In the previous post, we were introduced to the normality test. This time, we will see some phenomena related to this test. Download the following file: (click here) In the file there is a variable named norm200. Using SPSS, display the histogram, normal P-P plot, and box and whisker plot. The following is the SPSS output related to these things.
Figure 1
As can be seen in Figure 1, the resulting histogram resembles a normal curve.
Figure 2
Another diagram that supports the possibility that the sample was taken from a normally distributed population is the Normal P-P Plot, as shown in Figure 2. In the plot, many sample points are on the diagonal line or quite close to the line.
Figure 3
The box and whisker diagram of the data can be seen in Figure 3. It appears that the median line is in the middle of the box, and both whiskers are approximately the same length. This diagram is quite supportive of the statement that the sample comes from a normally distributed population, although there is a little doubt about it because in the diagram, there is an outlier below the lowest point of the whisker. Based on the three diagrams above, we are quite confident that the sample comes from a normally distributed population. To further confirm this conclusion, we perform the Shapiro-Wilk and Kolmogorov-Smirnov tests on the data. The following is the SPSS output of the data.
Table 1
From Table 1, it is obtained that the p-value of both tests is more than 0.05 (i.e., 0.200 and 0.699); both tests give insignificant results. With a fairly large size (i.e., 200), both tests are sensitive to shifts from the normal distribution. Because of its sensitivity, there should be at least one of the two tests that are significant if the original population of the sample is not normally distributed. However, it turns out that neither test is significant, even though the sample size is large. This strengthens the belief that the sample comes from a normally distributed population.
Let’s test the second data, namely the following data: (click here). In the file, there is a variable named unif20. Using SPSS, display the histogram, normal P-P plot, and box and whisker plot. The following is the SPSS output related to these things.
Figure 4
From Figure 4, it is rather difficult to determine whether the histogram resembles a normal curve. Let’s look at the Normal P-P Plot.
Figure 5
The Normal P-P Plot in Figure 5 does not indicate that the sample comes from a normal population. Almost none of the sample points are located on the diagonal line, and many tend to be quite far from the diagonal line. So, the results of observations of this diagram tend to indicate deviations from the normal distribution.
Figure 6
The box and whisker diagram (Figure 6) shows a median line that is not located in the middle of the box. This image tends to suggest that the population of origin of the sample is not normally distributed, even though both whiskers have almost the same length. What about the formal test?
Table 2
As can be seen in Table 2, the p-values of both tests are greater than 0.05. This test gives an insignificant result. Based on this table, there is not enough evidence to refute the hypothesis that the sample comes from a normally distributed population.
The results of observing the diagrams and formal tests provide different conclusions. The box and whisker diagram and the Normal P-P Plot suggest that the sample does not come from a normally distributed population, while the formal test does not suggest this. However, because the sample size is small, both formal tests do not have enough power to reject the statement that the population is normally distributed, so the results of this test are not very reliable. Therefore, in this case, the more likely conclusion is that the original population is not normally distributed.
PROBLEMS
Download the sampling result using the following link: (click here).
- Display the histogram, Normal P-P Plot, and Box and Whisker plot of the data.
- Based on these diagrams, does the sample come from a normally distributed population?
- Display the results of the normality test using the Kolmogorov-Smirnov and Shapiro-Wilk tests.
- Based on these formal tests, does the conclusion support answer number 2?
- Conclude whether the sample comes from a normally distributed population.
