1. The Normal Distribution

A normal distribution is symmetric and bell-shaped, with a single peak at its center. The mean, median, and mode are all equal at that peak. The standard deviation sets the width: a larger standard deviation gives a wider, flatter bell.

When to expect a normal shape

1. Most values cluster near the mean, giving one well-defined peak.
2. Values spread evenly around the mean, so the distribution is symmetric.
3. Values far from the mean are rare, so the tails thin out.
4. Many small factors (genes, environment, chance) shape the variable. Example: foot length is close to normal; scores on an easy test are not, because they pile up near 100 and are left-skewed.

2. The 68–95–99.7 Rule

In a normal distribution:

• about 68% of values lie within 1 standard deviation of the mean;
• about 95% lie within 2 standard deviations;
• about 99.7% lie within 3 standard deviations.

The pieces of the bell

Split each band in half, since the curve is symmetric:

• mean to 1 SD: 68 ÷ 2 = 34%
• 1 SD to 2 SD: (95 − 68) ÷ 2 = 13.5%
• 2 SD to 3 SD: (99.7 − 95) ÷ 2 = 2.35%
• beyond 3 SD on one side: (100 − 99.7) ÷ 2 = 0.15%
• each half of the curve: 50%

Method: draw the bell; label the mean, then mean ± 1, 2 and 3 SD; write 34, 13.5, 2.35 and 0.15 in the regions; add the regions the question asks for.

Example: legal quarters have a mean weight of 5.67 g with SD 0.07 g. A vending machine accepts only 5.53 to 5.81 g, which is mean ± 2 SD. It accepts about 95% and rejects about 5% of legal quarters.

3. Standard Scores (z-scores)

The standard score is the number of standard deviations a value lies above or below the mean:

• Positive z: above the mean. Negative z: below the mean. z = 0: at the mean.
• Example (IQ, mean 100, SD 15): 85 → z = (85 − 100)/15 = −1.00; 100 → z = 0; 125 → z = 25/15 ≈ 1.67.
• Working backward: data value = mean + z × standard deviation.

4. Percentiles

The percentile of a data value is the percentage of all values that are less than it. Find it from the z-score with the standard score table.

• z = −1 → 15.87th; z = 0 → 50th; z = 1 → 84.13th; z = 2 → 97.72nd.
• Percent above a value = 100% − its percentile. (Above z = −1: 100 − 15.87 = 84.13%.)
• Percent between two values = the larger percentile − the smaller one.

Example: cholesterol (mean 178, SD 41)

1. Level 190: z = (190 − 178)/41 ≈ 0.29, which the table puts at about the 62nd percentile.
2. 90th percentile: the table gives z ≈ 1.3 (90.32%), so the level is 178 + 1.3 × 41 ≈ 231.

5. Excel Online

• =STANDARDIZE(x, mean, sd) gives the z-score.
• =NORM.DIST(x, mean, sd, TRUE) gives the fraction of values below x (multiply by 100 for the percentile).
• =NORM.INV(p, mean, sd) gives the value at a percentile (p as a decimal, such as 0.90).

Excel and the table can differ slightly in the second decimal place, because the table rounds z.

Resources

Lesson: andrewhvolk.github.io/courses/math114-module10.html  ·  Slides: andrewhvolk.github.io/pdfs/math114-module10-slides-2026-10-26.pdf  ·  Practice: Normal curve explorer and z-score calculator  ·  Desmos Scientific and Excel Online