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Lab #1 -Data Screening7. Normality Below, I describe five steps for determining and dealing with normality. However, the bottom line is that almost no one checks their data for normality; instead they assume normality, and use the statistical tests that are based upon assumptions of normality that have more power (ability to find significant results in the data). First, what is normality? A normal distribution is a symmetric bell-shaped curve defined by two things: the mean (average) and variance (variability). Second, why is normality important? The central idea behind statistical inference is that as sample size increases, distributions will approximate normal. Most statistical tests rely upon the assumption that your data is “normal”. Tests that rely upon the assumption or normality are called parametric tests. If your data is not normal, then you would use statistical tests that do not rely upon the assumption of normality, call nonparametric tests. Non-parametric tests are less powerful than parametric tests, which means the non-parametric tests have less ability to detect real differences or variability in your data. In other words, you want to conduct parametric tests because you want to increase your chances of finding significant results. Third, how do you determine whether data are “normal”？ There are three interrelated approaches to determine normality, and all three should be conducted. First, look at a histogram with the normal curve superimposed. A histogram provides useful graphical representation of the data. SPSS can also superimpose the theoretical “normal” distribution onto the histogram of your data so that you can compare your data to the normal curve. To obtain a histogram with the superimposed normal curve:1. Select Analyze - Descriptive Statistics - Frequencies.2. Move all variables into the “Variable(s)” window.3. Click “Charts”, and click “Histogram, with normal curve”.4. Click OK Output below is for “system1”. Notice the bell-shaped black line superimposed on the distribution. All samples deviate somewhat from normal, so the question is how much deviation from the black line indicates “non-normality”？ Unfortunately, graphical representations like histogram provide no hard-and-fast rules. After you have viewed many (many!) histograms, over time you will get a sense for the normality of data. In my view, the histogram for “system1” shows a fairly normal distribution.HistogramMean =7.32Std. Dev. =2.054N=327system 14 Second, look at the values of Skewness and Kurtosis. Skewness involves the symmetry of the distribution. Skewness that is normal involves a perfectly symmetric distribution. A positively skewed distribution has scores clustered to the left, with the tail extending to the right. A negatively skewed distribution has scores clustered to the right, with the tail extending to the left. Kurtosis involves the peakedness of the distribution. Kurtosis that is normal involves a distribution that is bell-shaped and not too peaked or flat. Positive kurtosis is indicated by a peak. Negative kurtosis is indicated by a flat distribution. Descriptive statistics about skewness and kurtosis can be found by using either the Frequencies, Descriptives, or Explore commands. I like to use the “Explore” command because it provides other useful information about normality, so1. Select Analyze - Descriptive Statistics - Explore.2. Move all variables into the “Variable(s)” window.3. Click “Plots”, and unclick “Stem-and-leaf”4. Click OK- Descriptives box tells you descriptive statistics about the variable, including the value of Skewness and Kurtosis, with accompanying standard error for each. Both Skewness and Kurtosis are 0 in a normal distribution, so the farther away from 0, the more non-normal the distribution. The question is “how much” skew or kurtosis render the data non-normal? This is an arbitrary determination, and sometimes difficult to interpret using the values of Skewness and Kurtosis. Luckily, there are more objective tests of normality, described next.Descript ivesStatisticStd. Errorsys 柜 mlMean7.32.11495% Confidence In柜rv日ILower Bound7.10for MeanUpper Bound7.555% Trimmed Mean7.43Median0.00Variance4.220Std. Deviation2.054Minimum1Maximum11Rang e10Interquartile Range3Skewness-786.135Kurtosis.271.269Third, the descriptive statistics for Skewness and Kurtosis are not as informative as established tests for normality that take into account both Skewness and Kurtosis simultaneously. The Kolmogorov-Smirnov test (K-S) and Shapiro-Wilk (S-W) test are designed to test normality by comparing your data to a normal distribution with the same mean and standard deviation of your sample:1. Select Analyze - Descriptive Statistics - Explore.2. Move all variables into the “Variable(s)” window.3. Click “Plots”, and unclick “Stem-and-leaf”, and click “Normality plots with tests”.4. Click OK-“Test of Normality” box gives the K-S and S-W test results. If the test is NOT significant, then the data are normal, soany value above .05 indicates normality. If the test is significant (less than .05), then the data are non-normal. In this case, both tests indicate the data are non-normal. However, one limitation of the normality tests is that the larger the sample size, the more likely to get significant results. Thus, you may get significant results with only slight deviations from normality. In this case, our sample size is large (n=327) so the significance of the K-S and S-W tests may only indicate slight deviations from normality. You need to eyeball your data (using histograms) to determine for yourself 讦 the data rise to the level of non-normal.Tests of NormalityKolmogorov-SmiShapiro-WilkStatisticdfSia.StatisticdfSia.syst.139327.000.930327.000a. Lilliefors Significance Correction“Normal Q-Q Plot” provides a graphical way to determine the level of normality. The black line indicates the values your sample should adhere to if the distribution was normal. The dots are your actual data. If the dots fall exactly on the black line, then your data are normal. If they deviate from the black line, your data are nonnormal. In this case, you can see substantial deviation from the straight black line.Normal Q-Q Plot of systemlObserved Value Fourth, f your data are non-normal, what are your options to deal with non-normality? You have four basic options.a. Option 1 is to leave your data non-normal, and conduct the parametric tests that rely upon the assumptions of normality. Just because your data are non-normal, does not instantly invalidate the parametric tests. Normality (versus non-normality) is a matter of degrees, not a strict cut-off point. Slight deviations from normality may render the parametric tests only slightly inaccurate. The issue is the degree to which the data are non-normal.b. Option 2 is to leave your data non-normal, and conduct the non-parametric tests designed for nonnormal data.c. Option 3 is to conduct “robust” tests. There is a growing branch of statistics called “robust” tests that are just as powerful as parametric tests but account for non-normality of the data.d. Option 4 is to transform the data. Transforming your data involving using mathematical formulas to modify the data into normality. Fifth, how do you transform your data into “normal” data? There are different types of transformations based upon the type of non-normality. For example, see handout “Figure 8.1” on the last page of this document that shows six types of non-normality (e.g., 3 positive skew that are moderate, substantial, and severe; 3 negative skew that are moderate, substantial, and severe). Figure 8.1 also shows the type of transformation for each type of non-normality. Transforming the data involves using the “Compute” function to create a new variable (the new variable is the old variable transformed by the mathematical formula):1. Select Transform - Compute Variable2. Type the name of the new variable you want to create, such as “transform_system1”.3. Select the type of transformation from the “Functions” list, and double-click.4. Move the (non-normal) variable name into the place of the question mark “?”.5. Click OK. The new variable is reproduced in the last column in the “Data view”. Now, check that the variable is normal by using the tests described above. If the variable is normal, then you can start conducting statistical analyses of that variable. If the variable is non-normal, then try other transformations.
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