Normal distribution and fitting data with mean & SD
The Normal Distribution
The Normal Distribution is a topic in Statistics & Probability in the Common Core State Standards. It is aligned to Standard S-ID.4, which requires students to use the mean and standard deviation of a data set to fit it to a normal distribution and estimate population percentages.
The normal distribution is a symmetric bell curve; the 68-95-99.7 rule gives those percents within 1, 2, and 3 standard deviations of the mean.
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 \(3\sigma\).
The z-score:
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.
Within \(1\sigma\) of the mean.
| \(100\pm15\) | \(=\) | \(85\ \text{to}\ 115\) |
Within \(2\sigma\).
| \(100\pm30\) | \(=\) | \(70\ \text{to}\ 130\) |
\(115\) is \(1\sigma\) above; \(68\%\) are within, leaving \(32\%\) split.
| \(\dfrac{100-68}{2}\) | \(=\) | \(16\%\) |
Use \(z=\dfrac{x-\mu}{\sigma}\).
| \(z\) | \(=\) | \(\dfrac{130-100}{15}=2\) |
Common pitfalls
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.