Section2.3 – Measures of Central Tendency SWBAT: Identify and analyze patterns of distributions using shape, center and spread.

Measures of Central Tendency (MCT) A value that represents a typical, or central, entry of a data set. The most commonly used are: Mean: average Median: middle value of an ordered data set Mode: the data item occurring most frequently

MEAN In statistics, mean is designated differently depending on if the dataset is a sample or population: Sample Mean Population Mean “x bar” μ (pronounced mu) n = sample size N = population size

Mean Sensitive to the influence of outliers When extreme outliers are present, median is a better measure of central tendency EX) Calculate mean and median for: 45, 83, 90, 79, 81, 83, 90, 88 and for: 83, 90, 79, 81, 83, 90, 88

Let’s try some less obvious MEAN calculations: The mean scores for a statistics course (by major) are given: What is the mean score for the class? 9 engineering majors: 85 5 math majors: 90 13 business majors: 81 9(85) + 5(90) + 13(81) = 84 27

MEAN calculations, cont’d Find the mean of the frequency distribution: Height (in inches) Frequency 60 - 62 4 63 – 65 5 66 – 48 8 69 – 71 1

Finding the MEAN of a Frequency Distribution MEAN calculations, cont’d Finding the MEAN of a Frequency Distribution In Words In symbols Find the midpoint of each class. 2. Find the sum of the products of the midpoints and the frequencies. ∑(x • f) 3. Find the sum of the frequencies n = ∑ f 4. Find the mean of the frequency distribution

MEAN calculations, cont’d Height (in inches) Frequency Midpt (x) (x • f) 60 - 62 4 61 244 63 – 65 5 64 320 66 – 48 8 67 536 69 – 71 1 70 ______ 18 ___________ 1170

More bars on the left of the peak More bars on the right of the peak MEAN calculations, cont’d: 3) Find the mean of the histograms: More bars on the left of the peak More bars on the right of the peak Skewed Right Skewed Left Symmetric Since a histogram is just a graphical representation of a distribution, you use the same process as finding the mean of the distribution…. but…..

MEAN calculations cont’d: …there are some generalities based on shape Symmetric Uniform Mean (as well as median) will be at the center.

MEAN calculations, cont’d Skewed histogram means…. Skewed Left Skewed Right Mean Mode Mode Mean Median Median **Mean always fall in the direction the distribution is skewed**

YOU TRY…. Find the mean of the histogram below:

Sample Weighted Mean Pop. Weighted Mean The mean of a data set whose entries have varying weights. Sample Weighted Mean Pop. Weighted Mean w = weight of each entry --weights may not sum to 100% --if weights sum to 100%, then ∑w = 1

Weighted Mean, cont’d Ex 1) A class is graded based on weighted mean as follows: Homework: 20% Quizzes: 35% Tests: 45% Let’s say scores are 95 on homework, 82 on quizzes and 79 on tests. What is the weighted mean? What if there were no test scores yet?

Weighted Mean, cont’d 70.9 + .2x = 90 .2x = 19.1 X = 95.5 Ex 2) Rachel is taking a class in which her grade is determined as follows: 50% from her test mean 15% from her midterm 20% from her final 15% from her homework Her scores are 86 (tests), 96 (midterm), and 100 (homework). What does Rachel need to get on her final exam to receive a 90% in class? 70.9 + .2x = 90 .2x = 19.1 X = 95.5   Rachel needs to get a 95.5% on the final to earn a 90% in the class.

Weighted Mean, cont’d Ex 3) For the month of April, a checking account has a balance of $523 for 24 days, $2415 for 2 days and $250 for 4 days. What is the account’s mean daily balance for April?

For large datasets: MEDIANS…. There’s an easier way to do it! Arrange in order The location of the median is found by counting data items up from the bottom of the set. Ex) Ages at concert: 24, 27, 19, 21, 18, 23, 21, 20, 19, 33, 30, 29, 21 18, 24, 26, 38, 19, 35, 34, 33, 30, 21, 27, 30 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 23, 24, 24, 26, 27, 27, 29, 30, 30, 30, 33, 33, 34, 35, 38

MEDIAN calculations, cont’d: Works for stem-and-leaf plots too:

MEDIAN calculations, cont’d: Works for histograms too: