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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 Mean Population Mean “x bar” μ (pronounced mu) n = sample size N = 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:**

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) = 27

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**MEAN calculations, cont’d**

Find the mean of the frequency distribution: Height (in inches) Frequency 4 63 – 65 5 66 – 48 8 69 – 71 1

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**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

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**MEAN calculations, cont’d**

Height (in inches) Frequency Midpt (x) (x • f) 4 61 244 63 – 65 5 64 320 66 – 48 8 67 536 69 – 71 1 70 ______ 18 ___________ 1170

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**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…..

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**MEAN calculations cont’d:**

…there are some generalities based on shape Symmetric Uniform Mean (as well as median) will be at the center.

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**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**

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

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**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

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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 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? 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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**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

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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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1 M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 2-4 Measures of Center.

1 M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 2-4 Measures of Center.

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