Turning Spread into a Margin of Error

Lesson 2 of 2: The Interval, and Its Limits

In this lesson:

• Turn the simulated spread into a margin of error and interval
• See how sample size shrinks it — and what it can't fix

The SD of the Simulated Estimates

Our sampling distribution centers at 0.65. Summarize its spread with one number.

• This captures how much the estimate typically wanders

One number for the spread — the raw material for the margin of error.

Margin of Error Is About Two SDs

• Margin of error ≈ 2 standard deviations
• That band captures the middle 95% of estimates

The Shaded Middle-95% Band of Estimates

With center 0.65 and SD ≈ 0.06:

• Two SDs ≈ 0.12
• The band runs from 0.53 to 0.77

That shaded band is "± 0.12" in visual form — the plausible values for .

The Proportion Interval: 0.65 ± 0.12

Interval estimate = point estimate ± margin of error

• Point estimate in the center, margin reaching both ways

This is exactly "52%, ± 3 points" — and you just computed it.

Interpret in Context: 53% to 77%

The interval in plain English:

• "We estimate that between 53% and 77% of all students would re-enroll."

Not "65% will" — a range, with the true value plausibly inside. The sentence is the conclusion.

The Same Procedure on the Mean

Commute estimates: center 27 min, SD ≈ 1.5 min.

• Margin ≈ 2 × 1.5 = 3 minutes
• Interval: minutes

"The true mean commute is between 24 and 30 minutes." Same recipe.

The Formula as a Check, Not the Method

There's a formula:

• It gives nearly the same 0.12 — a useful check
• But simulation is the named method; it shows why

Keep the formula as verification. Understanding beats plugging in.

Compute the Margin and Write the Interval

A poll of 400 adults: , simulated SD ≈ 0.05.

1. Margin of error? (≈ 2 SD)
2. Write the interval
3. State it in context

Do all three — margin, interval, sentence — before advancing.

Now Make It Tighter and Bounded

You can now report an interval. Two questions remain:

• How do we make it tighter? (more data — but how much?)
• What can the margin never fix?

The second answer is the warning the whole unit builds toward.

Comparing Two Sample Sizes Side by Side

• Quadrupling n (80 → 320) halved the margin
• Not the proportional shrink you'd expect

Quadruple n to Halve the Margin

The margin shrinks like .

• To halve the margin → quadruple n
• Doubling n shrinks it only by about 1.4×

Precision gets expensive fast. Each extra digit costs far more data.

Precision Is Expensive to Buy

Halving the margin costs 4× the data — time and money.

• Why polls settle near n = 1,000, margin ≈ 3 points
• This is all about precision — how tight the interval is

But tight says nothing about whether the interval is even centered on truth.

The Bias Caution: A Shifted Distribution

• A biased sample centers the whole distribution on the wrong value
• Narrow spread → small margin → but it's in the wrong place

A Tight Margin Can Be Precisely Wrong

A biased sample → a tight margin and a wrong interval.

• It may never even contain the truth
• Tight-and-wrong is worse than wide-and-honest

Precision is not accuracy. The margin assumes random sampling.

Margin of Error Is Not Bias Protection

The margin is precision under random sampling — nothing more.

• It never protects against a bad sampling method
• Same lesson as "size doesn't fix bias," now in interval form

Ask "was the sample random?" first. Only then does the margin mean anything.

Predict the Shrink; Flag the Bias

1. Margin is 4% at n = 100. What n gives a 2% margin?
2. A 100,000-person self-selected poll reports a margin under 1%. Why misleading?

Do both — predict the shrink, then flag the bias — before advancing.

Four Errors About Margins of Error

"Error" → it's sampling variability, not mistakes
"Bigger margin = worse data" → reflects n, not quality
"Double n halves it" → you must quadruple n
"It covers bias" → bias shifts the whole distribution; margin is blind

Four traps, four corrections.

Key Takeaways From Lesson Two

✓ Margin ≈ 2 SD → a range of plausible truths
✓ Interval = estimate ± margin; state it in context
✓ Margin shrinks like — quadruple to halve

The margin never fixes bias — it can be precisely wrong
Precision is not accuracy

Coming Up Next: Comparing Two Treatments

In the next lesson, the same tool tackles a difference between two groups.

When one treatment beats another by some gap — is it real, or just the bounce of random assignment?