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CP Math Lesson 12-6

CP Math Lesson 12-6. Measures of Spread. Quiz 11-1. 1. Find the mean of the following data. { 11, 3, 6, 2, 9, 4}. 2. Find the median of the following data. { 10, 15, 6, 27, 9}Ums. 3. Find the range of the following data. { 16, 12, 14, 17, 14}.

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CP Math Lesson 12-6

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  1. CP MathLesson 12-6 Measures of Spread

  2. Quiz 11-1 1. Find the mean of the following data. { 11, 3, 6, 2, 9, 4} 2. Find the median of the following data. { 10, 15, 6, 27, 9}Ums 3. Find the range of the following data. { 16, 12, 14, 17, 14} 4. Find the mode of the following data. { 16, 12, 14, 17, 14, 17, 16, 14}

  3. Frequency Distribution curves Standarddeviation: a measurement of spread of the data from the mean. • Smaller standard deviation • higher peak in the middle • less spread. Larger standard deviation  wider the spread of the data  Greater spread

  4. 34% 34% 13.5% 13.5% 2.25% 2.25% Data Distribution Standard deviation a number that describes the spread of the data. Standard deviation 68% of the data will be within one standard deviation of the mean. probability of a data point being within two standard deviations of the mean. = 13.5 + 13.5 + 34 + 34 = 95% probability of a data point being within three standard deviations of the mean. mean = 68 + 27 + 4.5 = 99.5 %

  5. Your turn: 1. What is the name of the variable that combines (1) The height of the peak and (2) spread of the data.: “standard deviation” 2. What do we call the number that is the center of the peak? “mean” http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html Go to: http://www.shodor.org/interactivate/activities/NormalDistribution

  6. 34% 34% 13.5% 13.5% 2.25% 2.25% Probability that an event is more than 2 standard deviations ABOVE the mean?

  7. 34% 34% 13.5% 13.5% 2.25% 2.25% Probability that an event is between the mean and 2 stddevABOVE the mean?

  8. 34% 34% 13.5% 13.5% 2.25% 2.25% Your turn: 3. What is the probability that a data point will be between one standard deviation and 2 standard deviations below the mean? 4. What is the probability that a data point will be within 2 standard deviations of the mean?

  9. 34% 34% 13.5% 13.5% 2.25% 2.25% Probabilities and Standard Deviation The standard deviation for some data is 10. The mean of the data is 50 What is the probability that the data falls between 40 and 60? 68% What is the probability that the data falls between 50 and 70? = 34% + 13.5% = 47.5% Is it very likely that a data point falls above 90? 30 40 50 60 70

  10. Your turn: 5. The standard deviation for some data is 7. The mean for this data is 42. Draw a bell curve and label the x-axis up to 3 standard deviations above and below the mean. 6. What is the probability that a data point will be in the range between 28 and 42? 7. What is the probability that a data point will be in the range between 21 and 28?

  11. Standard Deviation Standarddeviation: a measurement of spread of the data from the mean. Step one: calculate the mean ( ). Step two: subtract the mean from each data point.

  12. Standard Deviation Standarddeviation: a measurement of spread of the data from the mean. Step four: add the last row togetther. Step three: square each of the terms Step five: divide by the number of data points: Step six: take the square root of the number.

  13. Your turn: 8. Find the standard deviation for the test scores:

  14. Two different Standard Deviations 1. Data set is all of the data (the “population”) Example: test scores of a single class 2. Data set is just a sample of a HUGE data Example: the opinions of 1000 individuals randomly selected

  15. Let’s use the calculator Enter the data into a list  “Stat” then “edit” option. Scroll until the cursor has “L1” highlighted, then clear the list (don’t delete it). Enter the data into list “L1”. “stat” then scroll over to “calc” Tell the calculator where the data is :“2nd” “1” (List 1) Option 1: 1-var stats then “enter”

  16. Let’s use the calculator Mean: Standard Deviation: Number of data points: Minimum data point: Q1 (we’ll talk about this later) Median: Q3 (we’ll talk about this later) Maximum data point:

  17. Let’s use the calculator Go to the catalog (“2nd then “0” buttons) Find “s” (“ln” button) Then scroll down to “stDev(“ Hit “enter” button Left bracket (“2nd” and “(“ buttons), enter the data with commas, (“2nd” and “)” buttons) This gives the standard deviation of the “population” if the data is just a “sample”.

  18. Your turn: 9. Find the standard deviation using the “stDev” function on your calculator for the following data. (this data is the entire population of the data set, it not just a sample)

  19. Quartiles: are medians 90 = Quartile 3 Third Quartile: the median of the upper half of the data. 75 = Quartile 2 Second Quartile: the median of the data 55 = Quartile 1 First Quartile: the median of the lower half of the data.

  20. Inter-Quartile range: the distance between quartile 1 and quartile 3. Vocabulary 90 = Quartile 3 75 = Quartile 2 Inter-quartile range = 35 55 = Quartile 1

  21. Vocabulary First Quartile: the median of the lower half of the data. Second Quartile: the median of the data Third Quartile: the median upper half of the data Inter-Quartile range:

  22. Quartiles Median = quartile 2 • Even number of data points: • Must average the middle two points. • Top half will be an odd # of points • Bottom half will be an odd # of points 22 terms 75 = Quartile 2

  23. Quartiles Quartile 1 = median of the upper half of the data points. Find the center point of the data 11 terms 90 = Quartile 3 How do you find the median of a data set with an odd number of data points?

  24. Quartiles Quartile 3 = median of the lower half of the data points. How do you find the median of a data set with an odd number of data points? Find the center point of the data 11 terms 55 = Quartile 1

  25. Your turn: For the following data set, find: 10. First Quartile 11. Second Quartile 12. Third Quartile 13. Inter-Quartile range

  26. Use the calculator Number of data points: Minimum data point: Q1 (we’ll talk about this later) Median: Q3 (we’ll talk about this later) Maximum data point:

  27. Your turn: Use the “power of the calculator” to find the following statistics for the data set given. 14. First Quartile 15. Second Quartile 16. Third Quartile 17. Inter-Quartile range

  28. Quartiles Box Plot: visually shows: 110 100 90 80 70 60 50 40 30 20 10 0

  29. Quartiles Box Plot: visually shows: 110 100 90 80 70 60 50 40 30 20 10 0 Inter-Quartile range: Range

  30. Using the data from problems 3 – 5, build a box plot next to the number line that shows: Your turn: 110 100 90 80 70 60 50 40 30 20 10 0 18. Largest data point. 19. Third Quartile 20. Second Quartile 21. First Quartile 22. Smallest data point 23. Are there any “outliers” ? If so, what are their values?

  31. Box Plots: Allow visual comparisons of the data. 110 100 90 80 70 60 50 40 30 20 10 0

  32. 10 14 18 22 26 30 34 38 42 46 50 • 24. What is the range of the data? • 25. Lower quartile = ? • 26. Upper quartile = ? • 27. Median = ? • 28. Interquartile range = ? • 29. Are there any outlier(s). • If so, what are they ? 46 – 10 = 36 Lower quartile: 14 Upper quartile: 22 Median: 20 Interquartile range: 8 Yes; 46 is an outlier

  33. Skewed right Skewed left symetrical

  34. HOMEWORK Section 11-1 12, 14, 26 (find standard deviation only) Section 11-3 4, 6, 12, 14, 32a, 32b Page 1009: problems: 7, 8, 14 – 19 (all) 18 problems

  35. the 65th percentile can be defined as the lowest score that is greater than 65% of the scores. This is the way we defined it above and we will call this "Definition 1." The 65th percentile can also be defined as the smallest score that is greater than or equal to 65% of the scores. This we will call "Definition 2." Unfortunately, these two definitions can lead to dramatically different results, especially when there is relatively little data. Moreover, neither of these definitions is explicit about how to handle rounding A third way to compute percentiles (presented below), is a weighted average of the percentiles computed according to the first two definitions. This third definition handles rounding more gracefully than the other two and has the advantage that it allows the median (discussed in Chapter 3) to be defined conveniently as the 50th percentile.

  36. HOMEWORK • Section 11-1 • 2-8, 12, 14, 18-22, 26, 32 • (for 12 & 14: find range only) • (for 18-22: find the outlier only) • (for 26: don’t find standard deviation) Section 7-6: 18, 20, 42, 44 15 problems

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