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. 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

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). 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

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: 1.2.3: Assessing Normality

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. 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 . 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

removing outliers without justification 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 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

Recent U.S. History Test Scores Summary statistics n 40 μ 80.5 Median 85 σ 18.1 Minimum Maximum 98 Number of students Test score 1.2.3: Assessing Normality

Guided Practice: Example 2, continued 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. 1.2.3: Assessing Normality

Guided Practice: Example 2, continued 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. 1.2.3: Assessing Normality

Guided Practice: Example 2, continued 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. ✔ 1.2.3: Assessing Normality

Guided Practice: Example 2, continued http://www.walch.com/ei/00625 1.2.3: Assessing Normality

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} 1.2.3: Assessing Normality

Guided Practice: Example 4, continued 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. 1.2.3: Assessing Normality

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. 1.2.3: Assessing Normality

Step 4: Press [Y=]. Press [CLEAR] to delete any equations. 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. 1.2.3: Assessing Normality

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]. 1.2.3: Assessing Normality

Guided Practice: Example 4, continued Your graph should show the general shape of the plot as follows. 1.2.3: Assessing Normality

Guided Practice: Example 4, continued 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. 1.2.3: Assessing Normality

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. ✔ 1.2.3: Assessing Normality

Guided Practice: Example 4, continued http://www.walch.com/ei/00626 1.2.3: Assessing Normality