Presentation on theme: "Section2.3 – Measures of Central Tendency"— Presentation transcript:
1Section2.3 – Measures of Central Tendency SWBAT:Identify and analyze patterns of distributions using shape, center and spread.
2Measures of Central Tendency (MCT) A value that represents a typical, or central, entry of a data set. The most commonly used are:Mean: averageMedian: middle value of an ordered data setMode: the data item occurring most frequently
3MEANIn 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
4Mean Sensitive to the influence of outliers When extreme outliers are present, median is a better measure of central tendencyEX) Calculate mean and median for:45, 83, 90, 79, 81, 83, 90, 88and for:83, 90, 79, 81, 83, 90, 88
5Let’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: 855 math majors: 9013 business majors: 819(85) + 5(90) + 13(81) =27
6MEAN calculations, cont’d Find the mean of the frequency distribution:Height(in inches)Frequency463 – 65566 – 48869 – 711
7Finding the MEAN of a Frequency Distribution MEAN calculations, cont’dFinding the MEAN of a Frequency DistributionIn WordsIn symbolsFind 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 frequenciesn = ∑ f4. Find the mean of the frequency distribution
9More 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 peakMore bars on the right of the peakSkewed RightSkewed LeftSymmetricSince a histogram is just a graphical representation of a distribution, you use the same process as finding the mean of the distribution…. but…..
10MEAN calculations cont’d: …there are some generalities based on shapeSymmetricUniformMean (as well as median) will be at the center.
11MEAN calculations, cont’d Skewed histogram means….Skewed LeftSkewed RightMeanModeModeMeanMedianMedian**Mean always fall in the direction the distribution is skewed**
13Sample Weighted Mean Pop. Weighted Mean The mean of a data set whose entries have varying weights.Sample Weighted Mean Pop. Weighted Meanw = weight of each entry--weights may not sum to 100%--if weights sum to 100%, then ∑w = 1
14Weighted Mean, cont’dEx 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?
15Weighted 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 mean15% from her midterm20% from her final15% from her homeworkHer 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 = 19.1X = 95.5Rachel needs to get a 95.5% on the final to earn a 90% in the class.
16Weighted Mean, cont’dEx 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?
17For large datasets: MEDIANS…. There’s an easier way to do it! Arrange in orderThe location of the median is found by countingdata items up from the bottom of the set.Ex) Ages at concert:24, 27, 19, 21, 18, 23, 21, 20, 19, 33, 30, 29, 2118, 24, 26, 38, 19, 35, 34, 33, 30, 21, 27, 3018, 18, 19, 19, 19, 20, 21, 21, 21, 21, 23, 24, 24, 26, 27, 27, 29, 30, 30, 30, 33, 33, 34, 35, 38
18MEDIAN calculations, cont’d: Works for stem-and-leaf plots too:
19MEDIAN calculations, cont’d: Works for histograms too: