28  More About the Normal Distribution

Builds on the normal-approximation chapter. This chapter assumes you’ve already met the normal distribution as it was developed in the normal-approximation chapter — the \(N(\mu, \sigma)\) notation, the 68-95-99.7 rule, z-scores, and finding tail probabilities and percentiles. Here we look at two extensions that didn’t fit earlier: how to check whether data actually are approximately normal, and which real-world variables the normal model fits well (and which it doesn’t).

The normal distribution is powerful precisely because so many real-world variables are approximately normal. But approximately is a word doing a lot of work — some variables really are close to normal, others are clearly not, and it matters which is which before you use the normal model to compute probabilities or run a formula-based test. This chapter gives you two tools for making that call.

28.1 Checking whether data are normal

Before using the normal distribution as a model, we should check whether the assumption of normality is reasonable. Two visual methods are commonly used, and they complement each other.

28.1.1 Method 1: Histogram with a normal curve overlay

The simplest approach is to plot a histogram of the data and overlay the best-fitting normal curve (using \(\bar{x}\) for \(\mu\) and \(s\) for \(\sigma\)). If the histogram closely matches the curve, the normal model is reasonable. If the histogram is skewed, bimodal, or has long tails the curve doesn’t capture, the model is a poor fit.

This works well when \(n\) is large enough for the histogram to reveal the true shape — typically \(n \geq 50\) or so. With smaller samples, the histogram is noisy and even genuinely normal data can look ragged.

28.1.2 Method 2: Normal probability plots (QQ plots)

A normal probability plot (also called a QQ plot, short for “quantile-quantile plot”) plots the observed data values against the values we would expect if the data were perfectly normal. If the data are approximately normal, the points will fall close to a straight line.

How to read a QQ plot:

  • Points on or near the line: The data are consistent with a normal distribution.
  • Points curving up at the right end: The distribution is right-skewed (a long upper tail).
  • Points curving down at the left end: The distribution is left-skewed (a long lower tail).
  • Points curving away from the line at both ends: The distribution has heavier tails than normal — more extreme values than the normal predicts.

Figure 28.1 shows the four patterns you will see most often.

A two-by-two grid of QQ plots. Top-left shows points following the diagonal line closely. Top-right shows points curving above the line at the right end. Bottom-left shows points curving below the line at the left end. Bottom-right shows points curving away from the line at both the lower-left and upper-right ends.
Figure 28.1: Four QQ plots illustrating common patterns: (a) approximately normal data — points fall close to the line; (b) right-skewed data — points curve upward at the right; (c) left-skewed data — points curve downward at the left; (d) heavy-tailed data — points curve away from the line at both ends.

Sample size matters. With small sample sizes (\(n < 30\)), we should expect more variability in the QQ plot even when the data truly come from a normal distribution. Deviations must be more severe to be considered evidence against normality when \(n\) is small.

A researcher collects data and creates a QQ plot. The points follow the diagonal line closely in the middle of the distribution but curve upward at the right end. What does this suggest about the shape of the data?

Show answer The QQ plot suggests the data are right-skewed. The bulk of the data are approximately normal, but there are larger values in the upper tail than the normal distribution would predict.

28.2 When to use — and not use — the normal model

The normal distribution is a good model for data that are:

  • Symmetric: The histogram is roughly mirror-image around the center.
  • Unimodal: There is a single peak.
  • Bell-shaped: The histogram has the characteristic tapered shape of the normal curve.
  • No extreme outliers: Outliers distort both the mean and standard deviation, making the normal model a poor fit.

Many variables are approximately normal:

  • Standardized test scores (SAT, ACT, GRE)
  • Heights of adults (within a sex)
  • Measurement errors in scientific instruments
  • Many biological measurements (blood pressure, cholesterol levels)
  • Averages of many small independent effects — which is why sample means behave normally by the Central Limit Theorem

Many variables are not normal:

  • Income distributions — strongly right-skewed; a few very high earners pull the tail out.
  • Waiting times — typically right-skewed, often bounded below by zero.
  • Count data — usually better modeled by the binomial or Poisson, not the normal.
  • Proportions and percentages — bounded between 0 and 1, so they can’t be strictly normal.
  • Ages — bounded below by 0 and often bounded above by natural limits, so the tails don’t behave like a normal.

When the normal model doesn’t fit, don’t force it. For skewed data, consider a transformation (e.g., log for right-skewed positive data), a different model (binomial for counts of successes, Poisson for rare event counts), or a simulation-based method like the bootstrap or a randomization test. Those simulation methods (developed in the randomization-tests and bootstrap-CI chapters) don’t require normality — they let the data speak for themselves.

A hospital measures the time patients wait to be seen in the emergency room. The data range from 3 minutes to 6 hours, with most patients waiting under 45 minutes but a long right tail. A researcher wants to run a one-sample \(t\)-test to see if the mean wait time exceeds 30 minutes. Is a \(t\)-test appropriate?

The data are strongly right-skewed and bounded below by zero, so they are clearly not normal. A one-sample \(t\)-test may still be reasonable if the sample size is large — the CLT says the sampling distribution of \(\bar{x}\) will be approximately normal even when the population is skewed. But with a small sample (say, \(n < 30\)), the \(t\)-test could give misleading results. A safer approach is a bootstrap confidence interval for the mean wait time, which doesn’t rely on normality of the population.

See it in action. Open the Descriptive Statistics Explorer on any bundled dataset — switch to the histogram view to eyeball symmetry and unimodality, then check the summary statistics for extreme outliers. For a formal look, StatLens’s Normal Distribution Explorer’s overlay mode fits a normal curve to your data so you can see where the fit is good and where it fails.

28.3 Chapter review

28.3.1 Summary

  • Before applying the normal model, check whether the data are approximately normal. Two visual tools help: a histogram with a normal-curve overlay, and a QQ plot (points near the diagonal = approximately normal; systematic curvature = departure from normality).
  • The four QQ-plot patterns worth recognizing: normal (straight), right-skewed (curves up at the right), left-skewed (curves down at the left), and heavy-tailed (curves away at both ends).
  • The normal model fits symmetric, unimodal, bell-shaped variables without extreme outliers. Standardized test scores, adult heights, and measurement errors are canonical examples.
  • The normal model does not fit strongly skewed data (income, waiting times), bounded data (proportions, ages), or count data (better handled by the binomial or Poisson).
  • When the normal model doesn’t fit, use a simulation-based method (bootstrap CI, randomization test) or a different distribution family appropriate for the data.

28.3.2 Key terms

QQ plot (normal probability plot), normality assessment, histogram-with-normal-overlay, symmetric / unimodal / bell-shaped, heavy-tailed distribution, right-skewed / left-skewed, bootstrap alternative to normal-based inference.

28.4 Exercises

Answers to odd-numbered exercises are provided in the Exercise Solutions appendix at the back of the book.

  1. Reading a QQ plot. For each described pattern below, name the shape of the underlying data distribution and briefly explain how you can tell.

  1. Points fall very close to the reference line across the whole range.

  2. Points hug the line in the middle but curve upward away from the line at the right end.

  3. Points hug the line in the middle but curve downward away from the line at the left end.

  4. Points fall below the line at the left end and above the line at the right end.

Show answer

  1. Approximately normal — a straight-line QQ plot is exactly what we hope to see; the observed quantiles match the normal quantiles at every point.

  2. Right-skewed — the upper tail of the data extends farther than a normal distribution would predict, so the largest observed values are bigger than the normal quantiles → points rise above the line.

  3. Left-skewed — the lower tail extends farther than normal, so the smallest observed values are smaller than the normal quantiles → points fall below the line on the left.

  4. Heavy-tailed — both extremes are more extreme than a normal would predict. Common with a \(t\)-distribution with low df or with data that have occasional extreme outliers on both sides.

  1. Which of these variables would you model as normal? For each variable, decide whether the normal distribution is a reasonable model for individual observations. Answer yes, no, or depends, and explain in one sentence.

  1. Adult male heights in a large city.

  2. Household incomes in the United States.

  3. The number of car accidents at a given intersection per day.

  4. The average of many random draws from any distribution.

  5. The proportion of a small sample that responds “yes” to a survey question.

  6. Systolic blood pressure of healthy adults.

Show answer

  1. Yes — heights within a sex are approximately normal (well-documented; symmetric, unimodal, bell-shaped, no natural boundary issues in the observed range).

  2. No — income is strongly right-skewed with a long upper tail from high earners; use a log-transformation or a different family.

  3. No — count data are usually modeled by the Poisson distribution, not the normal. (For high rates the normal can be a rough approximation, but not the primary model.)

  4. Yes — by the Central Limit Theorem, the sampling distribution of the mean is approximately normal for large enough \(n\), regardless of the population.

  5. Depends — the sampling distribution of a proportion is approximately normal only when the success-failure condition holds (\(np \ge 10\) and \(n(1-p) \ge 10\)). Small \(n\) or extreme \(p\) makes the normal a poor model.

  6. Yes — blood pressure in healthy adults is roughly symmetric, unimodal, and bell-shaped (a standard example in medical statistics).

  1. Small-sample QQ plot interpretation. A researcher has a sample of \(n = 15\) measurements. The QQ plot shows some scatter around the reference line — the points don’t fall perfectly on it, but there’s no clear systematic curve either.

  1. Is this stronger evidence for or against the data being approximately normal? Explain.

  2. If the same amount of scatter appeared in a QQ plot with \(n = 500\), would your conclusion change? Why?

  3. A colleague says “the points don’t fall exactly on the line, so we can’t use a \(t\)-test.” Is this reasoning sound? Explain.

Show answer

  1. Consistent with normality. With small \(n\), we expect some scatter around the line even when the data truly are normal — random sampling variability alone produces wobble. Absence of a systematic curve (S-shape, monotonic curve at one end) is the signal that matters, not the amount of wobble.

  2. Yes. With \(n = 500\), random sampling variability is much smaller and the plot should look much cleaner. Persistent scatter at \(n = 500\) would indicate the data actually depart from normal in some way (mixture distribution, heteroscedasticity, etc.).

  3. No — this misunderstands what we’re checking. The \(t\)-test is robust to moderate departures from normality, especially with modest-to-large \(n\). Perfect linearity of the QQ plot is not required. The right questions are: is the departure severe (clear skew, heavy tails)? and is \(n\) small enough that departures matter? Modest wobble at any \(n\) is not a reason to abandon the \(t\)-test.

  1. Pick a method. For each scenario, decide whether a one-sample \(t\)-test is reasonable, or whether a bootstrap CI (or different approach) would be safer. Explain in one sentence.

  1. You want to estimate the mean commute time in a city. A histogram of your \(n = 40\) commute-time observations is roughly symmetric with no outliers.

  2. You want to estimate the mean insurance claim amount. Your \(n = 25\) observations are strongly right-skewed with three extreme outliers.

  3. You want to estimate the mean systolic blood pressure of adult women. Your \(n = 200\) observations look approximately normal.

  4. You want to test whether the mean wait time at an emergency room exceeds 30 minutes. Your \(n = 8\) observations are strongly right-skewed and bounded below by 0.

Show answer

  1. \(t\)-test reasonable. With \(n = 40\) and symmetric-no-outliers, the CLT plus \(t\)’s robustness handles the mild departures from perfect normality.

  2. Bootstrap safer. \(n = 25\) is not big enough for the CLT to fully rescue strongly skewed + outlier-driven data; \(\bar{x}\) and \(s\) are both distorted, so the \(t\)-test standard error is unreliable. A percentile bootstrap CI makes fewer assumptions.

  3. \(t\)-test reasonable. Large \(n\) + approximately normal — this is exactly the setting the \(t\)-test was designed for.

  4. Bootstrap safer. \(n = 8\) is too small for the CLT with strong right skew and a hard boundary at zero. The sampling distribution of \(\bar{x}\) won’t be close to normal here, and the \(t\)-test could give misleading results. A bootstrap CI is more trustworthy.

  1. Two normality checks, one dataset. You have a dataset of daily rainfall (inches) for 300 days at a weather station. You plot both a histogram with a fitted normal overlay and a QQ plot. The histogram is strongly right-skewed with many days near 0 and a few days with heavy rain; the QQ plot’s points curve sharply upward at the right end.

  1. Do the two plots agree or disagree on normality?

  2. Suggest one transformation of the data that might make the transformed values more nearly normal, and explain why it would help.

  3. Suppose you compute a 95% one-sample \(t\) confidence interval for the mean daily rainfall from these 300 observations. Given the CLT and \(n = 300\), do you trust it? What’s the caveat?

Show answer

  1. Agree. Both diagnostics point to strong right skew. The histogram makes the shape visible directly; the QQ plot’s upward curve at the right end confirms an over-long upper tail versus a normal.

  2. Log transformation (\(\log(x + \varepsilon)\) or \(\sqrt{x}\) for zero-heavy data) is a natural choice for right-skewed positive data — it compresses the long upper tail and expands the crowded region near zero, often producing an approximately symmetric transformed distribution.

  3. Cautiously trust it, with the caveat that “the mean” may not be the interesting quantity. With \(n = 300\) the CLT makes the sampling distribution of \(\bar{x}\) approximately normal even from skewed data, so the \(t\) CI’s coverage is roughly right. The deeper caveat: for very skewed data the mean is not always the best summary — the median or another percentile may answer the substantive question better.