1 / 51

Chapter 3

Chapter 3. Averages and Variation. Table of Contents. 3.1 Measures of Central Tendency: Mode, Median, and Mean 3.2 Measures of Variation 3.3 Percentiles and Box-and-Whisker Plots. 3.1 Measures of Central Tendency: Mode, Median, and Mean. Definition.

art
Download Presentation

Chapter 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 3 Averages and Variation

  2. Table of Contents 3.1 Measures of Central Tendency: Mode, Median, and Mean 3.2 Measures of Variation 3.3 Percentiles and Box-and-Whisker Plots

  3. 3.1 Measures of Central Tendency: Mode, Median, and Mean Definition The mode of a data set is the value that occurs most frequently.

  4. Example 1 5 3 7 2 4 4 2 4 8 3 4 3 4 ‘4’ occurs most frequently in this data set, so it is the mode. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  5. Definition The median is the central value of an ordered distribution. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  6. Sum of middle two values Median = 2 How to find the median 1. Order the data from smallest to largest. 2. For an odd number of data values in the distribution, Median = Middle data value 3. For an even number of data values in the distribution, 3.1 Measures of Central Tendency: Mode, Median, and Mean

  7. Calculator Function We will be using the STAT|CALC|1­Var StatsList function, which returns the results of many different calculations on the given List. But first, we must edit a list. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  8. Example 2a 1. Enter the data into a list. [ENTER] Hit STAT|EDIT|1:Edit… [STAT] Button Menu Menu Item 3.1 Measures of Central Tendency: Mode, Median, and Mean

  9. Example 2a 1. Enter the data into a list. Hit STAT|EDIT|1:Edit… If you are not in List 1 (L1), use the left arrow [◄] to move to List 1. If List 1 is not empty, use the up arrow [▲] to move into List 1’s header (L1), which will highlight the header. Then press [CLEAR][ENTER]. WARNING: DO NOT PRESS [DEL][ENTER] INSTEAD OF [CLEAR][ENTER]! 3.1 Measures of Central Tendency: Mode, Median, and Mean

  10. Example 2a 1. Enter the data into a list. Hit STAT|EDIT|1:Edit… Enter data from Example 2 part (a) into the list by typing each value and pressing [ENTER] after each value. After completing the list, press [2nd][MODE] to QUIT out of the list editor. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  11. Example 2a 1. Enter the data into a list. Hit STAT|EDIT|1:Edit… Enter data into a list. 2. Hit STAT|CALC|1:1-Var Stats [ENTER] [STAT] Button Menu Item Use the right arrow [ ►] to select the CALC Menu 3.1 Measures of Central Tendency: Mode, Median, and Mean

  12. Example 2a 1. Enter the data into a list. Hit STAT|EDIT|1:Edit… Enter data into a list. 2. Hit STAT|CALC|1:1-Var Stats [ENTER] [ENTER] 1-Var Stats L1 Hit [2nd][1]. We now need to tell the function which list to process. Since our data is in L1, we will tell it to process L1. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  13. 1-Var Stats x=24.33333333 Σx=146 Σx²=3784 Sx=6.801960502 σx=6.209312003 ↓n=6 ¯ Example 2a The TI-83/84 should now display 1-Var Stats ↑n=6 minX=18 Q1=19 Med=23 Q3=28 maxX=35 Use the down arrow [ ▼] to scroll to the bottom of the output. The median is 23. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  14. Example 2b We can now use the TI-83/84 to do part (b) as well. Instead of entering a new list, we will simply remove the ‘35’ from L1. 1. Hit STAT|EDIT|Edit… 2. Use the down arrow [ ▼] to scroll the highlight down to the ‘35’. 3. Hit [DEL] 4. QUIT out of the list editor. 5. Run STAT|CALC|1-Var Stats L1 6. Scroll down until you can see the result for the median, which is 19. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  15. Recap Remember this: You clear[CLEAR] a list; you delete[DEL] an element from a list. If you [DEL] a list, then the entire list will become unavailable in the list editor. What to do if you accidentally delete a list Lists can be easily restored if they are deleted. Just select STAT|EDIT|5:SetUpEditor . After you select it (either by pressing [5] or by scrolling down to it and pressing [ENTER]), it will appear in the main screen as a command waiting to be run. Press [ENTER] to run the SetUpEditor command, and your list will be restored. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  16. Calculator Function Although the TI-83/84 orders lists internally when determining the median, it is sometimes convenient for us to have the list itself ordered. The STAT|EDIT|SortA function sorts a list in ascending order (starting with the lowest value and progressing to the highest). The STAT|EDIT|SortD function sorts a list in descending order (starting with the highest value and progressing to the lowest). We won’t have a need for SortD in this class. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  17. Calculator Function For example, let’s say we want to order the list given in Example 2. Select the STAT|EDIT|SortA function. [ENTER] L1 ) SortA( We will enter L1 because that is the list we want to sort. We can now view L1 in the list editor by going to STAT|EDIT|Edit… . 3.1 Measures of Central Tendency: Mode, Median, and Mean

  18. n + 1 Position of the middle value = 2 Formula for finding the position of the middle value For an ordered data set of size n, For an ordered data set of size 99, (99+1)/2 = 50, so the 50th data value is the middle value. For an ordered data set of size 100, (100+1) / 2 = 50.5, so the 50th and 51st data values are the middle values. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  19. Sum of all entries Mean = Number of entries Definition This is what most people think of when they hear the word ‘average’. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  20. 1-Var Stats x=80 Σx=560 Σx²=46762 Sx=18.08314132 σx=16.7417357 ↓n=7 ¯ Example 3 Let’s use the TI-83/84 to do Example 3. 1. Enter data into L2. 2. QUIT out of the list editor. 3. Run 1-Var Stats L2 The TI-83/84 should now display The mean is 80. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  21. Σx ¯ Sample mean = x = n Σx Population mean = μ = N Definitions Sample statistics: Sample size = n Population parameters: Population size = N 3.1 Measures of Central Tendency: Mode, Median, and Mean

  22. How to compute a 5% trimmed mean 1. Order the data from smallest to largest. 2. Delete the bottom 5% of the data and the top 5% of the data. Note: If the calculation of 5% of the number of data values does not produce a whole number, round to the nearest integer. 3. Compute the mean of the remaining 90% of the data. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  23. Σxw Weighted average = Σw Definition where x is a data value and w is the weight assigned to that data value. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  24. How to find the weighted average using the TI­83/84 To find the weighted average using the TI­83/84, we must first enter both an x-List and a weight-List. Then we use the function 1-Var Stats x-List,weight-List In Example 4, we could enter the test scores into L5 and their respective weights into L6. Then we could run 1-Var Stats L5,L6. 3.1 Measures of Central Tendency: Mode, Median, and Mean

  25. 3.2 Measures of Variation Definition The range is the difference between the largest and smallest values of a data distribution.

  26. Σ(x − x)2 ¯ Sample variance = s2 = n− 1 Σ(x − x)2 ¯ Sample standard deviation = s = n− 1 Definitions Sample statistics: 3.2 Measures of Variation

  27. 1-Var Stats x=6 Σx=36 Σx²=286 Sx=3.741657387 σx=3.415650255 ↓n=6 ¯ Example 6 Let’s enter our data into L1. If there is already data in L1, we will have to first clear the list before entering the data. Now we can enter 1-Var Stats L1. The display should now show The sample standard deviation is s = 3.74. The sample variance is s2 = 3.742≈ 14. Use the [x2] button 3.2 Measures of Variation

  28. Σ(x − x)2 ¯ Population variance = σ2 = N Σ(x − x)2 ¯ Pop. standard deviation = σ = N Definitions Population parameters: 3.2 Measures of Variation

  29. 1-Var Stats x=6 Σx=36 Σx²=286 Sx=3.741657387 σx=3.415650255 ↓n=6 ¯ If the list represents a sample, then x = 6 and s = 3.74. ¯ Note The TI-83/84 cannot know whether the data entered into a list represents a sample or a population. E.g., let’s use the data from Example 6. Running 1-Var Stats L1 would still yield If the list represents a census, then μ= 6 and σ = 3.42. Data can either be statistical or parametric, but NEVER BOTH.

  30. If x and s represent the sample mean and sample standard deviation, respectively, then the sample coefficient of variation CV is defined to be ¯ s CV = · 100% ¯ x σ CV = · 100% μ Definitions If μ and σ represent the sample mean and sample standard deviation, respectively, then the sample coefficient of variation CV is defined to be

  31. 1 1 − k2 Chebyshev’s theorem For any set of data (either population or sample) and for any constant k greater than1, the proportion of data that must lie within k standard deviations on either side of the mean is at least 3.2 Measures of Variation

  32. Results of Chebyshev’s theorem For any set of data: ● at least 75% of the data fall in the interval μ– 2σ to μ + 2σ. ● at least 88.9% of the data fall in the interval μ– 3σ to μ + 3σ. ● at least 93.8% of the data fall in the interval μ– 4σ to μ + 4σ. 3.2 Measures of Variation

  33. not guaranteed anything guaranteed at least 75% of data guaranteed at least 88.9% of data Results of Chebyshev’s theorem μ–3σ μ–2σ μ–σ μ μ+σ μ+2σ μ+3σ 3.2 Measures of Variation

  34. Note In practice, you can use x and s as point estimates for μ and σ in Chebyshev’s theorem. ¯ 3.2 Measures of Variation

  35. Note/Warning Your trifold insert does not contain Chebyshev’s theorem. It would be wise to write it in yourself as you are guaranteed to encounter it on a test. 3.2 Measures of Variation

  36. Grouped data The term ‘grouped data’ refers to data sets in the form of classes with corresponding frequencies. Frequency tables and histograms are examples of grouped data. In practice, sometimes grouped data is the only data we have available to us. To estimate the mean and standard deviation of grouped data, we will treat the midpoints of each class as an x-List and the frequencies of each class as a weight-List, and then run the familiar 1-Var Stats x-List,weight-List. 3.2 Measures of Variation

  37. 15 15 13 10 8 Frequency 5 5 4 3 2 Midpoints 1 4 7 10 13 16 19 Grouped Data Example Enter midpoints for the x-List. Enter frequencies for the weight-List. 3.2 Measures of Variation

  38. 15 15 13 10 8 Frequency 5 5 4 3 2 Midpoints 1 4 7 10 13 16 19 x = 10.18, s≈ 4.43 ¯ Grouped Data Example Run 1-Var Stats x-List,weight-List 3.2 Measures of Variation

  39. 3.3 Percentiles and Box­and­Whisker Plots Definition For whole numbers P (where 1 ≤ P ≤ 99), the Pth percentile of a distribution is a value such that P% of the data fall at or below it and (100 – P)% of the data fall at or above it.

  40. 100 Biology Test Scores 40 30 Frequency 20 10 Score 39.5 49.5 59.5 69.5 79.5 89.5 99.5 Figure 3-3 60% of scores at or below 40% of scores at or above 60th Percentile 3.3 Percentiles and Box-and-Whisker Plots

  41. 1% 1% 1% 1% 1% 1% 1% 1% … Lowest 1st 2nd 3rd 4th 5th 98th 99th Highest Percentiles Figure 3-4 In an ideal situation, there are 99 percentiles which divide the data set evenly into 100 equal parts. If the size of a data set is not divisible by 100, then the percentiles will not divide the data evenly. We will not be concerned with such situations. Instead, we will focus on quartiles.

  42. 25% 25% 25% 25% Lowest Highest Definitions The first quartile, Q1, is the 25th percentile. The second quartile, Q2, is the 50th percentile, which is the same as the median. The third quartile, Q3, is the 75th percentile. Figure 3-5 Q1 Q2 Median 50th Percentile Q3 3.3 Percentiles and Box-and-Whisker Plots

  43. How to compute quartiles 1. Order the data from smallest to largest. 2. Find the median. This is the second quartile Q2. 3. The first quartile Q1 is then the median of the lower half of the data; i.e., it is the median of the data falling below the Q2 position (and not including Q2). 4. The third quartile Q3 is then the median of the upper half of the data; i.e., it is the median of the data falling above the Q2 position (and not including Q2). 3.3 Percentiles and Box-and-Whisker Plots

  44. Calculator Function The TI-83/84 can compute quartiles with its 1­Var Stats x-List function. 3.3 Percentiles and Box-and-Whisker Plots

  45. Definition Interquartile range = Q3− Q1 3.3 Percentiles and Box-and-Whisker Plots

  46. 1-Var Stats x=.6311111111 Σx=17.04 Σx²=14.0812 Sx=.3577207047 σx=.3510337468 ↓n=27 ¯ Example 9 Let’s enter our data into L3. If there is already data in L3, we will have to first clear the list before entering the data. Now we can enter 1-Var Stats L3. The display should now show 1-Var Stats ↑n=27 minX=.16 Q1=.33 Med=.5 Q3=1 maxX=1.23 Scroll down

  47. Example 9 Let’s enter our data into L3. If there is already data in L3, we will have to first clear the list before entering the data. Now we can enter 1-Var Stats L3. The display should now show 1-Var Stats ↑n=27 minX=.16 Q1=.33 Med=.5 Q3=1 maxX=1.23 Q1 = 0.33 Q2 = 0.50 Q3 = 1.00 IQR = Q3−Q1 =1.00 − 0.33 = 0.67

  48. Definition Five-number summary Lowest value, Q1, median, Q3, highest value 3.3 Percentiles and Box-and-Whisker Plots

  49. Note The 1-Var Stats function gives the five-number summary at the end of its output. 1-Var Stats ↑n=... minX=... Q1=... Med=... Q3=... maxX=... Five-number summary 3.3 Percentiles and Box-and-Whisker Plots

  50. How to make a box-and-whisker plot 1. Draw a vertical scale to include the lowest and highest data values. 2. To the right of the scale, draw a box from Q1 to Q3. 3. Include a solid line through the box at the median level. 4. Draw solid lines, called whiskers, from Q1 to the lowest value and from Q3 to the highest value. 3.3 Percentiles and Box-and-Whisker Plots

More Related