 Roughly describes where the center of the data is in the set.  2 Important measures of center: a) Mean b) Median

SamplePopulation

23, 25, 26, 29, 39, 42, 50 Show Your Work – Including the Formula that you used!

 The 50 states plus the District of Columbia have a total of 3137 counties. There are a total of 248,709,873 people in each of these counties. Find the average population per county.

Sample 1Sample 2Sample 3 20,09528,89516, ,97810, ,38416,17473,478 13,931959,27514,798 24,96030,79713,859 Do you suppose these are close to the values we’d get if we could use the population?

 We will study this further later on to see how to be able to use samples to predict the populations better.

The mean tells us how large each observation in the data set would be if the total were split equally among all the observations.

 A group of elementary school children was asked how many pets they have. Here are there responses. Find the mean and explain what it means We can look at it this way: If every child in the group had the same number of pets, each would have 5 pets.

 In the last unit, we introduced the median as an informal measure of center that described the “midpoint” of a distribution.  Now, it is time to offer an official “rule” for calculating the median.

 The median is the value in the middle!  50% of the data is above and below this value.  Steps: 1. Put numbers in order. 2. If the # of numbers is odd – median is the middle number. 3. If the # of numbers is even – median is the average of the two #’s in the middle.

Median = 12

“In the sample of New York workers, about half of the people reported traveling less than 22.5 minutes to work, and about half reported traveling more.”

Stemplot:

 The median travel time is 20 minutes.  The mean travel time is higher, 22.5 minutes.  The mean is pulled toward the right tail of this right-skewed distribution.  The median, unlike the mean, is resistant.  If the longest travel time were 600 minutes rather than 60 minutes, the mean would get higher, but the median would not change at all! Mean = 22.5 Median = 20

 The mean and median of a roughly symmetric distribution are close together.  If the distribution is exactly symmetric, the mean and median are exactly the same.  In a skewed distribution, the mean is usually farther out in the long tail than is the median.

 The mean is greatly affected by outliers – it is very sensitive to them – which means it is pulled towards the outlier.  The median is insensitive to outliers. It is often used more because it is more stable.

 Salaries for MLB players or NFL players  Scores on a test when there’s one that has NOT been made up yet.  Home prices  Personal Incomes  College tuitions

 Based only on the stemplot, would you expect the mean travel time to be less than, about the same as, or larger than the median? Why? Since the distribution is skewed to the right, we would expect the mean to be larger than the median.

 Use your calculator to find the mean and median travel time. Was your answer to Question 1 correct? The mean is minutes, which is bigger than the median of 22.5 minutes.

 Interpret your result from Question 2 in context without using the words “mean” or “average.” If we divided the travel time up evenly among all 20 people, each would have a minute travel time.

 Would the mean or median be a more appropriate summary of the center of this distribution of drive times? Justify your answer. Since the distribution is skewed, the median would be a better measure of the center.

 M F F F M F M F M M F F F M For a Population use “P”

 A disadvantage of the mean is that it can be affected by extremely high or low values.  One way to make the mean more resistant to exceptional values and still sensitive to specific data values is to do a trimmed mean.

 Order the data – delete a selected number of values from each end of the list then average the remaining values.  Trimming Percentage: The percent of values trimmed from the list.

Example a) Compute the mean for the entire sample. b) Compute a 5% trimmed mean.

Example c) Compute the median for the entire sample. d) Compute a 5% trimmed median. The median is still e) Is the trimmed mean or the original mean closer to the median? Trimmed Mean