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The Normal Probability Distribution

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The Normal Probability Distribution BIOSTATISTICS BIOL 4343 Assessing normality Not all continuous random variables are normally distributed. It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution. In this section some statistical tools will be presented to check whether a given set of data is normally distributed. 1. Previous knowledge of the nature of the distribution Problem: A researcher working with sea stars needs to know if sea star size (length of radii) is normally distributed. What do we know about the size distributions of sea star populations? 1. Has previous work with this species of sea star shown them to be normally distributed? 2. Has previous work with a closely related species of seas star shown them to be normally distributed? 3. Has previous work with seas stars in general shown them to be normally distributed? If you can answer yes to any of the above questions and you do not have a reason to think your population should be different, you could reasonably assume that your population is also normally distributed and stop here. However, if any previous work has shown non-normal distribution of sea stars you had probably better use other techniques. 2. Construct charts For small- or moderate-sized data sets, the stem-and-leaf display and box-and- whisker plot will look symmetric. For large data sets, construct a histogram or polygon and see if the distribution bell-shaped or deviates grossly from a bell-shaped normal distribution. Look for skewness and asymmetry. Look for gaps in the distribution – intervals with no observations. However, remember that normality requires more than just symmetry; the fact that the histogram is symmetric does not mean that the data come from a normal distribution. Also, data sampled from normal distribution will sometimes look distinctly different from the parent distribution. So, we need to develop some techniques that allow us to determine if data are significantly different from a normal distribution. 3. Normal Counts method Count the number of observations within 1, 2, and 3 standard deviations of the mean and compare the results with what is expected for a normal distribution in the 68-95- 99.7 rule. According to the rule, 68% of the observations lie within one standard deviation of the mean. 95% of observations within two standard deviations of the mean. 99.7% of observations within three standard deviations of the mean. Example: As part of a demonstration one semester, I collected data on the heights of sample of 25 IUG biostatistics students. These data are presented in the table below. Does the sample shown below have been drawn from normally distributed populations? 66 BIOSTATISTICS BIOL 4343 Table. Heights, in inches, of 25 IUG biostatistics students. 71.0 69.0 70.0 72.5 73.0 70.0 71.5 70.5 72.0 71.0 68.5 69.0 69.0 68.5 74.0 67.0 69.0 71.5 66.0 70.0 68.5 74.0 74.5 74.0 Solution: For normal Counts method, determine the following Heights, in Frequency inches 66 1 67 1 68.5 3 69 4 70 3 70.5 1 71 2 Total = 17 71.5 2 72 1 72.5 1 73 1 74 3 74.5 1 Total 24 x = 70.6; s = 2.3 xs is 72.9 to 68.3. 17 out of the 24 observations i.e. 17/24 = 0.70 = 70% fall within , i.e. between 72.9 and 68.3, which is approximately equal to 68%.There is no reason to doubt that the sample is drawn from a normal population. 4. Compute descriptive summary measures a. The mean, median and mode will have similar values. b. The interquartile range approximately equal to 1.33 s. c. The range approximately equal 6 s. 5. Evaluate normal probability plot If the data come from a normal or approximately normal distribution, the plotted points will fall approximately along a straight line (a 45 degree line). However, if your sample departs from normality, the points on the graph will deviate from that line. If they trail off from a straight-line pack in a curve at the “top” end, observed values bigger than expected, that‟s right skewed (see below).If the observed values trail off at the bottom end, that‟s left skewed. Realize that it is important note that any worthwhile computer statistics package will construct these graphs for you (see below). 67 BIOSTATISTICS BIOL 4343 6. Measure of Skewness and Kurtosis Skewness: The normal distribution is symmetrical. Asymmetrical distributions are sometimes called skewed. Skewness is calculated as follows: n 3 n(-) xi x skewness i 1 s3( n - 1)( n - 2) where x is the mean, s is the standard deviation, and n is the number of data points A perfectly normal distribution will have a skewness statistic of zero. If this statistic departs significantly from 0, then we lose confidence that our sample comes from a normally distributed population. If it is negative, then the distribution is skewed to the left or negatively skewed distribution. If it is positive, then the distribution is skewed right or positively skewed distribution. Negatively skewed distribution Normal distribution Positively skewed distribution or Skewed to the left Symmetrical or Skewed to the right Skewness <0 Skewness = 0 Skewness > 0 Kurtosis: A “bell curve” will also depart from normality if the “tails” fail to fall off at the proper rate. If they decrease too fast, the distribution ends up too “peaked.” If they don‟t decrease fast enough, the distribution is too flat in the middle and too fat in the tails. One statistic commonly used to measure kurtosis is typically calculated using the formula, 4 n( n 1)xx 3( n 1)2 kurtosis i (n 1)( n 2)( n 3) s ( n 2)( n 3) where is the mean, s is the standard deviation, and n is the number of data points A perfectly normal distribution will also have a kurtosis statistic of zero. If kurtosis is significantly less than zero, then our distribution is „flat‟, it is said to be platykurtic. If kurtosis is significantly greater than 0, the distribution is „pointed‟ or peaked, it is called leptokurtic. Platykurtic distribution Normal distribution Leptokurtic distribution Low degree of peakedness Mesokurtic distribution High degree of peakedness Kurtosis <0 Kurtosis = 0 Kurtosis > 0 68 BIOSTATISTICS BIOL 4343 You won’t have you calculate it by hand. The calculation itself is sensitive to rounding errors because they are raised to the third and fourth powers. Using SPSS to Evaluate Data for Normality Before the advent of good computers and statistical programs, users could be forgiven for trying to avoid any surplus calculations. Now that both are available and much easier to use, tests for normality should always be carried out as a best practice in statistics. SPSS offers a variety of methods for evaluating normality. Normal probability plot (P-P plot) The P-P plot graphs the expected cumulative probability against the observed cumulative probability. 1. Open the SPSS file containing your data. 2. From the main menu, select Graph and then P-P… From the list of available variables, move the variables you wish to analyze to the variable window. If you select multiple variables then SPSS will create separate plots for each. 3. In the box for Test Distribution be sure that the pop-up menu is set for a Normal distribution. In addition, be sure that the Estimate from data box is checked. 4. In the box for Proportion Estimation Formula, select the radio button for the Rankit method. 5. Finally, in the Ranks Assigned to Ties box, select the radio button for High. 6. Click on OK to obtain the plot and complete your analysis Normal P-P Plot of Student's heights (inches) 1.00 .75 .50 .25 Expected Cum Prob 0.00 0.00 .25 .50 .75 1.00 Observed Cum Prob Q-Q plot Repeat the above steps, but in this time select Q-Q (main menu, Graph and then Q-Q) 69 BIOSTATISTICS BIOL 4343 Normal Q-Q Plot of Student's heights (inches) 2.0 1.5 1.0 .5 0.0 -.5 -1.0 -1.5 Expected Normal -2.0 64 66 68 70 72 74 76 Observed Value Limitation of visual method One limitation to any visual approach for evaluating normality is that your conclusion is open to some uncertainty. How, for example, can you put a quantitative statement on the confidence of your conclusion? How linear is linear and how much deviation from linearity is acceptable? One approach to obtaining a more quantitative determination of whether a data set is normally distributed is the Kolmogorov-Smirnov test or Shapiro- Wilk‟s tests. Use data set to tests of normality 1. Open the SPSS file containing your data and from the main menu select Analyze 2. Descriptive Statistics 3. Explore 4. Move your variable from the “Variable List” window to the “Dependent List” window. 5. Under “Display” click “Both” 6. Click “Plots” 7. Under “Boxplots” check “Factor levels together” 8. Under “Descriptive” check " Histogram" and “stem-and-leaf” 9. Check “Normality plots with tests” 10. Click “continue” 11. Click “OK” 12. Evaluate the plot for evidence of normality. 70 BIOSTATISTICS BIOL 4343 Descriptives Statistic Std. Error Student's Mean 70.583 .4698 heights (inches) 95% Conf idence Lower Bound 69.611 Interv al for Mean Upper Bound 71.555 5% Trimmed Mean 70.616 Median 70.250 Variance 5.297 Std. Dev iation 2.3015 Minimum 66.0 Maximum 74.5 Range 8.5 Interquartile Range 3.375 Skewness .076 .472 Kurtosis -.647 .918 Assessment of skewness and kurtosis In fact (like all estimates), we are unlikely to ever see the values of zero in either skewness or kurtosis statistics.
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