Key takeaway: A normal distribution is a symmetric, bell-shaped distribution fully determined by its mean μ and standard deviation σ, with most values clustered near the center.
Grade 11 – Algebra 2S-ID.A.4Statistics and Data
Definition of Normal Distribution
A normal distribution describes data that pile up around a typical value and thin out evenly on both sides. Heights, measurement errors and manufacturing sizes often follow this pattern. Its graph is the familiar bell curve.
Two numbers pin the curve down. The mean \(\mu\) sets where the center sits, and the standard deviation \(\sigma\) sets how wide the bell spreads. The curve is perfectly symmetric about the mean, so the mean, median and mode coincide. The total area under it is 1, and areas stand for proportions of the data.
Formula and notation
A variable \(X\) that follows this pattern is written
\[X\sim N(\mu,\sigma),\]
where \(\mu\) is the mean and \(\sigma\) the standard deviation. Any value can be converted to a standard score with \(z=\frac{x-\mu}{\sigma}\), which places it on the standard bell with mean 0 and spread 1.
Key facts
- The curve is symmetric, so exactly half of the area lies on each side of the mean.
- A larger σ gives a wider, flatter bell; a smaller σ gives a taller, narrower one.
- The curve never touches the horizontal axis, though it comes extremely close.
- Inflection points, where the bend changes direction, sit one σ from the mean.
Normal Distribution example
Battery life for a tablet is normally distributed with mean 410 minutes and standard deviation 25 minutes. Between which two values do the middle stretch of one standard deviation on each side fall, and what share of tablets last longer than 410 minutes?
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Go one \(\sigma\) below and above the mean: \(410-25=385\) and \(410+25=435\).
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The curve is symmetric about 410, so half the tablets last longer than the mean.
The band is 385 to 435 minutes, and exactly 50% of tablets last longer than 410 minutes.
Common mistake
Watch out
Tessa says a normal distribution with a standard deviation of 25 has all of its values between 385 and 435.
Why it is wrong
The curve has no hard edges. About 68% of the values fall within one standard deviation, and the rest lie farther out in both tails. Some tablets last under 385 minutes and some over 435.
Related terms and lessons
- Algebra 2 · Data, Normal Distributions and Statistical InferenceLessonWorksheetQuiz
- AP Statistics · Binomial and Normal DistributionsLessonWorksheetQuiz
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