Validate, Graph, and Score the Distribution

Lesson 2 of 2: From Table to Expected Grade

In this lesson:

• Check the distribution sums to 1
• Graph its right-skewed shape
• Compare expected grades across schemes

What You Will Be Able to Do

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

1. Verify the distribution sums to 1 and graph it as a probability histogram
2. Compute the expected number correct and the expected grade under different schemes

Does It Hold Together, and What Grade?

You built the guessing table last lesson.

Does it sum to 1? What does it look like? What grade does a guesser expect?

Sum Check: 1024 Over 1024

So the probabilities total . ✓

A Failed Sum Means a Wrong Count

• The values of cover every outcome, so they must total 1
• If your sum isn't 1, a count was likely wrong
• Treat the check as a safeguard, not a ritual

Graph the Distribution: Strongly Skewed

Most probability sits at 0 and 1 correct.

Why Guessing Almost Never Scores High

• Getting many right by chance needs several lucky guesses to coincide
• Each lucky guess has probability only
• So the probability piles up at the low end

Quick Check: Where Is the Probability?

Look at the histogram.

In one sentence: where is the probability concentrated, and what does that say about guessing?

Answer before advancing.

The Table Is Correct: What Grade?

The distribution is valid and graphed.

Now: what grade does this guessing produce — and can a scheme discourage it?

Expected Number Correct Is 1.25

A Quick Way to Confirm It

• Five questions, each correct with probability
• The expected count is just (number) × (chance each)

Scheme A: Plain Scoring, No Penalty

20 points per correct answer.

Expected grade out of 100.

Scheme B: Penalty Per Question

+20 per correct, −5 per wrong. Per question:

Scheme B Total: Far Below Plain

Over 5 questions: points.

Your Turn: Recommend a Scheme

A teacher wants to discourage guessing.

Which scheme should they use? Justify with the expected grades.

Commit to a recommendation before advancing.

A Penalty-Scheme Trap to Avoid

Weight the penalty by — the −5 is multiplied by , not counted once

Keep its negative sign — a punishing scheme must lower the expected grade

What This Lesson Gave You

✓ Validate with the sum-to-1 check, then graph the skew

✓ Compute the expected number correct (1.25)

✓ The same distribution scores very differently by scheme

Coming Up Next: Probabilities From Data

Here the probabilities came from a model.

Next standard: the same expected-value work when probabilities must come from data — the empirical route.