1. Introduction Previous lessons have demonstrated that the normal distribution provides a useful model for many situations in business and industry, as well as in the physical and social sciences. Determining whether or not it is appropriate to use normal distributions in calculating probabilities is an important skill to learn, and one that will be discussed in this lesson. 1 1.1.3: Assessing Normality

2. Introduction, continued There are many methods to assess a data set for normality. Some can be calculated without a great deal of effort, while others require advanced techniques and sophisticated software. Here, we will focus on three useful methods: Rules of thumb using the properties of the standard normal distribution (including symmetry and the 68–95–99.7 rule). Visual inspection of histograms for symmetry, clustering of values, and outliers. Use of normal probability plots. 2 1.1.3: Assessing Normality

3. Introduction, continued With advances in technology, it is now more efficient to calculate probabilities based on normal distributions. With our new understanding of a few important concepts, we will be ready to conduct research that was formerly reserved for a small percentage of people in society. 3 1.1.3: Assessing Normality

4. Key Concepts Although the normal distribution has a wide range of useful applications, it is crucial to assess a distribution for normality before using the probabilities associated with normal distributions. Assessing a distribution for normality requires evaluating the distribution’s four key components: a sample or population size, a sketch of the overall shape of the distribution, a measure of average (or central tendency), and a measure of variation. It is difficult to assess normality in a distribution without a proper sample size. When possible, a sample with more than 30 items should be used. 4 1.1.3: Assessing Normality

5. Key Concepts, continued Outliers are values far above or below other values of a distribution. The use of mean and standard deviation is inappropriate for distributions with outliers. Probabilities based on normal distributions are unreliable for data sets that contain outliers. Some outliers, like those caused by mistakes in data entry, can be eliminated from a data set before a statistical analysis is performed. 5 1.1.3: Assessing Normality

6. Key Concepts, continued Other outliers must be considered on a case-by-case basis. Histograms and other graphs provide more efficient methods to assess the normality of a distribution. If a histogram is approximately symmetric with a concentration of values near the mean, then using a normal distribution is reasonable (assuming there are no outliers). 6 1.1.3: Assessing Normality

7. 7 Key Concepts, continued If a histogram has most of its weight on the right side of the graph with a long “tail” of isolated, spread-out data points to the left of the median, the distribution is said to be skewed to the left, or negatively skewed: In a negatively skewed distribution, the mean is often, but not always, less than the median.

8. Key Concepts, continued If a histogram has most of its weight on the left side of the graph with a long tail on the right side of the graph, the distribution is said to be skewed to the right, or positively skewed: 8 1.1.3: Assessing Normality

9. Key Concepts, continued In a positively skewed distribution, the mean is often, but not always, greater than the median. Histograms should contain between 5 and 20 categories of data, including categories with frequencies of 0 (empty catagories). Recall that the 68–95–99.7 rule, also known as the Empirical Rule, states percentages of data under the normal curve are as follows:,, and. 9 1.1.3: Assessing Normality

10. Key Concepts, continued The 68–95–99.7 rule can also be used for a quick assessment of normality. For example, in a sample with less than 100 items, obtaining a z-score below –3.0 or above +3.0 indicates possible outliers or skew. Graphing calculators and computers can be used to construct normal probability plots, which are a more advanced system for assessing normality. In a normal probability plot, the z-scores in a data set are paired with their corresponding x-values. 10 1.1.3: Assessing Normality

11. Key Concepts, continued If the points in the normal plot are approximately linear with no systematic pattern of values above and below the line of best fit, then it is reasonable to assume that the data set is normally distributed. 11 1.1.3: Assessing Normality

12. Common Errors/Misconceptions treating a data set that has outliers as if it were a normal distribution removing outliers without justification adhering too strictly to the rules of thumb for assessing normality deeming a distribution as normal when it is actually skewed left or right 12 1.1.3: Assessing Normality

13. 13 1.1.3: Assessing Normality

14. 14 1.1.3: Assessing Normality

15. So we now can calculate the z-scores of the upper bound and the lower bound… 15 1.1.3: Assessing Normality

16. Then table lookup… *Percent of data from z-score of the upper bound:_______ *Percent of data from the z-score of the lower- bound:_______ Then subtract the percent of the lower bound from the percent of the upper bound: 16 1.1.3: Assessing Normality

17. *Is the percent of students within one standard deviation of the mean close to 68%? *Ok, now do this for 2 standard deviations from the mean….and then 3 standard deviations from the mean... *Does the percent of students within 2 standard deviations from the mean fall close to 95%? *Does the percent of students within 3 standard deviations from the mean fall close to 99.7%? What can you conclude from your findings? IOW…Is the data normally distributed? 17 1.1.3: Assessing Normality

18. Guided Practice Example 2 In order to constantly improve instruction, Mr. Hoople keeps careful records on how his students perform on exams. The histogram on the next slide displays the grades of 40 students on a recent United States history test. The table next to it summarizes some of the characteristics of the data. Use the properties of a normal distribution to determine if a normal distribution is an appropriate model for the grades on this test. 18 1.1.3: Assessing Normality

19. 19 1.1.3: Assessing Normality Summary statistics n40 μ80.5 Median85 σ18.1 Minimum0 Maximum98 Recent U.S. History Test Scores Test score Number of students

20. Guided Practice: Example 2, continued 1.Analyze the histogram for symmetry and concentration of values. The histogram is asymmetric; there is a skew to the left (or a negative skew). The mean is 85.0 – 80.5 = 4.5 less than the median. Also, there appears to be a higher concentration of values above the mean (80.5) than below the mean. 20 1.1.3: Assessing Normality

21. Guided Practice: Example 2, continued 2.Examine the distribution for outliers and evaluate their significance, if any outliers exist. There is one negative outlier (0) on this test. There may be outside factors that affected this student’s performance on the test, such as illness or lack of preparation. 21 1.1.3: Assessing Normality

22. Guided Practice: Example 2, continued 3.Determine whether a normal distribution is an appropriate model for this data. 22 1.1.3: Assessing Normality

23. Guided Practice: Example 2, continued 3.Determine whether a normal distribution is an appropriate model for this data. Because of the outlier, the normal distribution is not an appropriate model for this population. 23 1.1.3: Assessing Normality ✔

24. Guided Practice: Example 2, continued 24 1.1.3: Assessing Normality

25. 25 1.1.3: Assessing Normality Example 3 Rent at the Cedar Creek apartment complex includes all utilities, including water. The operations manager at the complex monitors the daily water usage of its residents. The following table shows water usage, in gallons, for residents of 36 apartments. To better assess the data, the manager sorted the values from lowest to highest. Does the data show an approximate normal distribution?

26. 26 1.1.3: Assessing Normality

27. 1 – Create a histogram… How many categories? Lets make an educated choice…. What is the range of the data? Range = Maximum Value – Minimum Value Range = 431 – 181 = 250 36 data points (apartments) span this range Lets make 6 categories. 27 1.1.3: Assessing Normality

28. 2 – What’s the category width? 28 1.1.3: Assessing Normality

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30. 3 – Create a frequency table… 30 1.1.3: Assessing Normality

31. 3 – Create a frequency table… 31 1.1.3: Assessing Normality

32. 4 – Sketch a graph of this table… 32 1.1.3: Assessing Normality

33. 5 – Describe the overall shape of the distribution. Is there any kind of skew? Where is the highest concentration of values? Are there any outliers? 33 1.1.3: Assessing Normality

34. 6 – Drawing Conclusions… Statistical analysis require your judgment. Can you assume normality? Things to think about: What is the context of the problem? What will the calculations be used for? Is the data to be used to make an informed decision? OR Is it life impacting? 34 1.1.3: Assessing Normality

35. 6 – Drawing Conclusions, Continued… In Example 3… Why is one apartment using more water than another? Number of residents? Do they have a washer, dish washer? Do we know any of this information? Can we assume normality? 35 1.1.3: Assessing Normality

36. Guided Practice Example 4 Use a graphing calculator to construct a normal probability plot of the following values. Do the data appear to come from a normal distribution? {1, 2, 4, 8, 16, 32} 36 1.1.3: Assessing Normality

37. Guided Practice: Example 4, continued 1.Use a graphing calculator or computer software to obtain a normal probability plot. Different graphing calculators and computer software will produce different graphs; however, the following directions can be used with TI-83/84 or TI-Nspire calculators. 37 1.1.3: Assessing Normality

38. Guided Practice: Example 4, continued On a TI-83/84: Step 1: Press [STAT] to bring up the statistics menu. The first option, 1: Edit, will already be highlighted. Press [ENTER]. Step 2: Arrow up to L1 and press [CLEAR], then [ENTER], to clear the list. Repeat this process to clear L2 and L3 if needed. Step 3: From L1, press the down arrow to move your cursor into the list. Enter each number from the data set, pressing [ENTER] after each number to navigate down to the next blank spot in the list. 38 1.1.3: Assessing Normality

39. Guided Practice: Example 4, continued Step 4: Press [Y=]. Press [CLEAR] to delete any equations. Step 5: Set the viewing window by pressing [WINDOW]. Enter the following values, using the arrow keys to navigate between fields and [CLEAR] to delete any existing values: Xmin = 0, Xmax = 35, Xscl = 5, Ymin = –3, Ymax = 3, Yscl = 1, and Xres = 1. Step 6: Press [2ND][Y=] to bring up the STAT PLOTS menu. 39 1.1.3: Assessing Normality

40. Guided Practice: Example 4, continued Step 7: The first option, Plot 1, will already be highlighted. Press [ENTER]. Step 8: Under Plot 1, press [ENTER] to select “On” if it isn’t selected already. Arrow down to “Type,” then arrow right to the normal probability plot icon (the last of the six icons shown) and press [ENTER]. Step 9: Press [GRAPH]. 40 1.1.3: Assessing Normality

41. Guided Practice: Example 4, continued On a TI-Nspire: Step 1: Press the [home] key. Step 2: Arrow over to the spreadsheet icon and press [enter]. Step 3: The cursor will be in the first cell of the first column. Enter each number from the data set, pressing [enter] after each number to navigate down to the next blank cell. 41 1.1.3: Assessing Normality

42. Guided Practice: Example 4, continued Step 4: Arrow up to the topmost cell of the column, labeled “A.” Name the column “exp1” using the letters and numbers on your keypad. Press [enter]. Step 5: Press the [home] key. Arrow over to the data and statistics icon and press [enter]. Step 6: Press the [menu] key. Arrow down to 2: Plot Properties, then arrow right to bring up the sub-menu. Arrow down to 4: Add X Variable, if it isn’t already highlighted. Press [enter]. 42 1.1.3: Assessing Normality

43. Guided Practice: Example 4, continued Step 7: Arrow down to {…}exp1 if it isn’t already highlighted. Press [enter]. This will graph the data values along an x-axis. Step 8: Press [menu]. The first option, 1: Plot Type, will be highlighted. Arrow right to bring up the next sub-menu. Arrow down to 4: Normal Probability Plot. Press [enter]. 43 1.1.3: Assessing Normality

44. Guided Practice: Example 4, continued Your graph should show the general shape of the plot as follows. 44 1.1.3: Assessing Normality

45. Guided Practice: Example 4, continued 2.Analyze the graph to determine whether it follows a normal distribution. Do the points lie close to a straight line? If the data lies close to the line, is roughly linear, and does not deviate from the line of best fit with any systematic pattern, then the data can be assumed to be normally distributed. If any of these criteria are not met, then normality cannot be assumed. 45 1.1.3: Assessing Normality

46. Guided Practice: Example 4, continued The data does not lie close to the line; the data is not roughly linear. The data seems to curve about the line, which suggests a pattern. Therefore, normality cannot be assumed. The normal distribution is not an appropriate model for this data set. 46 1.1.3: Assessing Normality ✔

47. Guided Practice: Example 4, continued 47 1.1.3: Assessing Normality