Probability and Statistics Mrs. Leahy 1 Chapter 2: Descriptive Statistics Part 2: 2.3B, 2.4, 2.5 Assignments and Schedule Pd 2 Pd 4,7 Lesson Homework Fri 8/19 Fri 8/19 2.3B Measures of Central Tendency (Mean/Median/Mode/Weighted Averages) HW 2.3B pg 75-80 #19,22,41,44,49,50,66a Mon 8/22 Mon 8/22 2.4A Measures of Variation (Sample Variance and Standard Deviation) HW 2.4A pg 93-99 #13, 50a, 21, 22 Tues 8/23 A Wed 8/24 B HW 2.3B DUE TODAY 2.4 B Measures of Variation Worksheet 2.4 - Part 1 Wed 8/24 B Fri 8/26 Mon 8/29 Tues 8/30 A Wed 8/31 B Thurs 8/25 C Fri 8/26 Mon 8/29 Wed 8/31 B Thurs 9/1 C (Coefficient of Variation, St. Dev. Of Frequency Dist.) HW 2.4A DUE TODAY 2.4 C Measures of Variation (Empirical Rule, Chebyshev s Rule, Standard z Scores) Worksheet 2.4 DUE TODAY 2.5 Measures of Position (Percentiles, 5-number summary, Box- and- Whisker) HW 2.5 DUE TODAY Review Chapter 2B Review Guide Worksheet DUE TODAY TEST CHAPTER 2 In Class Activity: TiNspires 1 variable statistics, linear transformations, making graphs Worksheet 2.4 - Part 2 HW 2.5 pg 109 # 15,16,17,18 Review Guide Worksheet No homework In Class Handout no homework Academic Standards: PS.DA.1 Create, compare, and evaluate different graphic displays of the same data, using histograms, frequency polygons, cumulative frequency distribution functions, pie charts, scatterplots, stem-and-leaf plots, and box-and-whisker plots. Draw these with and without technology. PS.DA.2 PS.DA.5 Compute and use mean, median, mode, weighted mean, geometric mean, harmonic mean, range, quartiles, variance and standard deviation. Use tables and technology to estimate areas under the normal curve. Fit a data set to a normal distribution and estimate population percentages. Recognize that there are data sets not normally distributed for which such procedures are inappropriate. Recognize how linear transformations of univariate data affect shape, center, and spread.
Probability and Statistics Mrs. Leahy 2 Chapter 2: Descriptive Statistics Part 2: 2.3B, 2.4, 2.5 2.3 B: Measures of Central Tendency (Day 1) While the individual man is an insolvable puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. -- Arthur Conan Doyle, The Sign of Four What does it mean to be average? The average price of gold is $920 per ounce. My car averages 28 miles per gallon. The average shoe size for women is a size 8. The average test score was an 85%. An average is a way to describe an data set using only. Sometimes we will be working with a population and sometimes with a sample. A population is possible cases in the situation we are studying. A sample is a set of cases from the population use to represent the entire population. Example: Population: Every carton of orange juice manufactured by Tropicana this year. Sample: Ten cartons of Tropicana orange juice randomly selected from a grocery store shelf. Most Commonly Used Averages : Mode, Median, and Mean MODE The mode of a set of data is the number that occurs frequently. It is possible to have more than one mode. It is possible to have no mode. Example 1: Find the mode of this data: {1, 7, 8, 4, 4, 4, 6, 3, 8, 7} Example 2: Sixteen students are asked how many college math classes they have completed. There responses are shown at the right. What is the mode?
Probability and Statistics Mrs. Leahy 3 MEDIAN The median of a set of data is the number that is exactly in the middle of a set of ordered values. Sometimes the median is called the central value. Example 3a: Find the median: To find the median: 1. Order the data from smallest to largest 2. For an odd number of data values: Mean = Middle value 3. For an even number of data values: Mean = average of two middle values (Sum two middle values, then 2) 23 27 19 21 18 23 21 20 19 33 30 29 21 Example 3b: Find the median of this data: 30 27 21 29 33 34 35 19 38 26 MEAN The mean of a set of data is the arithmetic of all data values. Symbols: = Summation, the sum of the following Mean of a sample: Mean of a population: n = the number of values in the sample N = the number of values in the population x x x n N x-bar mew Example 4: Find the mean of the following sample data set: {3, 8, 5, 4, 8, 4, 10 }
Probability and Statistics Mrs. Leahy 4 Trimmed Mean: The mean of the data values left after trimming a specified percentage of the smallest and largest data values from a data set. Common trims are 5% or 10%. Example 5: A sample of 20 colleges showed class sizes for intro courses to be: 14 20 20 20 20 23 25 30 30 30 35 35 35 40 40 42 50 50 80 80 a) What is the mean for the entire sample. b) Compute a 5% trimmed mean for the sample. Weighted Averages (Weighted Mean) Example 6: In this class you were told the following information about grade weighting: Homework 17% Tests 68% Final Exam 15% You currently have a 100% on homework and think you got an 85% on the Chapter 2 test. What is your current grade in the class? Weighted Mean x = (x w) w 1. Take every value times its weight. Find the sum of these products. 2. Find the sum of all the weights. 3. Divide. This type of mean is called a weighted average or weighted mean because some values are considered more important than others.
Probability and Statistics Mrs. Leahy 5 Example 7: Suppose your midterm test score is 80, your project score is 85 and your final exam score is 98. Suppose the weights are 30% for midterm, 30% for projects, and 40% for final. If the minimum average of an A is 90%, will you earn an A? Example 8: For the month of August, your checking account has a balance of $300 for 20 days, $450 for 10 days, and $25 for 1 day. What is the account s mean daily balance in August? Mean of a Frequency Distribution Mean of a Frequency Distribution Example 9: Find the weighted (group) average for the following data. x = (x f) n 1. Let x = the midpoint of each class 2. n = f 3. Take each midpoint times the class frequency. Find the sum. 4. Divide.
Probability and Statistics Mrs. Leahy 6 2.4 Measures of Variation (Day 2) Any set of measurements has two important properties: The central/typical/average value (Lesson 2.3) The SPREAD about that value. Spread tells us how far from the center the data ranges. Example: You survey 2 groups of 50 students asking them to report their weight. Group 1: Mean weight 145lbs Group 2: Mean weight 145lbs. Today, we will learn about standard deviation, a measurement of spread about the. Small standard deviation means most values are to the mean. Large standard deviation means more values are to the mean. In lesson 2.5, we will learn about interquartile range, a measurement of spread about the. Example 1: You are asked to compare three data sets. Which has the largest standard deviation? Which has the smallest standard deviation?
Probability and Statistics Mrs. Leahy 7 Example 2: A large bakery regularly orders cartons of Maine blueberries. The average weight of the cartons is supposed to be 22 ounces. Random samples of cartons from two suppliers were weighed. The weights in ounces per cartons were: Supplier A: 17 22 22 22 27 Supplier B: 17 19 20 27 27 a) What is the range of each set of data? b) What is the mean of each set of data? c) The bakery uses 1 carton of blueberries per blueberry muffin recipe. Which supplier should they choose? Variance and Standard Deviation of a Sample (Ungrouped Data --- no classes.) Sample Variance and Standard Deviation Sample Variance: Sample Standard Deviation: s 2 = (x x )2 n 1 s = s 2 = (x x )2 n 1 How to use these formulas: 1. If unknown, determine x, the mean of the data 2. Find (x x ) for each data value and square the result. 3. Find the sum of the squares. 4. Divide by (n 1) This is the variance. 5. Square root the variance. This is the standard deviation. Supplier A: x = Supplier B: x = x x x (x x ) 2 x x x (x x ) 2 Sample Variance (s 2 ): Sample Standard Deviation (s) =
Probability and Statistics Mrs. Leahy 8 Example 3: Big Blossom Greenhouse measured a sample of rose blooms for diameters in inches. 2 3 3 8 10 10 Compute the sample variance and the sample standard deviation. Example 4: For the sample below, find the sample variance and the sample standard deviation. A study examining the health risks of smoking measured cholesterol levels of people who had smoked for at least 25 years.
Probability and Statistics Mrs. Leahy 9 Example 5: Variance and Standard Deviation of a POPULATION In most statistics applications, we work with a random sample of data rather than the entire population. If you have the data for a population, you can determine the population mean, population variance, and population standard deviation. N = size of the population μ = the mean of a population Population Mean = μ = x N Population Standard Deviation = σ = (x μ)2 N (Day 3) Coefficient of Variation Which is MORE variable? The Coefficient of Variation expresses standard deviation as a percentage of the sample or population mean. This allows us to compare data from different populations that may use different units of measurement. CV = standard deviation mean x 100 Example 6: Mrs. Leahy and Mr. Johnson decide to compare the heights of the students in their classes. Mrs. Leahy s class had a mean height of 67 inches, with a standard deviation of 2.13 inches. Mr. Johnson s class had a average height of 165cm with a standard deviation of 5cm. Use the coefficient of variation to compare the two classes.
Example 7: The following height and weight data was collected from the members of a basketball team: Probability and Statistics Mrs. Leahy 10 Heights: 72 74 68 76 74 69 72 79 70 69 77 73 Weights: 180 168 225 201 189 192 197 162 174 171 185 210 The mean height is μ 72.8 inches with a standard deviation of σ 3.3 inches. The mean weight is μ 187.8 lbs with a standard deviation of σ = 17.7 lbs. Use the coefficient of variation to compare the results. Standard Deviation for Grouped Data (Classes) To find the standard deviation for a set of grouped (class) data -----------------> Mean of a Frequency Distribution: x = xf Sample Standard Deviation: n s = (x x )2 f n 1 Where x is the midpoint of the class and n is the sample size. Example 6: Find the mean and standard deviation of the following frequency distribution.
Probability and Statistics Mrs. Leahy 11 The Empirical Rule (68-95-99.7 RULE) (Day 4) Standard deviation can do more than tell us how varied our data is it can help us estimate what percentage of data lies within certain boundaries. Whenever data is Mound Shaped and Symmetrical we can use the Empirical Rule to make some estimates. The Empirical Rule For data sets that are approximately symmetric and bell-shaped, the standard deviation has these characteristics. 1. About 68% of the data lie within one standard deviation of the mean. 2. About 95% of the data lie within two standard deviations of the mean. 3. About 99.7% of the data lie within three standard deviation of the mean. 0.15% 0.15% Example 7: The scores of a recent SAT test are a bell shaped distribution with a mean of µ=1100 and a standard deviation σ=200. a) Estimate the percentage of scores between 900 and 1300. b) Between what two values do approximately 95% of the scores lie? c) A score beyond three standard deviations is considered to be unusual. Is a score of 600 unusual for this data set? If not, give an example of an unusual score.
Probability and Statistics Mrs. Leahy 12 What if data is not bell shaped? Can we still make an estimate if we know the mean and standard deviation? It isn t as accurate, but we can use: Chebyshev s Theorem The portion of any data set lying within k standard deviations of the mean is at least 1 1 k 2. Example 8: The histogram on the right shows Pittsburgh s Daily High Temperature last year. a) What temperature range represents within two standard deviations of the mean? b) Use Chebyshev s theorem to estimate the percentage of data in this range. c) Repeat step a) and b) for within three standard deviations of the mean. Standard Scores (Z-Scores) A standard score (Z-score) represents the number of standard deviations from the mean. The z-score tells you how far away from the average the score is the bigger the z-score (positive or negative), the more unusual or extreme the score is. Example 9: Use the Pittsburgh example above to determine the z-score for the following temperatures. A) 95 degrees B) 22 degrees C) 65 degrees
Probability and Statistics Mrs. Leahy 13 Example 10: Tony Stark and Steve Rogers are in separate Finite Math courses. On the Chapter 2 test, Tony scored an 82% and Steve scored a 92%. In Tony s class, the average was a 75% with a standard deviation of 7.5. In Steve s class, the average was an 89% with a standard deviation of 3.6. Who performed better with respect to his individual class? (Use a z-score to help you) 2.5 Measures of Position (Day 5) I took my son to the doctor and was told that he was in the 85th percentile for height and the 56th percentile for weight. You took a standardized test and received notice that you scored in the 90 th percentile. On the website for the college you want to attend, you see they are accepting applications from students in the 75 th percentile of their graduating class. WHAT DOES THIS MEAN? Percentiles There are 100 percentiles. If P = the Pth Percentile then P% of the data is to P and (100 P)% of the data is to P. Example 1: You took the English achievement test to obtain college credit in freshman English by examination. a) If your score is at the 89 th percentile, what percentage of scores are at or below yours? b) What percentage of scores are higher than yours? c) If the scores ranged from 1 to 100 and your raw score is 95, does this necessarily mean that your score is at the 95 th percentile?
Probability and Statistics Mrs. Leahy 14 Quartiles/Interquartile Range Quartiles divide the data into. Q 1 Q 2 first quartile = percentile. second quartile = percentile and is also the of the data Q 3 third quartile = percentile The Interquartile Range (IQR) is the difference between Q 3 and Q 1. The IQR you the range of values in the middle of the data set. The Interquartile Range is a measure of SPREAD about the MEDIAN of a set of data. Example: Data Set 1: Median = 6, IQR = 4 Data Set 2: Median = 6, IQR = 10
Probability and Statistics Mrs. Leahy 15 Box-and-Whisker Plots Smallest Value Largest Value Quartiles: Q 1, Q 2, Q 3 give us a very useful summary of the data and their spread A box-and-whisker plot is a graphical representation of these values: Outliers occur when any value is beyond 1.5 x IQR PROCEDURE: How to make a box-and-whisker plot 1. Draw a horizontal (or vertical) scale to include the lowest and highest values. Steps 1, 2, 3: 2. Above (or to the right) of the scale, draw a box from Q 1 to Q 3. 3. Include a solid line through the box at the median level. Step 4: 4. Check for outlier and draw them in as individual points. 5. Draw horizontal (or vertical) lines (called whiskers), from Q 1 to the lowest value and from Q 3 to the highest value. Step 5:
Probability and Statistics Mrs. Leahy 16 Example 4: For the sample {42,77,19,53,95,34,94,86} a) Find the five-number summary b) Make a box and whisker plot. Example 6: Consider the data {1, 2, 3, 3, 3, 5, 6, 7, 7, 10, 20} a) Find the five-number summary for the data. b) Draw a box and whisker plot. Example 7: Three classes (A, B, C) took the same test
Probability and Statistics Mrs. Leahy 17 Example 8: a) What is the range of the sugar content of these cereals? b) Describe the shape of the distribution of the: histogram: adult box-whisker-plot children box-whisker-plot c) Are all children s cereal higher in sugar than adult cereals? d) Which group varies more in sugar content? Example 9: