choice

A call center receives an average of 5 calls per hour. Which distribution models the number of calls?

Explanation: Poisson distribution models the count of rare, independent events over a fixed interval. Call arrivals per hour is a classic Poisson scenario.
Q9 multiple-choice

The exponential distribution is best suited for modeling:

Explanation: Exponential distribution models continuous waiting time until an event occurs. It's related to Poisson—if events follow Poisson, waiting times are exponential.
Q10 multiple-choice

What is the 'memoryless' property of the exponential distribution?

Explanation: Memoryless means P(wait > 5 more minutes | already waited 10) = P(wait > 5 minutes). The past doesn't inform the future—like radioactive decay.
Q11 multiple-choice

The Central Limit Theorem states that:

Explanation: CLT is fundamental: sample means approach a normal distribution as n increases, even if the population is skewed or non-normal. This justifies many statistical tests.
Q12 multiple-choice

Why is the t-distribution used instead of normal distribution for small samples?

Explanation: With small samples, we're less certain about the true population SD. The t-distribution has heavier tails, accounting for this extra uncertainty. As n increases, t approaches normal.
Q13 multiple-choice

A normal distribution has excess kurtosis of:

Explanation: Normal distribution has kurtosis of 3, but EXCESS kurtosis (kurtosis - 3) is 0. This is the baseline—positive excess means heavier tails, negative means lighter.
Q14 multiple-choice

Easy exam scores are typically:

Explanation: Easy exams produce left-skewed distributions: most students score high (clustering right), with a tail of lower scores stretching left. Mean < Median < Mode.

Quiz Results

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