Statistics

1. Statistics Statistics is the branch of mathematics that deals with collecting, organizing, analyzing, and reporting quantitative information.

2. Data Recorded observations that can be measured or counted

3. Population A population is an entire set of objects sharing similar characteristics, such as human beings, automobiles, or measurements, from which data can be collected and analyzed.

4. Sample A sample is a portion of a population from which data is collected to estimate the characteristics of the entire population.

5. Statistic A statistic is a measure calculated from a sample of data.

6. Parameter A parameter is a measure calculated from data for an entire population.

7. Example 1 Identify as a sample or a population: the snacks dispensed from a vending machine during its existence. populationAll snacks are being considered.

8. Example 1 Identify as a sample or a population: 100 freshman from among those enrolled at a local college for the fall of 2010. sampleA small number of students is being considered.

9. Example 1 Identify as a sample or a population: the automobiles built by General Motors in the 1990s. populationAll of the automobiles produced by the company are being considered.

10. Example Identify as a sample or a population: a school official gathers data about all students at Union University when studying the majors offered at the university. population

11. Example Identify as a sample or a population: a magazine wishes to determine the political viewpoint of college students, so it sends a questionnaire to all students at Union University. sample

12. Four Types of Population Sampling

13. Random—A random number generator is used to determine page number, column, and row of each person in the sample. • Systematic—Every thirtieth person listed in the phone book is included in the sample.

14. 3. Convenience—A questionnaire with a return envelope is mailed out with the phone bill. 4. Cluster—Everyone whose address indicates that he lives in a particular section of town is included in the sample.

15. Example 2 Identify the sample as random, systematic, convenience, or cluster: math teachers who attend a workshop at the regional conference. convenience

16. Example 2 Identify the sample as random, systematic, convenience, or cluster: every fifth person on the class roster. systematic

17. Example 2 Identify the sample as random, systematic, convenience, or cluster: 500 voters, based on phone numbers chosen by computer-generated selection of page number and column in the phone book. random

18. Example Identify the sample as random, systematic, convenience, or cluster: A worker selects the first phone number from every page of the phone book for a phone survey. systematic

19. Example Identify the sample as random, systematic, convenience, or cluster: To evaluate rates for the countywide school system, the school board randomly selects two high schools and evaluates the data at those two schools. cluster

20. Range The range is the difference between the largest and smallest numbers in a set of data.

21. Mean The mean is the arithmetic average of a set of numbers (sum of data divided by the number of data).

22. Median The median is the middle number in a set of data arranged in numerical order. If there is an even number of data, the median is the average of the two middle numbers.

23. Mode The mode is the number or numbers that occur most frequently in a set of data. If no number occurs more than once, there is no mode. Data sets in which two values occur most frequently are called bimodal, while other data sets may have no mode.

24. Mean Average Median Middle Mode Most

25. Example 3 Find the range, mean, median, and mode of the following set of data: {6, 9, 7, 3, 6, 7, 10, 5}. 10, 9, 7, 7, 6, 6, 5, 3 range: 10 – 3 = 7 mean: sum of data = 53 average = 53 ÷ 8 ≈ 6.6

26. 7 + 62 132 = Find the range, mean, median, and mode of the following set of data: {6, 9, 7, 3, 6, 7, 10, 5}. median: 10, 9, 7, 7, 6, 6, 5, 3 7 6 = 6.5 mode: 6 and 7

27. Example 4 Find the range, mean, median, and mode of the following set of data: {58, 38, 60, 44, 45, 42, 49, 50, 41}. 38, 41, 42, 44, 45, 49, 50, 58, 60 range: 60 – 38 = 22

28. Find the range, mean, median, and mode of the following set of data: {58, 38, 60, 44, 45, 42, 49, 50, 41}. mean: sum of data = 427 average = 427 ÷ 9 ≈ 47.4 median: 38, 41, 42, 44, 45, 49, 50, 58, 60 45 mode: none

29. Example 5 The following are daily low temperatures from the preceding week. Find the range, mean, median, and mode of the temperatures.

30. Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday Temperature (°F) 25° 21° 20° 29° 25° 22° 19°

31. 29, 25, 25, 22, 21, 20, 19 range: 29 – 19 = 10° mean: sum of data = 161 average = 161 ÷ 7 = 23° 22 median: 29, 25, 25, 22, 21, 20, 19 mode: 25°

32. Example Use this data to solve the following problems: A = {13, 25, 22, 18, 17, 17, 14, 16, 22} B = {19, 22, 35, 39, 35, 37, 40, 100}

33. Example Arrange the data for each set in ascending order. A = {13, 14, 16, 17, 17, 18, 22, 22, 25} B = {19, 22, 35, 35, 37, 39, 40, 100}

34. Example Find the range of each set. A = {13, 14, 16, 17, 17, 18, 22, 22, 25} B = {19, 22, 35, 35, 37, 39, 40, 100} A: 12; B: 81

35. Example Find the median of each set. A = {13, 14, 16, 17, 17, 18, 22, 22, 25} B = {19, 22, 35, 35, 37, 39, 40, 100} A: 17; B: 36

36. Example Find the mode of each set. A = {13, 14, 16, 17, 17, 18, 22, 22, 25} B = {19, 22, 35, 35, 37, 39, 40, 100} A: 17 and 22; B: 35

37. Example Find the mean of each set. A = {13, 14, 16, 17, 17, 18, 22, 22, 25} B = {19, 22, 35, 35, 37, 39, 40, 100} A: 18.22; B: 40.875

38. Example Would changing a single value in A always change the mean? Is this true for any set of data? A = {13, 14, 16, 17, 17, 18, 22, 22, 25} yes; yes

39. Example Would changing a single value in A always change the median? A = {13, 14, 16, 17, 17, 18, 22, 22, 25} no

40. Example If C = {12, 13, 14}, does C have three modes or no modes? no modes