Algebra 2
Statistics and probability
Normal distribution and fitting data with mean & SD
20 practice questions
0 video lessons
Theory + worked examples
Theory
The normal distribution is a symmetric bell curve described by its mean \(\mu\) and standard deviation \(\sigma\). The 68-95-99.7 rule gives:
- \(68\%\) of data within \(1\sigma\) of the mean,
- \(95\%\) within \(2\sigma\),
- \(99.7\%\) within \(3\sigma\).
A \(z\)-score \(z=\dfrac{x-\mu}{\sigma}\) counts standard deviations from the mean.
The bell curve and the 68-95-99.7 rule.
The empirical rule and z-scores.
The z-score:
\[z=\dfrac{x-\mu}{\sigma}\]
The curve is symmetric, so each tail beyond a \(z\) is half the outside.
How to use the normal model
- Identify the mean and standard deviation.
- Count standard deviations from the mean.
- Apply the 68-95-99.7 percentages.
- Use \(z\)-scores for other values.
Example 1 — One standard deviation
Test scores are normal with mean \(100\), SD \(15\). Between what values are \(68\%\)?
Solution
Within \(1\sigma\) of the mean.
| \(100\pm15\) | \(=\) | \(85\ \text{to}\ 115\) |
Example 2 — Two standard deviations
For the same scores, between what values are \(95\%\)?
Solution
Within \(2\sigma\).
| \(100\pm30\) | \(=\) | \(70\ \text{to}\ 130\) |
Example 3 — Tail percentage
What percent of scores are above \(115\)?
Solution
\(115\) is \(1\sigma\) above; \(68\%\) are within, leaving \(32\%\) split.
| \(\dfrac{100-68}{2}\) | \(=\) | \(16\%\) |
Example 4 — z-score
Find the \(z\)-score of \(130\) for mean \(100\), SD \(15\).
Solution
Use \(z=\dfrac{x-\mu}{\sigma}\).
| \(z\) | \(=\) | \(\dfrac{130-100}{15}=2\) |
Common pitfalls
The percentages are within \(\pm\) the SD, split evenly on each side.
A tail beyond \(1\sigma\) is \(16\%\), not \(32\%\).
The \(z\)-score subtracts the mean before dividing.
Frequently asked questions
What is the 68-95-99.7 rule?
Those percents of data lie within 1, 2, and 3 SDs of the mean.
What is a z-score?
The number of standard deviations a value is from the mean.
What percent is above one standard deviation?
About \(16\%\).
Is the normal curve symmetric?
Yes — it is symmetric about the mean.
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