Statistics and Probability for 11th Graders — Part 2

Probability distribution (continuation and light mode)

Joshua Marie

2025-12-05

Hands-on Quiz

What defines the population and sample?

• Population refers to a subset of data while sample refers to all data
• Population refers to estimates while sample refers to parameters
• Population refers to small groups while sample refers to large groups
• Population refers to the entire group being studied while sample refers to a subset selected from that group

Hands-on Quiz

What is the probability of drawing a “king” from the deck of card?

• \(\frac{1}{13}\)
• \(\frac{1}{2}\)
• \(\frac{2}{13}\)
• \(\frac{1}{5}\)

Hands-on Quiz

Consider \(A\) and \(B\) defines the two events. Which of the following defines mutually exclusive events?

• Two events occur simultaneously
• Two events are disjoint
• \(P(A \cap B) = 0\)
• \(P(A \cap B) \neq 0\)

Hands-on Quiz

A binomial distribution requires which conditions?

• Trials are dependent and probability changes
• Fixed number of independent trials with constant probability
• Continuous outcomes with no upper limit
• Only two trials are needed

Hands-on Quiz

What is the expected value of rolling a fair six-sided die?

• 3.2
• 3.5
• 4.5
• 5.3

Probability distribution

Let us continue…

Probability distribution

Let us continue…

Remember: For \(n\) independent trials with probability \(p\):

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Side question: What shape do you notice as \(n\) gets larger?

Probability distribution

What happens when you increase the number of trials \(n\)?

Probability distribution

What happens when you increase the number of trials \(n\)?

import { plotBinomial } from "./binomial_sim.js"

// Sample Size
viewof sampleSize = Inputs.range([50, 100000], {
    label: "Size",
    value: 50,
    step: 10
})

// Number of Trials
viewof n = Inputs.range([1, 300], {
    label: "(n)",
    value: 1,
    step: 1
})

// Probability of Success
viewof p = Inputs.range([0.001, 0.999], {
    label: "(p)",
    value: 0.5,
    step: 0.001
})

plotBinomial(sampleSize, n, p)

Because of this pattern, we define what we called the normal distribution.

Normal distribution

A normal distribution is a continuous probability distribution that describes the population that has mean \(\mu\) and variance \(\sigma^2\) forms a bell-shaped curve.

Normal distribution

A normal distribution is a continuous probability distribution that describes the population that has mean \(\mu\) and variance \(\sigma^2\) forms a bell-shaped curve.

Key characteristics:

• Bell-shaped and symmetric
• Mean = Median = Mode (all at the center)
• Total area under curve = 1 (or 100%)
• Extends indefinitely in both directions (left and right of the mean \(\mu\))

Normal distribution

It has the closed mathematical form that defines the normal distribution:

\[ f(x)=\frac{1}{\sigma\sqrt{2\pi}} \exp\!\left( -\frac{(x-\mu)^2}{2\sigma^2} \right) \]

The \(f(x)\) there refers to the density function. Some textbooks denotes the density function as \(f_X(x)\).

The random variable for this distribution \(X \sim \mathcal{N}(\mu, \sigma)\) (reads as “X follows the normal distribution with parameters mean \(\mu\) and standard deviation \(\sigma\)”).

Visualization of the normal distribution

viewof mu = Inputs.range([-7, 7], {
    label: "Mean (μ)",
    value: 0,
    step: 0.1
})

viewof sigma = Inputs.range([0.001, 5], {
    label: "Standard Deviation (σ)",
    value: 1,
    step: 0.01
})

{
    const width = 800;
    const height = 400;
    const margin = {top: 20, right: 30, bottom: 40, left: 50};
    
    const xMin = -10;
    const xMax = 10;
    
    const data = [];
    for (let x = xMin; x <= xMax; x += 0.1) {
        const y = (1 / (sigma * Math.sqrt(2 * Math.PI))) * 
                  Math.exp(-0.5 * Math.pow((x - mu) / sigma, 2));
        data.push({x, y});
    }
    
    const svg = d3.create("svg")
        .attr("width", width)
        .attr("height", height)
        .attr("viewBox", [0, 0, width, height]);
    
    const xScale = d3.scaleLinear()
        .domain([xMin, xMax])
        .range([margin.left, width - margin.right]);
    
    const yScale = d3.scaleLinear()
        .domain([0, 0.9])
        .range([height - margin.bottom, margin.top]);
    
    const line = d3.line()
        .x(d => xScale(d.x))
        .y(d => yScale(d.y));
    
    svg.append("path")
        .datum(data)
        .attr("fill", "none")
        .attr("stroke", "#4CAF50")
        .attr("stroke-width", 2.5)
        .attr("d", line);
    
    svg.append("g")
        .attr("transform", `translate(0,${height - margin.bottom})`)
        .call(d3.axisBottom(xScale));
    
    svg.append("g")
        .attr("transform", `translate(${margin.left},0)`)
        .call(d3.axisLeft(yScale));
    
    svg.append("line")
        .attr("x1", xScale(mu))
        .attr("x2", xScale(mu))
        .attr("y1", margin.top)
        .attr("y2", height - margin.bottom)
        .attr("stroke", "red")
        .attr("stroke-width", 2)
        .attr("stroke-dasharray", "5,5");
    
    return svg.node();
}

Observe

As you adjust \(\mu\) and \(\sigma\), notice how the curve shifts and spreads!

Characteristics of the normal distribution

Visual Properties

1. Bell-shaped with highest peak at center
2. Symmetrical about the mean
3. Continuous on both sides
4. Horizontally Asymptotic Never touches the horizontal axis

Characteristics of the normal distribution

Visual Properties

1. Bell-shaped with highest peak at center
2. Symmetrical about the mean
3. Continuous on both sides
4. Horizontally Asymptotic Never touches the horizontal axis

Mathematical Properties

• Mean = Median = Mode
• Total area = 1 (100%)
• Determined by two parameters:
  • \(\mu\) (mean): location
  • \(\sigma\) (standard deviation): spread

Standard Normal Distribution

The standard normal distribution is a special case where:

• Mean \(\mu = 0\)
• Standard deviation \(\sigma = 1\)

Most textbooks denotes the random variable \(Z\) of the standard normal distribution as \(Z \sim \mathcal{N}(0, 1)\)

We can scale down \(X \sim \mathcal{N}(\mu, \sigma)\) to \(Z \sim \mathcal{N}(0, 1)\). We called it “standardization”.

The importance of z-scores

Compare different distributions

It is much easier to compare different distribution (still normal) on the same scale

Dealing with probabilities

Although we have softwares for us to calculate such probabilities, interpreting probabilities under z-score scale is much easier.

Identify outliers

Standardization has a huge advantage to the application of outlier identification.

The Empirical Rule (68-95-99.7 Rule)

viewof mu_empirical = Inputs.range([-10, 10], {
    label: "Mean (μ)",
    value: 0,
    step: 0.5
})

viewof sigma_empirical = Inputs.range([0.001, 5], {
    label: "Std. Dev. (σ)",
    value: 1,
    step: 0.01
})

viewof ruleSelect = Inputs.radio(
    new Map([
        ["68% Rule (±1σ)", 1],
        ["95% Rule (±2σ)", 2],
        ["99.7% Rule (±3σ)", 3],
        ["All Rules", 4]
    ]),
    {label: "Show Rule:", value: 1}
)

import { plotEmpiricalRule } from "./normal_empirical_rule.js"

plotEmpiricalRule(mu_empirical, sigma_empirical, ruleSelect)

html`<hr><h4>Current Values:</h4>`

md`${
    ruleSelect === 1 ? `**68% of data falls between ${(mu_empirical - sigma_empirical).toFixed(2)} and ${(mu_empirical + sigma_empirical).toFixed(2)}**` :
    ruleSelect === 2 ? `**95% of data falls between ${(mu_empirical - 2*sigma_empirical).toFixed(2)} and ${(mu_empirical + 2*sigma_empirical).toFixed(2)}**` :
    ruleSelect === 3 ? `**99.7% of data falls between ${(mu_empirical - 3*sigma_empirical).toFixed(2)} and ${(mu_empirical + 3*sigma_empirical).toFixed(2)}**` :
    `**μ ± 1σ:** [${(mu_empirical - sigma_empirical).toFixed(2)}, ${(mu_empirical + sigma_empirical).toFixed(2)}]
    
**μ ± 2σ:** [${(mu_empirical - 2*sigma_empirical).toFixed(2)}, ${(mu_empirical + 2*sigma_empirical).toFixed(2)}]

**μ ± 3σ:** [${(mu_empirical - 3*sigma_empirical).toFixed(2)}, ${(mu_empirical + 3*sigma_empirical).toFixed(2)}]`
}`

md`${
    ruleSelect === 1 ? `**68% Rule:** Approximately 68% of all data values fall within ONE standard deviation of the mean (μ ± σ).` :
    ruleSelect === 2 ? `**95% Rule:** Approximately 95% of all data values fall within TWO standard deviations of the mean (μ ± 2σ).` :
    ruleSelect === 3 ? `**99.7% Rule:** Approximately 99.7% of all data values fall within THREE standard deviations of the mean (μ ± 3σ).` :
    `**Complete Empirical Rule:** This shows all three rules simultaneously:
- 68% within μ ± σ (purple)
- 95% within μ ± 2σ (orange)  
- 99.7% within μ ± 3σ (red)`
}`

The Empirical Rule (68-95-99.7 Rule)

The Empirical Rule (also called the 68-95-99.7 Rule) tells us what percentage of data falls within certain distances from the mean in a normal distribution.

68% Rule

Approximately 68% of data falls within 1 standard deviation of the mean:

\[\mu - \sigma \leq X \leq \mu + \sigma\]

95% Rule

Approximately 95% of data falls within 2 standard deviations of the mean:

\[\mu - 2\sigma \leq X \leq \mu + 2\sigma\]

99.7% Rule

Approximately 99.7% of data falls within 3 standard deviations of the mean:

\[\mu - 3\sigma \leq X \leq \mu + 3\sigma\]

Practice Problem

Scenario: Heights of adult males in a population are normally distributed with a mean of 175 cm and a standard deviation of 8 cm.

Questions:

1. What percentage of men have heights between 167 cm and 183 cm?
2. What percentage of men are taller than 191 cm?
3. If a man is 159 cm tall, how many standard deviations is he from the mean?
4. Would a height of 159 cm be considered unusual? Why or why not?

Solutions:

1. 167 to 183 cm = \(\mu \pm 1\sigma\). It should be 68%

2. 191 cm = \(\mu + 2\sigma\)

   • Beyond \(2\sigma\) above mean
   • \((100\% - 95\%) / 2\) = 2.5%

3. \(z = \frac{159 - 175}{8} = \frac{-16}{8} = -2\) — 2 standard deviations below

4. Yes, unusual because it’s exactly \(2\sigma\) below the mean (only 2.5% are this short or shorter)

Assignment

Instructions:

1. Open the link for the exercise: