• Use counts and proportions to describe categorical data (e.g. number orange)
• For continuous variables, usually use the sample mean (e.g. years of service in senate)

Example Years of service in senate

- What is the standard deviation of the years of service of the 10 senators in your sample?

- Is this close to the population standard deviation?

- What is the standard deviation of the averages we obtained in our many samples?

- Is this close to the population standard deviation?

- What happens to the shape, center, and variability if we increase the sample size?

• Case 1: Population is normal
• Case 2: Central Limit Theorem - For any population, if the sample size is large enough, the sampling distribution of the sample mean

• Center: Expected value, E() = m
• Spread: Standard deviation, SD()=
• Shape: When will follow a Normal distribution?

1) The population is Normal

2) The population has any distribution but n is large (Central Limit Theorem)

• Averages are less variable than individual observations
• Averages are more normal that individual observations

Example Why do people use stock "portfolios" instead of investing in a single stock?

Example A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a Normal distribution with mean m=298 ml and standard deviation s=3 ml.

- What is the probability that an individual bottle contains less than 295 ml?

- What is the probability that the mean contents of the bottles in a six-pack is less than 295 ml?