Chapter 2 Descriptive Statistics Larson Farber Elementary Statistics

Chapter 2 Descriptive Statistics

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Range Larson & Farber, Elementary Statistics: Picturing the World, 3 e 4

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§ 2. 1 Frequency Distributions and Their Graphs

LEARNING OBJECTIVES 1. 2. Construct a frequency distribution that includes classes, frequencies, midpoints, relative frequencies, and cumulative frequencies. Construct frequency histograms, frequency polygons, relative frequency histograms, and ogives. Larson & Farber, Elementary Statistics: Picturing the World, 3 e 8

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Histogram Larson & Farber, Elementary Statistics: Picturing the World, 3 e 10

Frequency Distributions A frequency distribution is a table that shows classes or intervals of data with a count of the number in each class. The frequency f of a class is the number of data points in the class. Lower Upper Class Limits Class 1– 4 5– 8 9 – 12 Frequency, f 4 5 3 13 – 16 17 – 20 4 2 Larson & Farber, Elementary Statistics: Picturing the World, 3 e Frequencies 11

Frequency Distributions The class width is the distance between lower (or upper) limits of consecutive classes. 5– 1=4 9– 5=4 13 – 9 = 4 17 – 13 = 4 Class 1– 4 5– 8 9 – 12 Frequency, f 4 5 3 13 – 16 4 17 – 20 2 The class width is 4. The range is the difference between the maximum and minimum data entries. Larson & Farber, Elementary Statistics: Picturing the World, 3 e 12

Constructing a Frequency Distribution Guidelines 1. Decide on the number of classes to include. The number of classes should be between 5 and 20; otherwise, it may be difficult to detect any patterns. 2. Find the class width as follows. Determine the range of the data, divide the range by the number of classes, and round up to the next convenient number. 3. Find the class limits. You can use the minimum entry as the lower limit of the first class. To find the remaining lower limits, add the class width to the lower limit of the preceding class. Then find the upper class limits. 4. Make a tally mark for each data entry in the row of the appropriate class. 5. Count the tally marks to find the total frequency f for each class. Larson & Farber, Elementary Statistics: Picturing the World, 3 e 13

Constructing a Frequency Distribution Example: The following data represents the ages of 30 students in a statistics class. Construct a frequency distribution that has five classes. Ages of Students 18 20 21 27 29 20 19 30 32 19 34 19 24 29 18 37 38 22 30 39 32 44 33 46 54 49 18 51 21 21 Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3 e 14

Constructing a Frequency Distribution Example continued: 1. The number of classes (5) is stated in the problem. 2. The minimum data entry is 18 and maximum entry is 54, so the range is 36. Divide the range by the number of classes to find the class width. Class width = 36 = 7. 2 5 Round up to 8. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3 e 15

Constructing a Frequency Distribution Example continued: 3. The minimum data entry of 18 may be used for the lower limit of the first class. To find the lower class limits of the remaining classes, add the width (8) to each lower limit. The lower class limits are 18, 26, 34, 42, and 50. The upper class limits are 25, 33, 41, 49, and 57. 4. Make a tally mark for each data entry in the appropriate class. 5. The number of tally marks for a class is the frequency for that class. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3 e 16

Constructing a Frequency Distribution Example continued: Ages of Students Class 18 – 25 26 – 33 34 – 41 42 – 49 50 – 57 Tally Number of students Frequency, f 13 8 4 3 2 Larson & Farber, Elementary Statistics: Picturing the World, 3 e Check that the sum equals the number in the sample. 17

Midpoint The midpoint of a class is the sum of the lower and upper limits of the class divided by two. The midpoint is sometimes called the class mark. Midpoint = (Lower class limit) + (Upper class limit) 2 Class 1– 4 Frequency, f 4 Midpoint 2. 5 Midpoint = Larson & Farber, Elementary Statistics: Picturing the World, 3 e 18

Midpoint Example: Find the midpoints for the “Ages of Students” frequency distribution. Ages of Students Class 18 – 25 26 – 33 34 – 41 42 – 49 50 – 57 Frequency, f 13 8 4 3 2 Midpoint 21. 5 29. 5 37. 5 45. 5 53. 5 Larson & Farber, Elementary Statistics: Picturing the World, 3 e 18 + 25 = 43 43 2 = 21. 5 19

Relative Frequency The relative frequency of a class is the portion or percentage of the data that falls in that class. To find the relative frequency of a class, divide the frequency f by the sample size n. Class frequency Relative frequency = Sample size Class Frequency, f Relative Frequency 1– 4 4 0. 222 Relative frequency Larson & Farber, Elementary Statistics: Picturing the World, 3 e 20

Relative Frequency Example: Find the relative frequencies for the “Ages of Students” frequency distribution. Class Frequency, f Relative Frequency 18 – 25 26 – 33 34 – 41 42 – 49 50 – 57 13 8 4 3 2 0. 433 0. 267 0. 133 0. 1 0. 067 Portion of students Larson & Farber, Elementary Statistics: Picturing the World, 3 e 21

Cumulative Frequency The cumulative frequency of a class is the sum of the frequency for that class and all the previous classes. Ages of Students Class Frequency, f Cumulative Frequency 18 – 25 26 – 33 34 – 41 42 – 49 50 – 57 13 +8 +4 +3 +2 13 21 25 28 30 Larson & Farber, Elementary Statistics: Picturing the World, 3 e Total number students of 22

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Frequency Histogram A frequency histogram is a bar graph that represents the frequency distribution of a data set. 1. The horizontal scale is quantitative and measures the data values. 2. The vertical scale measures the frequencies of the classes. 3. Consecutive bars must touch. Class boundaries are the numbers that separate the classes without forming gaps between them. The horizontal scale of a histogram can be marked with either the class boundaries or the midpoints. Larson & Farber, Elementary Statistics: Picturing the World, 3 e 24

Class Boundaries Example: Find the class boundaries for the “Ages of Students” frequency distribution. Ages of Students The distance from the upper limit of the first class to the lower limit of the second class is 1. Half this distance is 0. 5. Class Frequency, f Class Boundaries 18 – 25 26 – 33 34 – 41 42 – 49 50 – 57 13 8 4 3 2 17. 5 25. 5 33. 5 41. 5 49. 5 57. 5 Larson & Farber, Elementary Statistics: Picturing the World, 3 e 25

Frequency Histogram Example: Draw a frequency histogram for the “Ages of Students” frequency distribution. Use the class boundaries. 14 Ages of Students 13 12 10 8 8 f 6 4 4 3 2 2 0 Broken axis 17. 5 25. 5 33. 5 41. 5 49. 5 57. 5 Age (in years) Larson & Farber, Elementary Statistics: Picturing the World, 3 e 26

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Relative Frequency Histogram A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram. Relative frequency (portion of students) 0. 5 0. 433 Ages of Students 0. 4 0. 3 0. 267 0. 2 0. 133 0. 1 0. 067 0 17. 5 25. 5 33. 5 41. 5 49. 5 57. 5 Age (in years) Larson & Farber, Elementary Statistics: Picturing the World, 3 e 31

Frequency Polygon A frequency polygon is a line graph that emphasizes the continuous change in frequencies. 14 Ages of Students 12 10 Line is extended to the x-axis. 8 f 6 4 2 0 Broken axis 13. 5 21. 5 29. 5 37. 5 45. 5 53. 5 Age (in years) Larson & Farber, Elementary Statistics: Picturing the World, 3 e 61. 5 Midpoints 32

Cumulative Frequency Graph A cumulative frequency graph or ogive, is a line graph that displays the cumulative frequency of each class at its upper class boundary. Cumulative frequency (portion of students) 30 Ages of Students 24 The graph ends at the upper boundary of the last class. 18 12 6 0 17. 5 25. 5 33. 5 41. 5 49. 5 57. 5 Age (in years) Larson & Farber, Elementary Statistics: Picturing the World, 3 e 33