Math 111 - Week 6 Notes

Mon, Feb 21

Today we introduced the normal distribution. This is a model for histograms where the data looks roughly like a bell. The normal distribution is

  1. Symmetric
  2. The mean equals the median (denoted by \(\mu\)).
  3. There are two inflection points (steepest points on the curve).
  4. The distance from the mean to either inflection point is the standard deviation (denoted by \(\sigma\)).

Any normal distribution is completely described by the numbers \(\mu\) and \(\sigma\).

We looked at these two examples in detail:

  1. The heights of men in the USA are roughly normally distributed with mean \(\mu = 70\) inches and standard deviation \(\sigma = 3\) inches.

  2. The annual rainfall totals for Farmville, VA are roughly normal with mean \(\mu = 44\) inches and standard deviation \(\sigma = 7\) inches.

Remember: not all data is normally distributed, the number of siblings people is a quantitative variable that is skewed right. Skewed data won’t be normal.

We finished by talking about standardizing data. A standardized value (also known as a z-value) is what you get when you describe a data point by figuring out how many standard deviations it is above or below the mean. A formula to find the standardized value is \[z = \frac{x - \mu}{\sigma}.\] For example in 2020, Farmville had 61 inches of rain (which is the second highest rainfall total on record). The z-value for that number is \[\frac{61 - 44}{7} = 2.43 \text{ standard deviations above average.}\]

Wed, Feb 23

Today we talked about the 68-95-99.7 rule for the normal distribution. We did this workshop in class:

Fri, Feb 25

Today we talked about how to find percentiles on the normal distribution using software (see the software tab on my website). We did this workshop in class.