1. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Work through the notes Then complete the class work next to this document on the website For the last page of the class work you will have to roll two dice. There are several websites that allow you to do this virtually two of them are below https://www.math.duke.edu/education/postcalc/probability/di ce/index.html http://www.math.csusb.edu/faculty/stanton/m262/intro_prob _models/intro_prob_models.html

2. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Find measures of central tendency and measures of variation for statistical data. Examine the effects of outliers on statistical data. Objectives

3. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation expected value probability distribution variance standard deviation outlier Vocabulary

4. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Recall that the mean, median, and mode are measures of central tendency—values that describe the center of a data set. The mean is the sum of the values in the set divided by the number of values. It is often represented as x. The median is the middle value or the mean of the two middle values when the set is ordered numerically. The mode is the value or values that occur most often. A data set may have one mode, no mode, or several modes.

5. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Example 1: Finding Measures of Central Tendency Find the mean, median, and mode of the data. deer at a feeder each hour: 3, 0, 2, 0, 1, 2, 4 Mean: Median: Mode: deer 0 0 1 2 2 3 4= 2 deer The most common results are 0 and 2.

6. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Check It Out! Example 1a Find the mean, median, and mode of the data set. {6, 9, 3, 8} Mean: Median: Mode: 3 6 8 9 None

7. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Check It Out! Example 1b Find the mean, median, and mode of the data set. {2, 5, 6, 2, 6} Mean: Median: Mode: 2 2 5 6 6 2 and 6 = 5

8. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation A weighted average is a mean calculated by using frequencies of data values. Suppose that 30 movies are rated as follows: weighted average of stars =

9. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation For numerical data, the weighted average of all of those outcomes is called the expected value for that experiment. The probability distribution for an experiment is the function that pairs each outcome with its probability.

10. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Example 2: Finding Expected Value The probability distribution of successful free throws for a practice set is given below. Find the expected number of successes for one set.

11. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Example 2 Continued Use the weighted average. Simplify. The expected number of successful free throws is 2.05.

12. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Check It Out! Example 2 The probability distribution of the number of accidents in a week at an intersection, based on past data, is given below. Find the expected number of accidents for one week. expected value = 0(0.75) + 1(0.15) + 2(0.08) + 3(0.02) Use the weighted average. = 0.37Simplify. The expected number of accidents is 0.37.

13. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation A box-and-whisker plot shows the spread of a data set. It displays 5 key points: the minimum and maximum values, the median, and the first and third quartiles.

14. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation The quartiles are the medians of the lower and upper halves of the data set. If there are an odd number of data values, do not include the median in either half. The interquartile range, or IQR, is the difference between the 1st and 3 rd quartiles, or Q3 – Q1. It represents the middle 50% of the data.

15. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Example 3: Making a Box-and-Whisker Plot and Finding the Interquartile Range Make a box-and-whisker plot of the data. Find the interquartile range. {6, 8, 7, 5, 10, 6, 9, 8, 4} Step 1 Order the data from least to greatest. 4, 5, 6, 6, 7, 8, 8, 9, 10 Step 2 Find the minimum, maximum, median, and quartiles. 4, 5, 6, 6, 7, 8, 8, 9, 10 Mimimum MedianMaximum First quartile 5.5 Third quartile 8.5

16. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Example 3 Continued Step 3 Draw a box-and-whisker plot. Draw a number line, and plot a point above each of the five values. Then draw a box from the first quartile to the third quartile with a line segment through the median. Draw whiskers from the box to the minimum and maximum.

17. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation The interquartile range is 3, the length of the box in the diagram. IRQ = 8.5 – 5.5 = 3 Example 3 Continued

18. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Check It Out! Example 3 Make a box-and-whisker plot of the data. Find the interquartile range. {13, 14, 18, 13, 12, 17, 15, 12, 13, 19, 11, 14, 14, 18, 22, 23} Step 1 Order the data from least to greatest. 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23 Step 2 Find the minimum, maximum, median, and quartiles. 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23 MimimumMedianMaximum First quartile 13 Third quartile 18

19. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Check It Out! Example 3 Continued Step 3 Draw a box-and-whisker plot. IQR = 18 – 13 = 5 The interquartile range is 5, the length of the box in the diagram.

20. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Example 1: Wildlife Application A researcher is gathering information on the gender of prairie dogs at a wildlife preserve. The researcher samples the population by catching 10 animals at a time, recording their genders, and releasing them.

21. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Since there were 24 males in the sample, and 16 females, he estimates that the ratio of males to females is 3:2. Continued Example 1: Wildlife Application How can he use this data to estimate the ratio of males to females in the population?

22. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Check It Out! Example 1 Identify the population and the sample. 1. A car factory just manufactured a load of 6,000 cars. The quality control team randomly chooses 60 cars and tests the air conditioners. They discover that 2 of the air conditioners do not work. population: 6,000 total cars sample: 60 cars

23. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation Example 2A: Identifying Potentially Biased Samples Decide whether each sampling method could result in a biased sample. Explain your reasoning. A. A survey of a city’s residents is conducted by asking 20 randomly selected people at a grocery store whether the city should impose a beverage tax. Residents who do not shop at the store are underrepresented, so the sample is biased.

24. Holt McDougal Algebra 2 8-1 Measures of Central Tendency and Variation B. A survey of students at a school is conducted by asking 30 randomly selected students in an all-school assembly whether they walk, drive, or take the bus to school. Example 2B: Identifying Potentially Biased Samples No group is overrepresented or underrepresented, so the sample is not likely to be biased.