4.3 Normal Distributions and Z-Scores

Why This Section Matters

Normal distributions model many real measurement variables — standardized test scores, errors in repeated measurement, and aggregated biological data. On Praxis Mathematics (5165), you should read a normal curve description, apply the empirical rule, and compute or interpret z-scores without relying on memorized tables alone.

Features of the Normal Model

A normal distribution is:

• Symmetric about the mean
• Bell-shaped with tapering tails
• Fully described by mean μ and standard deviation σ

The mean, median, and mode coincide at the center. Total area under the curve equals 1, representing 100% of the population or model.

The Empirical Rule (68–95–99.7)

For approximately normal data:

So about 32% lie more than one standard deviation from the mean (16% in each tail), and only about 2.5% lie above μ + 2σ in the upper tail alone.

These are approximate rules — exact normal probabilities need technology — but Praxis selected-response items are usually built around clean empirical-rule fractions.

Z-Scores: Standardized Position

The z-score of a value x is:

z = (x − μ) / σ

Interpretation:

• z = 0 → at the mean
• z = 1 → one standard deviation above the mean
• z = −2 → two standard deviations below the mean

Z-scores let you compare values from different distributions. A score of 82 in one class may be stronger or weaker than an 82 in another class depending on each distribution's mean and standard deviation.

Worked Example: Empirical Rule

Test scores are approximately normal with mean 72 and standard deviation 6.

Between 66 and 78? Those endpoints are μ − σ and μ + σ, so about 68% of scores fall in that interval.

Above 84? 84 = 72 + 2(6) = μ + 2σ. About 95% are within two standard deviations, so about 5% are outside that band, split between both tails. Only about 2.5% are above μ + 2σ.

Worked Example: Computing and Comparing Z-Scores

Class A: mean 75, SD 5. Maria scores 85.

z_A = (85 − 75) / 5 = 2.0 (two standard deviations above the class mean)

District exam: mean 70, SD 10. Maria scores 85.

z_D = (85 − 70) / 10 = 1.5

Maria's raw score is the same (85), but her relative standing is stronger in Class A (z = 2.0) than on the district exam (z = 1.5). Praxis items love this comparison — never assume equal raw scores mean equal percentile rank.

Worked Example: Finding a Value from a Z-Score

If adult heights are normal with μ = 170 cm and σ = 8 cm, what height corresponds to z = −1.25?

x = μ + zσ = 170 + (−1.25)(8) = 170 − 10 = 160 cm.

Normal Curves and Teaching Context

When a student says "nobody in our data set is exactly at the mean, so this cannot be normal," they confuse model with sample. A normal model describes population behavior; real samples can look slightly skewed yet still be reasonably modeled as normal.

Conversely, if a histogram is heavily skewed, applying empirical-rule percentages without checking shape is a reasoning error — a common distractor on teaching items.

Calculator and Graphing Support

On the computer-delivered test, the provided graphing calculator can evaluate normal probabilities when needed. Still estimate with the empirical rule first; many answers are designed so you can eliminate choices quickly when you know, for example, that only about 16% of data lie above μ + σ.

Praxis Traps

1. Forgetting which tail — "above μ + 2σ" is about 2.5%, not 5%.
2. Using range instead of σ when standardizing.
3. Comparing raw scores across different means and standard deviations without z-scores.
4. Applying the empirical rule to clearly non-normal, highly skewed data without question.

Worked Example: Percentile Language

If a student scores at z = 1.0 on a roughly normal exam, they are one standard deviation above the mean — higher than about 84% of scorers (50% below the mean plus about 34% between the mean and z = 1). Praxis items may phrase this as "above average" or "unusually high" depending on context; translate the words back to position on the normal curve before eliminating choices.

Connecting Normal Models to Regression

Many regression residuals are modeled as roughly normal when sample size is large. You will not derive that theory on Praxis 5165, but recognizing bell-shaped measurement error helps you understand why normal curves appear alongside scatterplot and regression topics in the same blueprint domain.

Worked Example: Finding a Percent from z

Scores on a statewide exam are approximately normal with μ = 500 and σ = 100. What percent of students scored below 600?

z = (600 − 500)/100 = 1.0. About 68% fall between 400 and 600, so 16% lie in the upper tail above 600 and 16% in the lower tail below 400. The area below 600 is 50% (below mean) + 34% (between mean and z = 1) ≈ 84%.

Praxis distractors often swap 16% and 84% — always sketch whether you want a tail or a cumulative region.