Expected Value and Scaling to a Population

Lesson 2 of 2: From Average to Population Total

In this lesson:

• Compute expected value from the data
• Handle an open-ended category
• Scale the average to a population total

What You Will Be Able to Do

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

1. Compute expected value from data, handling an open-ended category
2. Scale an expected value to a population total using
3. Explain how sample size and representativeness affect the estimate

Average Per Household, and Per Hundred?

We have the empirical distribution.

What's the average number of TV sets per household — and in 100 households?

Start the Sum, Hit "4 or More"

The last category is "4 or more" — what number goes there?

An Open-Ended Category Has No Single Value

• "4 or more" could mean 4, 5, 6, ...
• The formula needs one number per category
• So we must choose — and state — a representative value

Choose and State the Representative Value

We treat "4 or more" as 4, and note this biases slightly low.

Compute the Expected Value Now

Interpret: About Two Sets Per Household

means a typical household owns about 2 TV sets.

• It's the long-run average across households
• It need not be a whole number — like 2.3 people per household

Per Household Is 2.03: What About 100?

We know the average per household.

The standard asks: how many TV sets in 100 randomly selected households?

Scale Up: 100 Times the Average

In General: Expected Total Is n Times E(X)

• The expected value of a sum is the sum of expected values
• Each of households contributes on average

These Probabilities Are Only Estimates

Unlike the theoretical case, our probabilities came from a survey.

• A different or larger survey would give somewhat different numbers
• They are estimates, not exact truths

Sample Quality Drives Prediction Quality

• Small or biased sample → shaky prediction
• Large, representative sample → reliable estimate

Your Turn: Scale and Judge Confidence

A survey gives an average of 1.8 pets per household.

Find the expected total for 500 households, and rate your confidence given a 30-household survey.

Commit before advancing.

Two Scaling Traps to Watch For

Per-unit vs total — 2.03 is per household; 100 households is about 203

"4 or more" as 4 biases low — state the choice; the true average is a bit higher

What This Lesson Gave You

✓ Compute from data — state any open-ended choice

✓ Scale with to a population total

✓ The estimate is only as good as the sample

Coming Up Next: Using Expected Value to Decide

You can now build and summarize any distribution.

Next cluster: use expected value to evaluate decisions — games, insurance, and strategies.