Chapter 4 Lesson 4.1 Numerical Methods for Describing Data 4.1: Describing the Center of a Data Set

60% of the sample was satisfied with their cell phone service. What values are used to describe categorical data? Suppose that each person in a sample of 15 cell phone users is asked if he or she is satisfied with the cell phone service. Here are the responses: YNYYYNNYYYNYYYNNYY NYYYNNNYYYNN What would be the possible responses? Find the sample proportion of the people who answered “ yes ” : Pronounced p-hat The population proportion is denoted by the letter p.

Population Parameter - Fixed value about a population Typical unknown Suppose we want to know the MEAN length of all the fish in Lake Lewisville... Is this a value that is known? Can we find it out? At any given point in time, how many values are there for the mean length of fish in the lake?

Sample Statistic - calculatedValue calculated from a sample Suppose we want to know the MEAN length of all the fish in Lake Lewisville. What can we do to estimate this unknown population characteristic?

Measures of Central Tendency Median - the middle value of the data; it divides the observations in half To find: list the observations in numerical order Where n = sample size

Suppose we catch a sample of 5 fish from the lake. The lengths of the fish (in inches) are listed below. Find the median length of fish The numbers are in order & n is odd – so find the middle observation. The median length of fish is 5 inches.

Suppose we caught a sample of 6 fish from the lake. The median length is … The numbers are in order & n is even – so find the middle two observations. The median length is 5.5 inches. Now, average these two values. 5.5

Measures of Central Tendency Mean is the arithmetic average. –Use  to represent a population mean –Use x to represent a sample mean Formula: Population Parameter Sample Statistic  is the lower case Greek letter mu

Suppose we caught a sample of 6 fish from the lake. Find the mean length of the fish To find the mean length of fish - add the observations and divide by n.

What happens to the median & mean if the length of 10 inches was 15 inches? The median is The mean is What happened?

What happens to the median & mean if the 15 inches was 20? The median is The mean is What happened?

Some statistics that are not affected by extreme values... Is the median affected by extreme values? Is the mean affected by extreme values ? NO YES

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.) Mean = Median = Calculate the mean and median. 6.5 Look at the placement of the mean and median in this symmetrical distribution.

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width 1.) Mean = Median = Calculate the mean and median. Look at the placement of the mean and median in this skewed distribution

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.) Mean = Median = Calculate the mean and median. Look at the placement of the mean and median in this skewed distribution

Recap: In a symmetrical distribution, the mean and median are equal. In a skewed distribution, the mean is pulled in the direction of the skewness. In a symmetrical distribution, you should report the mean! In a skewed distribution, the median should be reported as the measure of center!

Summary p = Population Proportion (parameter) = Sample Proportion (statistic) = Population Mean (parameter) = Sample Mean (statistic) When describing center… Measure of Center Symmetric  Mean Skewed  Median

Homework Pg.182: #4.1, 4.6a, 4.7, 4.10a, 4.14