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- 1. Chapter 2 Descriptive Statistics Larson/Farber 4th ed.
- 2. Section 2.1 Frequency Distributions and Their Graphs Part 1: Frequency Distributions Larson/Farber 4th ed.
- 3. Frequency Distribution <ul><li>Frequency Distribution: A table that shows classes or intervals of data with a count of the number of entries in each class. It is used to organize data and helps to recognize patterns. </li></ul><ul><li>The frequency, f, of a class is the number of data entries in the class. </li></ul>Larson/Farber 4th ed. 4 26 – 30 5 21 – 25 8 16 – 20 6 11 – 15 8 6 – 10 5 1 – 5 Frequency Class
- 4. Frequency Distribution <ul><li>Each class has: </li></ul><ul><ul><li>A lower class limit , which is the least number that can belong to the class </li></ul></ul><ul><ul><li>(1, 6, 11, 16, 21, 26) </li></ul></ul><ul><ul><li>An upper class limit , which is the greatest number that can belong to the class </li></ul></ul><ul><ul><li>(5, 10, 15, 20, 25, 30) </li></ul></ul>4 26 – 30 5 21 – 25 8 16 – 20 6 11 – 15 8 6 – 10 5 1 – 5 Frequency Class
- 5. Frequency Distribution <ul><li>The class width is the distance between lower (or upper) limits of consecutive classes. </li></ul><ul><ul><li>Example: 6 – 1 = 5 </li></ul></ul>4 26 – 30 5 21 – 25 8 16 – 20 6 11 – 15 8 6 – 10 5 1 – 5 Frequency Class
- 6. Frequency Distribution <ul><li>The range is the difference between the maximum and the minimum data entries. </li></ul><ul><ul><li>Example: if the maximum data entry is 29 and the minimum data entry is 1, the range is 29 – 1 = 28 </li></ul></ul>4 26 – 30 5 21 – 25 8 16 – 20 6 11 – 15 8 6 – 10 5 1 – 5 Frequency Class
- 7. Constructing a Frequency Distribution from a Data Set <ul><li>Choose the number of classes (usually between 5 and 20) </li></ul><ul><li>Find the class width. </li></ul><ul><ul><li>Find the range of the data </li></ul></ul><ul><ul><li>Divide the range by the number of classes </li></ul></ul><ul><ul><li>Round up to the next convenient number </li></ul></ul><ul><li>Find the class limits. </li></ul><ul><ul><li>Use the minimum data entry as the lower limit of the first class </li></ul></ul><ul><ul><li>Find the remaining lower limits (add the class width to the lower limit of the preceding class). </li></ul></ul><ul><ul><li>Find the upper limit of the first class. Remember that classes cannot overlap. </li></ul></ul><ul><ul><li>Find the remaining upper class limits. </li></ul></ul><ul><li>Tally the data </li></ul><ul><li>Count the tally marks to find the total frequency for each class </li></ul>
- 8. Example: Constructing a Frequency Distribution <ul><li>The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes. </li></ul><ul><li>50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44 </li></ul>Larson/Farber 4th ed.
- 9. Solution: Constructing a Frequency Distribution <ul><li>Number of classes = 7 (given) </li></ul><ul><li>Find the class width </li></ul>Larson/Farber 4th ed. Round up to 12 50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44
- 10. Solution: Constructing a Frequency Distribution Larson/Farber 4th ed. Class width = 12 <ul><li>Find the class limits: </li></ul><ul><li>Use 7 (minimum value) as first lower limit. Add the class width of 12 to get the lower limit of the next class. </li></ul><ul><li>7 + 12 = 19 </li></ul><ul><li>Find the remaining lower limits. </li></ul>19 31 43 55 67 79 Lower limit Upper limit 7
- 11. Solution: Constructing a Frequency Distribution <ul><li>The upper limit of the first class is 18 (one less than the lower limit of the second class). </li></ul><ul><li>Add the class width of 12 to get the upper limit of the next class. </li></ul><ul><li>18 + 12 = 30 </li></ul><ul><li>Find the remaining upper limits. </li></ul>Larson/Farber 4th ed. Class width = 12 30 42 54 66 78 90 18 Lower limit Upper limit 7 19 31 43 55 67 79
- 12. Solution: Constructing a Frequency Distribution <ul><li>Make a tally mark for each data entry in the row of the appropriate class. </li></ul><ul><li>Count the tally marks to find the total frequency f for each class. </li></ul>Larson/Farber 4th ed. Σ f = 50 Class Tally Frequency, f 7 – 18 IIII I 6 19 – 30 IIII IIII 10 31 – 42 IIII IIII III 13 43 – 54 IIII III 8 55 – 66 IIII 5 67 – 78 IIII I 6 79 – 90 II 2
- 13. Example: Constructing a Frequency Distribution <ul><li>The following represents census data reporting the ages of the entire population of the 77 resdients of Akhiok, Alaska. Construct a frequency distribution with 6 classes. </li></ul><ul><li>28 6 17 48 63 47 27 21 3 7 12 39 50 54 33 45 15 24 1 7 36 53 46 27 5 10 32 50 52 11 42 22 3 17 34 56 25 2 30 10 33 1 49 13 16 8 31 21 6 9 2 11 32 25 0 55 23 41 29 4 51 1 6 31 5 5 11 4 10 26 12 6 16 8 2 4 28 </li></ul>Larson/Farber 4th ed.
- 14. Expanding the Frequency Distribution <ul><li>There are additional features that we can add to the frequency distribution that will provide a better understanding of the data </li></ul><ul><ul><li>Midpoint, relative frequency, and cumulative frequency </li></ul></ul>
- 15. Midpoint <ul><li>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 . </li></ul><ul><ul><li>Note: After you find one midpoint, you can find the following midpoints by adding the class width to the previous midpoint </li></ul></ul>
- 16. Relative Frequency <ul><li>The relative frequency of a class is the portion or percent of the data that falls in that class. </li></ul><ul><ul><li>To find the relative frequency of a class, divide the frequency, f , by the sample size, n . </li></ul></ul>Note: Relative frequency can be written as a decimal or as a percent.
- 17. Cumulative Frequency <ul><li>The cumulative frequency of a class is the sum of the frequency for that class and all previous classes. </li></ul><ul><ul><li>The cumulative frequency of the last class is equal to the sample size. </li></ul></ul>
- 18. Example: Midpoints, Relative and Cumulative Frequencies <ul><li>Using the frequency distribution, find the midpoint, relative frequency, and cumulative frequency. </li></ul>2 79 – 90 6 67 – 78 5 55 – 66 8 43 – 54 13 31 – 42 10 19 – 30 6 7 – 18 Frequency, f Class
- 19. Solution: Midpoints, Relative and Cumulative Frequencies Expanded Frequency Distribution Larson/Farber 4th ed. Σ f = 50 Class Frequency, f Midpoint Relative frequency Cumulative frequency 7 – 18 6 12.5 0.12 6 19 – 30 10 24.5 0.20 16 31 – 42 13 36.5 0.26 29 43 – 54 8 48.5 0.16 37 55 – 66 5 60.5 0.10 42 67 – 78 6 72.5 0.12 48 79 – 90 2 84.5 0.04 50
- 20. Example: Midpoints, Relative and Cumulative Frequencies <ul><li>Using the frequency distribution, find the midpoint, relative frequency, and cumulative frequency. </li></ul>3 55 – 65 11 44 – 54 7 33 – 43 16 22 – 32 13 11 – 21 5 0 – 10 Frequency, f Class
- 21. Solution: Midpoints, Relative and Cumulative Frequencies Expanded Frequency Distribution 55 – 65 44 – 54 33 – 43 22 – 32 11 – 21 0 – 10 Class Σ f = 50 3 11 7 16 13 27 Frequency, f Σ ≈ 1 77 0.0390 60 74 0.1429 49 63 0.0909 38 56 0.2078 27 40 0.1688 16 27 0.3506 5 Cumulative Frequency Relative Frequency Midpoint
- 22. Homework <ul><li>P41 #1, 3 – 6, 15, 16 </li></ul>

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75,35,41,41,16;28,75,45,55,25, 41,45,37,28,75,82,55,61,75,19,75,61, 19,28,19,61,28,25,16,16.

Construct an expanded frequency distribution table with seven classes