Topic 1.10 The Normal Distribution: use z-scores, the empirical (68-95-99.7) rule, and the standard normal model to find proportions and percentiles for approximately normal data.

What this topic is asking

The College Board (Topic 1.10) wants you to use the normal model for approximately bell-shaped data: standardize a value to a z-score, apply the empirical (68-95-99.7) rule, and use the standard normal distribution (table or technology) to find proportions and percentiles.

The normal model and z-scores

The z-score is the workhorse of the topic. It does two jobs: it lets you compare values from different distributions on a common scale (a z of $2$ is unusual whether it came from heights or test scores), and it is the key into the standard normal table, which is tabulated only for the standard scale.

The empirical rule

The empirical rule lets you answer many questions without a table whenever the cut-offs fall at whole numbers of standard deviations. Sketching the bell, marking $\mu$ and the $\pm 1, 2, 3\sigma$ points, and shading the region asked for is the reliable way to get these right and avoid sign slips.

Proportions and percentiles with the standard normal

For cut-offs that are not whole standard deviations, you use the standard normal table or technology to convert between a z-score and the area (proportion) to its left, which is the percentile. The table gives $P(Z < z)$, the area to the left. To find an upper-tail proportion, compute $1$ minus the table value; to find the area between two z-scores, subtract the two left areas. Going the other way, to find the value at a given percentile you look up the z-score with that left area (for example $z \approx 1.28$ for the $90$th percentile) and then unstandardise with $x = \mu + z\sigma$. This forward-and-backward fluency, value to z to area, and area to z to value, is the core computational skill the exam tests, and it returns constantly in the inference units where the same machinery describes sampling distributions.

When the normal model applies, and when it does not

A point the College Board is careful about is that the normal model is an approximation that is only appropriate when the data are themselves approximately normal, that is, roughly symmetric and bell-shaped with no strong skew or outliers. Applying z-scores and the empirical rule to a clearly skewed distribution (such as incomes) gives wrong answers, because the $68$-$95$-$99.7$ percentages simply do not hold there. So an exam answer that assumes normality should ideally note the assumption, and a question that shows a skewed histogram is often testing whether you will wrongly reach for the normal model. The honest workflow is: check (or be told) that the distribution is approximately normal, then standardize and use the table or empirical rule. Conversely, the normal model is genuinely powerful when it does apply, because a single pair of numbers, $\mu$ and $\sigma$, then determines the entire distribution and the proportion in any interval, which is why so much of later inference rests on it. Recognizing the boundary between "normal model is appropriate" and "data are too skewed for it" is the judgement Topic 1.10 is really training.

Try this

Q1. Data are normal with $\mu = 60$, $\sigma = 5$. About what percentage lie between $50$ and $70$? [1 point]

• Cue. $50$ to $70$ is $\mu \pm 2\sigma$, so by the empirical rule about $95\%$.

Q2. Find the z-score of $x = 82$ when $\mu = 70$ and $\sigma = 6$, and interpret it. [2 points]

• Cue. $z = \frac{82 - 70}{6} = 2$; the value is $2$ standard deviations above the mean (unusually high).