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Chapter 9

Survey of Mathematics – MM150 Unit 9 – Statistics Mr. Scott VanZuiden, Adjunct Professor Kaplan University svanzuiden@kaplan.edu Welcome to seminar!. Chapter 9. Statistics. WHAT YOU WILL LEARN. • Mode, median, mean, and midrange • Percentiles and quartiles • Range and standard deviation

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Chapter 9

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  1. Survey of Mathematics – MM150Unit 9 – StatisticsMr. Scott VanZuiden, Adjunct ProfessorKaplan Universitysvanzuiden@kaplan.eduWelcome to seminar!

  2. Chapter 9 Statistics

  3. WHAT YOU WILL LEARN • Mode, median, mean, and midrange • Percentiles and quartiles • Range and standard deviation •z-scores and the normal distribution • Correlation and regression

  4. 9.1 Measures of Central Tendency

  5. Definitions • An average is a number that is representative of a group of data. • The arithmetic mean, or simply the mean is symbolized by , when it is a sample of a population or by the Greek letter mu, , when it is the entire population.

  6. Mean • The mean, is the sum of the data divided by the number of pieces of data. The formula for calculating the mean is • where represents the sum of all the data and n represents the number of pieces of data.

  7. Example-find the mean • Find the mean amount of money parents spent on new school supplies and clothes if 5 parents randomly surveyed replied as follows: $327 $465 $672 $150 $230

  8. Median • The median is the value in the middle of a set of ranked data. • Example: Determine the median of $327 $465 $672 $150 $230.

  9. Example: Median (even data) • Determine the median of the following set of data: 8, 15, 9, 3, 4, 7, 11, 12, 6, 4.

  10. Mode • The mode is the piece of data that occurs most frequently. • Example: Determine the mode of the data set: 3, 4, 4, 6, 7, 8, 9, 11, 12, 15.

  11. Midrange • The midrange is the value halfway between the lowest (L) and highest (H) values in a set of data. • Example: Find the midrange of the data set$327, $465, $672, $150, $230.

  12. Example • The weights of eight Labrador retrievers rounded to the nearest pound are 85, 92, 88, 75, 94, 88, 84, and 101. Determine the a) mean b) median c) mode d) midrange e) rank the measures of central tendency from lowest to highest.

  13. Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101

  14. Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101

  15. Measures of Position • Measures of position are often used to make comparisons. • Two measures of position are percentiles and quartiles.

  16. To Find the Quartiles of a Set of Data 1. Order the data from smallest to largest. 2. Find the median, or 2nd quartile, of the set of data. If there are an odd number of pieces of data, the median is the middle value. If there are an even number of pieces of data, the median will be halfway between the two middle pieces of data.

  17. To Find the Quartiles of a Set of Data continued 3. The first quartile, Q1, is the median of the lower half of the data; that is, Q1, is the median of the data less than Q2. 4. The third quartile, Q3, is the median of the upper half of the data; that is, Q3is the median of the data greater than Q2.

  18. Example: Quartiles • The weekly grocery bills for 23 families are as follows. Determine Q1, Q2, and Q3. 170 210 270 270 280 330 80 170 240 270 225 225 215 310 50 75 160 130 74 81 95 172 190

  19. Example: Quartiles continued • Order the data: 50 75 74 80 81 95 130 160 170 170 172 190 210 215 225 225 240 270 270 270 280 310 330 Q2 is the median of the entire data set which is 190. Q1 is the median of the numbers from 50 to 172 which is 95. Q3 is the median of the numbers from 210 to 330 which is 270.

  20. 9.2 Measures of Dispersion

  21. Measures of Dispersion • Measures of dispersion are used to indicate the spread of the data. • The range is the difference between the highest and lowest values; it indicates the total spread of the data. Range = highest value – lowest value

  22. Example: Range • Nine different employees were selected and the amount of their salary was recorded. Find the range of the salaries. $24,000 $32,000 $26,500 $56,000 $48,000 $27,000 $28,500 $34,500 $56,750

  23. Standard Deviation • The standard deviation measures how much the data differ from the mean. It is symbolized with s when it is calculated for a sample, and with  (Greek letter sigma) when it is calculated for a population.

  24. To Find the Standard Deviation of a Set of Data 1. Find the mean of the set of data. 2. Make a chart having three columns: Data Data  Mean (Data  Mean)2 3. List the data vertically under the column marked Data. 4. Subtract the mean from each piece of data and place the difference in the Data  Mean column.

  25. To Find the Standard Deviation of a Set of Data continued 5. Square the values obtained in the Data  Mean column and record these values in the (Data  Mean)2 column. 6. Determine the sum of the values in the (Data  Mean)2 column. 7. Divide the sum obtained in step 6 by n  1, where n is the number of pieces of data. 8. Determine the square root of the number obtained in step 7. This number is the standard deviation of the set of data.

  26. Example • Find the standard deviation of the following prices of selected washing machines: $280, $217, $665, $684, $939, $299 Find the mean.

  27. Data Data  Mean (Data  Mean)2 217 280 299 665 684 939 Example continued, mean = 514

  28. Example continued, mean = 514 • The standard deviation is $

  29. 9.3 The Normal Curve

  30. Rectangular Distribution J-shaped distribution Types of Distributions

  31. Bimodal Skewed to right Types of Distributions continued

  32. Skewed to left Normal Types of Distributions continued

  33. Properties of a Normal Distribution • The graph of a normal distribution is called the normal curve. • The normal curve is bell shaped and symmetric about the mean. • In a normal distribution, the mean, median, and mode all have the same value and all occur at the center of the distribution.

  34. Empirical Rule • Approximately 68% of all the data lie within one standard deviation of the mean (in both directions). • Approximately 95% of all the data lie within two standard deviations of the mean (in both directions). • Approximately 99.7% of all the data lie within three standard deviations of the mean (in both directions).

  35. z-Scores • z-scores determine how far, in terms of standard deviations, a given score is from the mean of the distribution.

  36. Example: z-scores • A normal distribution has a mean of 50 and a standard deviation of 5. Find z-scores for the following values. • a) 55 b) 60 c) 43

  37. Example: z-scores continued

  38. To Find the Percent of Data Between any Two Values 1. Draw a diagram of the normal curve, indicating the area or percent to be determined. 2. Use the formula to convert the given values to z-scores. Indicate these z- scores on the diagram. 3. Look up the percent that corresponds to each z-score in Table 13.7.

  39. To Find the Percent of Data Between any Two Values continued 4. a) When finding the percent of data between two z-scores on opposite sides of the mean (when one z-score is positive and the other is negative), you find the sum of the individual percents. b) When finding the percent of data between two z-scores on the same side of the mean (when both z-scores are positive or both are negative), subtract the smaller percent from the larger percent.

  40. To Find the Percent of Data Between any Two Values continued c) When finding the percent of data to the right of a positive z-score or to the left of a negative z-score, subtract the percent of data between 0 and z from 50%. d) When finding the percent of data to the left of a positive z-score or to the right of a negative z-score, add the percent of data between 0 and z to 50%.

  41. Example Assume that the waiting times for customers at a popular restaurant before being seated for lunch are normally distributed with a mean of 12 minutes and a standard deviation of 3 min. a) Find the percent of customers who wait for at least 12 minutes before being seated. b) Find the percent of customers who wait between 9 and 18 minutes before being seated. c) Find the percent of customers who wait at least 17 minutes before being seated. d) Find the percent of customers who wait less than 8 minutes before being seated.

  42. a. wait for at least 12 minutes b. between 9 and 18 minutes Solution

  43. c. at least 17 min d. less than 8 min Solution continued

  44. 9.4 Linear Correlation and Regression

  45. Linear Correlation • Linear correlation is used to determine whether there is a relationship between two quantities and, if so, how strong the relationship is.

  46. Linear Correlation • The linear correlation coefficient, r, is a unitless measure that describes the strength of the linear relationship between two variables. • If the value is positive, as one variable increases, the other increases. • If the value is negative, as one variable increases, the other decreases. • The variable, r, will always be a value between –1 and 1 inclusive.

  47. Scatter Diagrams • A visual aid used with correlation is the scatter diagram, a plot of points (bivariate data). • The independent variable, x, generally is a quantity that can be controlled. • The dependent variable, y, is the other variable. • The value of r is a measure of how far a set of points varies from a straight line. • The greater the spread, the weaker the correlation and the closer the r value is to 0. • The smaller the spread, the stronger the correlation and the closer the r value is to 1.

  48. Correlation

  49. Correlation

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