Empirical Distributions From Data

Lesson 1 of 2: Probabilities as Relative Frequencies

In this lesson:

• Estimate probabilities from collected data
• Validate an empirical distribution (with rounding)
• Graph it as a probability histogram

What You Will Be Able to Do

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

1. Estimate each probability as a relative frequency, count over total
2. Verify the distribution sums to about 1, handling rounding
3. Graph the empirical distribution as a probability histogram

How Many TV Sets Does a Household Own?

There's no formula for this.

So where do the probabilities come from?

No Model: The Empirical Route

Recall the two routes from last standard:

• Theoretical — a model computes the probabilities
• Empirical — you estimate them from data

TV sets has no model, so we collect data.

Divide Each Count by the Total

Each value's probability = its count ÷ the total surveyed.

Each Probability Is a Relative Frequency

So — about 35% reported one set.

Quick Check: Convert These Counts

A survey of 200 pet owners: 0 pets: 40, 1 pet: 90, 2 pets: 50, 3+: 20.

Convert each count to a relative-frequency probability.

Work them out before advancing.

We Have Probabilities: Now Validate and Graph

We turned counts into probabilities.

Do they hold together — and what shape do they make?

The Sum Should Be About One

• A valid distribution's probabilities total 1
• For our table:
• With other data, small rounding can make it 0.99 or 1.01

Predict: Is a 1.01 Sum an Error?

A survey's probabilities sum to 1.01.

• A. A mistake — redo the work
• B. Something else is going on

Pick A or B before advancing.

It's Rounding — Just Renormalize

The cause is rounding of each reported percent, not an error.

• If you need an exact sum, renormalize: divide each probability by the total
• Example: each value divided by 1.01

Graph It: A Mildly Right-Skewed Shape

Most probability at 1 and 2 sets, a thin tail toward more.

It's a Data Histogram, Rescaled

• The bars came from survey counts, just divided by the total
• The y-axis reads probability instead of count
• An empirical distribution is essentially a rescaled data histogram

Your Turn: Validate and Describe

Take your pet-survey probabilities (0.20, 0.45, 0.25, 0.10).

Confirm they sum to 1, then describe the shape.

Commit before advancing.

Two Empirical Traps to Avoid

Raw counts are not probabilities — divide by the total; a probability is between 0 and 1

A sum of 0.99 or 1.01 is rounding — not an error; renormalize if you need exactly 1

What You Built In This Lesson

✓ Empirical probability = count ÷ total (relative frequency)

✓ Validate to about 1; rounding is normal

✓ It graphs as a rescaled data histogram

Coming Up Next: Expected Value and Scaling

Next lesson: compute the average TV sets per household, handle the open-ended "4 or more," and scale up.

How many sets would you expect in 100 households?