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

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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

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

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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

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Let’s try some less obvious MEAN calculations: 1)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: business majors: 81 9(85) + 5(90) + 13(81) = 84 27

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MEAN calculations, cont’d 2)Find the mean of the frequency distribution: Height (in inches)Frequency – – – 711

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Finding the MEAN of a Frequency Distribution In WordsIn symbols 1.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

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Height (in inches)FrequencyMidpt (x)(x f) – – – MEAN calculations, cont’d ___________ 1170 ______ 18

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

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

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

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

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Weighted Mean 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 The mean of a data set whose entries have varying weights.

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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?

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Weighted Mean, cont’d 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? x = 90.2x = 19.1 X = 95.5 Rachel needs to get a 95.5% on the final to earn a 90% in the class.

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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?

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MEDIANS…. There’s an easier way to do it! For large datasets: 1)Arrange in order 2)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

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

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

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