3.1 Measures of Center Objectives: By the end of this section, I will be able to… 1) Calculate the mean for a given data set. 2) Find the median, and describe why the median is sometimes preferable to the mean. 3) Find the mode of a data set. 4) Describe how skewness affects these measures of center.

Tuitions of Area High Schools St. Andrews St. Andrews St. Elizabeth’s St. Elizabeth’s St. Joseph's Prep, Philly St. Joseph's Prep, Philly St. Marks St. Marks St. Thomas More St. Thomas More Tatnall Tatnall Tower Hill Tower Hill Ursuline Ursuline Wilmington Christian Wilmington Christian Aquinas Aquinas Archmere Archmere Cardinal O'Hara, Phillly Cardinal O'Hara, Phillly Father Judge, Philly Father Judge, Philly Malvern Prep, Philly Malvern Prep, Philly Monsignor Bonner, Philly Monsignor Bonner, Philly Padua Padua Red Lion Red Lion Salesianum Salesianum Sanford Sanford Wilmington Friends Wilmington Friends Caravel Caravel GROUP 1 GROUP 2

Tuition Activity 1) Estimate the tuition of your school. Hint: All tuitions fall between $4,000 and $50,000. 2) Write your estimate on your slip of paper. 3) Get into the two groups and get in order from the lowest tuition to the highest tuition.

Find the Median 5) Each student on the end says “One” 6) The next two students say “Two” 7) The next two students say “Three” 8) Keep going until you find the middle person, or the median tuition. 9) Report your median tuition to the class. 10) Find the mean of your tuitions.

Three Measures of Central Tendency 1) Mean (sample mean): average of a data set 2) Median: middle number of a data set 3) Mode (modal): most frequent data point

Mean (or sample mean) Symbol is Symbol is Formula to find the mean Formula to find the mean n = total number of items n = total number of items Σ = Greek letter SIGMA and means SUM Σ = Greek letter SIGMA and means SUM

Mean (or sample mean) The mean is SENSITIVE to Extreme Values, called OUTLIERS. The mean is SENSITIVE to Extreme Values, called OUTLIERS. Why? Why? 90, 92, 91, 87, 90, 95, 96, 93, 88, 87, 46

Median Always put data items in ORDER FIRST! Always put data items in ORDER FIRST! Then find the middle term of the data set Then find the middle term of the data set If there is no exact middle, find the average of the TWO MIDDLE TERMS. If there is no exact middle, find the average of the TWO MIDDLE TERMS.

Median The median is NOT sensitive to OUTLIERS. The median is NOT sensitive to OUTLIERS. Why? Why? 90, 92, 91, 87, 90, 95, 96, 93, 88, 87, 46 46, 87, 87, 88, 90, 90, 91, 92, 93, 95, 96

Mode Order data first Order data first Find the data point that occurs the most Find the data point that occurs the most You do not have to have a mode You do not have to have a mode You can have more than one mode You can have more than one mode

Write down what you think the average tuition is for area private high schools. Write down what you think the average tuition is for area private high schools. Find the mean, median, and mode of the tuitions of area high schools. Find the mean, median, and mode of the tuitions of area high schools.tuitions

Skewness and Measures of Center For RIGHT-skewed distributions, the mean > median. For RIGHT-skewed distributions, the mean > median. For LEFT-skewed distributions, the mean < median. For LEFT-skewed distributions, the mean < median. For SYMMETRIC distributions, the mean = median (roughly) For SYMMETRIC distributions, the mean = median (roughly)

Finding values on Calculator 1) Input data into STAT → EDIT → LIST 2) Then Quit (2 nd MODE) and go to HOME SCREEN Find the MEAN 1) 2 nd STAT → MATH → 3.MEAN 2) 2 nd 1 to get L1. Hit ENTER. Find the MEDIAN 1) 2 nd STAT → MATH → 4.MEDIAN 2) 2 nd 1 to get L1. Hit ENTER