🔬 Grade 12 · Lesson 12 of 13
Bell Curve: The Normal Distribution
Heights, test scores, measurement errors — pile up enough of them and they settle into the same symmetric bell, controlled by just two numbers.
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The big idea: The normal distribution is the most important shape in statistics: a symmetric bell whose position and width are set entirely by the <strong>mean μ</strong> and the <strong>standard deviation σ</strong>. Once you know those two numbers, the 68–95–99.7 rule tells you how data spreads, and the z-score lets you compare any value against any normal curve on one universal scale.
The shape everything drifts toward
Measure the heights of thousands of adults, the errors in thousands of lab readings, or the scores on a big test, and a familiar shape appears again and again: a single hump, symmetric, tall in the middle and thinning smoothly toward both tails. This is the normal distribution, or bell curve.
What makes it so useful is its economy. The entire curve — where it sits and how spread out it is — is pinned down by only two numbers.
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Two numbers say it all
The mean μ locates the center of the bell — and because the curve is symmetric, the mean, median, and mode all sit there together. The standard deviation σ sets the width: a small σ gives a tall, narrow peak, a large σ a short, wide spread. Change μ and the bell slides; change σ and it stretches or squeezes.
\[ f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\;e^{-\tfrac{(x-\mu)^2}{2\sigma^2}} \]
The normal density. You rarely evaluate it by hand — but notice it depends only on μ and σ, and the total area beneath it is exactly 1.
The 68–95–99.7 rule
Because every normal curve has the same shape, the fraction of data within a given number of standard deviations is always the same. This is the empirical rule: about 68% of values fall within 1σ of the mean, about 95% within 2σ, and about 99.7% within 3σ.
The rule works in reverse too. Beyond 2σ lies only about 5% of the data, split into 2.5% in each tail — so a value more than two standard deviations above the mean is genuinely uncommon.
🎮 Bell Curve LIVE
Change the mean and standard deviation and shade the 68-95-99.7 regions.
📝 Worked example: Adult heights are roughly normal with mean μ = 170 cm and standard deviation σ = 10 cm. What fraction of people are between 160 cm and 180 cm?
- 160 cm is one σ below the mean (170 − 10) and 180 cm is one σ above (170 + 10).
- So the interval is exactly μ ± 1σ.
- By the empirical rule, about 68% of a normal distribution lies within one standard deviation of the mean.
✓ About <strong>68%</strong> of people are between 160 cm and 180 cm tall.
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The z-score is a universal ruler
To compare a value against its own curve, measure how many standard deviations it sits from the mean. That count is the z-score. A z-score of +2 means “two standard deviations above average,” whether we are talking about heights, test scores, or reaction times — it turns every normal curve into one shared scale.
\[ z = \dfrac{x - \mu}{\sigma} \]
The z-score. Subtract the mean to center, divide by σ to rescale — the result follows the standard normal N(0, 1).
📝 Worked example: A test is normal with mean μ = 500 and standard deviation σ = 100. Roughly what percent of scores are above 700?
- Find the z-score: \( z = \dfrac{700 - 500}{100} = 2 \). The score 700 is 2σ above the mean.
- By the empirical rule, about 95% of scores lie within 2σ of the mean, leaving about 5% outside.
- That 5% splits evenly between the two tails, so about 2.5% lies above +2σ.
✓ About <strong>2.5%</strong> of scores are above 700.
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Normal is a model, not a law
Many real datasets are approximately normal, but not all are. Strongly skewed data (like incomes) or data with hard limits do not fit the bell, and the 68–95–99.7 rule then gives poor estimates. Also, standard deviation is a spread, never negative — if you compute a “negative σ,” you have made a sign error.
Check your understanding
1. In a normal distribution, about what percent of the data lies within one standard deviation of the mean?
The empirical rule: roughly 68% within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.
2. A value x equals μ + 2σ. What is its z-score?
z = (x − μ)/σ = (μ + 2σ − μ)/σ = 2σ/σ = 2. The value sits two standard deviations above the mean.
3. For a normal distribution, the mean, median, and mode are…
Because the curve is perfectly symmetric about μ, the mean, median, and mode all coincide at the center.
4. Roughly what percent of a normal distribution lies MORE than two standard deviations from the mean (both tails combined)?
About 95% lies within 2σ, so about 5% lies beyond it — roughly 2.5% in each tail.
5. The standard normal distribution has which mean and standard deviation?
Standardizing with z = (x − μ)/σ produces the standard normal N(0, 1): mean 0, standard deviation 1.
✅ Key takeaways
- The normal distribution is a symmetric bell curve fixed entirely by its mean μ and standard deviation σ.
- The mean sets the center (where mean, median, and mode coincide); σ sets the width.
- The 68–95–99.7 rule: about 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations.
- A z-score z = (x − μ)/σ measures how many standard deviations a value is from the mean.
- Standardizing turns any normal curve into the standard normal N(0, 1), giving one universal comparison scale.