z-Scores and When the Normal Model Fails

Lesson 2 of 2: Fit Data to a Normal Model

In this lesson:

• Compute a z-score and find areas with technology
• Judge when a normal model does not apply

Learning Objectives for This Unit

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

1. Recognize the bell curve, fixed by mean and SD
2. Apply the empirical rule to estimate percentages
3. Compute a z-score and find areas with technology
4. Judge when a normal model is appropriate

A Question the Rule Can't Answer

Scores are normal: mean 500, SD 100. What percent scored above 650?

• 650 isn't a whole-SD boundary — it sits between 600 and 700
• The empirical rule can't handle the in-between value

We need a tool for any value. That tool is the z-score.

The z-Score: Distance in Standard Deviations

• The z-score is how many SDs a value sits from the mean

Compute the z-Score for 650

For 650 with mean 500, SD 100:

• 650 is 1.5 standard deviations above the mean
• It sits halfway between the 1-SD and 2-SD boundaries

Look Up the Area Beyond z

• Area below z = 1.5 is about 0.933 (93.3%)
• So area above is about 6.7% — scores above 650

The z-Score Is Not the Percentage

The z-score is a coordinate, not the answer.

• z = 1.5 is a position — a distance in SDs
• The percentage is the area you look up using z

Write both: "z = 1.5 → area 0.933 below → 6.7% above."

Below, Above, and Between Values

Three question types, all from the same two steps:

• Below: look up the area to the left
• Above: look up the area, subtract from 100%
• Between: look up both, subtract the smaller area

"Between 480 and 620" = area below 620 minus area below 480.

A Below-Mean Value Gives Negative z

For 420 with mean 500, SD 100:

• A negative z is not an error — it means below the mean
• 420 sits 0.8 SDs below the mean

Your Turn: Compute and Interpret z

Heights: normal, mean 170 cm, SD 8 cm.

Compute the z-score for 182 cm and say what it means.

Use the formula, then interpret.

Answer: — 182 cm is 1.5 SDs above the mean.

First, Always Check the Shape

Everything so far assumes a bell shape. Before any normal procedure, ask:

• Is the histogram roughly symmetric and bell-shaped?
• If yes, proceed; if no, stop — the model doesn't apply

This check separates correct analysis from confidently wrong answers.

What the Normal Model Requires

The model fits only symmetric, single-peaked, bell-shaped data.

• Skewed (incomes) → the symmetry is gone
• Bimodal (two groups) → no single center
• Outlier-heavy → mean and SD are distorted

For any of these, normal procedures give meaningless answers.

Three Cases Where the Bell Fails

• Skewed: predicts impossible negative incomes
• Bimodal: one bell fits neither peak
• Outlier-heavy: the curve describes almost nobody

Always Check the Shape First

The decision habit:

1. Look at the histogram or box plot
2. Ask: roughly symmetric and bell-shaped?
3. If yes, use the normal model; if no, stop

Fit a normal model only if the shape earns it.

Your Turn: Does the Bell Fit?

A histogram of household incomes: tall cluster low, long right tail.

Should you fit a normal model and use the empirical rule?

Apply the shape check, then advance.

Answer: No — it's right-skewed, not bell-shaped; the empirical rule would give impossible percentages.

Full Task: Check, Standardize, Estimate

Bolt diameters: normal, mean 10 mm, SD 0.2 mm.

1. Confirm the shape earns a normal model
2. Compute z for 10.3 mm
3. Estimate the percent larger than 10.3 mm

Do it all, then advance.

Answer: Shape OK; ; about 6.7% larger.

Key Takeaways From This Lesson

✓ The z-score is a distance in SDs from the mean

✓ The percentage is the area, not z

✓ A negative z means below the mean

Check shape first — the model needs bell data

Skewed or bimodal data breaks it

Next: two variables and scatter plots.