Graphing Distributions and Reading Them

Lesson 2 of 2: From Table to Probability Histogram

In this lesson:

• Graph a distribution as a probability histogram
• Read its shape, center, and spread
• Choose a random variable for a real situation

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Graph a probability distribution as a probability histogram
2. Describe its shape, center, and spread, and choose a random variable for a real quantity

Same Picture for Something Nobody Measured?

You drew histograms of data in the ID unit.

But this two-coin distribution isn't data — nobody tossed any coins.

Can the same picture still work?

Build the Two-Coin Probability Histogram

= number of heads. Bars at heights 0.25, 0.50, 0.25.

The One Change: Probability on the Y-Axis

• The bars are the same shape
• The y-axis reads probability, not frequency or count

Read It Like You Read Data

The two-coin distribution is:

• Symmetric — mirror image around the center
• Centered at 1 — the middle value carries the most probability
• Most probability piled in the middle

Graph the Three-Children Distribution Now

= number of girls, probabilities :

Describe This Distribution In Words

Look at the three-children histogram above.

In your own words: what is its shape, where is its center, and how spread out is it?

Commit to a description before advancing.

Shape Is Not Always Symmetric

= number of sixes in two die rolls:

• Heavy at 0, tiny at 2 — right-skewed

Predict: Do You Need Data First?

To graph a probability distribution, must you first collect data?

• A. Yes — measure, then plot
• B. No — the model gives the heights

Pick A or B before advancing.

Data Shows What Happened; This Shows What's Expected

• A data distribution shows what happened in a sample
• A probability distribution shows what's expected from the model

Every height here came from reasoning — zero coins were tossed.

Now: Choosing X for a Real Situation

We've read graphs. The harder real-world move is naming the variable.

Given a situation, what number is worth tracking?

Name X for Each Context

For each situation, the count or value is the natural object:

• Pull 3 parts from a line — = number defective
• Guess on a quiz — = number correct
• Spin a prize wheel — = points won

Worked: Defects in a Sample of Three

Pull 3 parts; each is defective or not.

= number defective. Possible values: 0, 1, 2, 3.

Three parts means at most three defects — four possible values.

A Distribution Describes the Long Run

A probability distribution is not a prediction of one trial.

• It describes what to expect over many trials
• One spin or one pull can land anywhere

Your Turn: Match Scenario to Graph

Match each scenario to where its probability should pile up.

Reason about shape before checking.

Two Graph-Reading Traps to Avoid

Y-axis is probability — never label it "count" or "frequency"

No data needed — these heights come from the model, not a sample

What This Lesson Gave You

✓ A distribution graphs as a probability histogram — same display, probability y-axis

✓ Read shape, center, spread as you do for data

✓ It shows the long run, not one trial

Coming Up Next: Expected Value

Every graph here has a balance point — one number for where the whole distribution sits.

Next standard: compute it. It's the expected value.