Data Distribution

5.7.1 Skewness

The distribution is a summary of the frequency of individual values or ranges of values for, in this case, an item. In order to measure the distribution of data pertaining to the 40 items it was necessary to test for skewness and kurtosis. Normal distributions produce a skewness statistic (SS) of about zero but small variations can occur by chance alone. As the skewness statistic departs further from zero, a positive value indicates the possibility of a positively skewed distribution, that is, with scores loading on the low end of the score scale or a negative value indicates the possibility of a negatively skewed

distribution that is, with scores loading on the high end of the scale (Brown, 1996, pp. 138-142). Values of twice the standard error of skewness (SES) or more regardless of whether they are positive or negative would be deemed as skewed to a significant degree (George & Mallery, 2010). SPSS was employed to calculate descriptive statistics for distribution including, mean, standard deviation, skewness, kurtosis and the standard error of both skewness and kurtosis. Observations made on the data revealed all items demonstrated a degree of skewness. Standard error was calculated using a formula derived by Tabachnick & Fidell (1996):

√6 !

To find out whether an item is skewed and within the acceptable range of +/- 2 SES, the skewness statistic from the data output was used in the above calculation. An

approximate estimate of the SES for this set of data is:

!"! = 6

Since two times the standard error of the skewness is .22 and the value of the skewness statistic for item 1 is -1.99, which is greater than .22, the assumption can be made that the distribution is significantly skewed. Since the sign of the skewness statistic is negative, the distribution is negatively skewed. Alternatively, item 35 has a skewness statistic of .48 and greater than .22 indicating that the skewness statistic is positive, and that the distribution was positively skewed. Item 40 however has a skewness statistic of -0.15 and falls within the range between - 0.22 and + 0.22, in which case, the

assumption is that the skewness was within the expected range of chance fluctuations in that statistic, which would further indicate a distribution with no significant skewness problem. Items 34 and 40 were within range +/- 0.22 and yielded a skewness statistic = -0.15 and -0.20 with the SES = 0.11for both items. Figure 2. below is provided as an example and indicates slight skewness compared to normal distribution.

Fig. 2. Example of Acceptable Level of Skewness.

All other items demonstrated scores that were outside the acceptable range for normal distribution range, with variables 33, SS = 0.437 and 35, SS = 0.476 displaying positive skewness. The remaining variables were negatively skewed with variable 28 yielding the highest level of skewness, SS =-1.354, SES=0.11. Figures 3 and 4 provide examples of the relative level of skewness in both positive, item 35 and negative, item 28.

Fig. 3. Example of Positive Skewness Fig. 4.Example of Negative Skewness 5.7.2 Kurtosis

The level of kurtosis characterises the overall shape of the data distribution relative to the shape of the curve. Negative kurtosis will have a sharp profile with lightness of tails while positive kurtosis has a flatter profile with heavier tails (DeCarlo, 1997, pp. 292- 307). Normal distribution produces a kurtosis of zero with an acceptable range being +/- twice the standard error of kurtosis (SEK). Values of twice SEK or more regardless of being positive or negative would be deemed to be out with the acceptable range (Brown, 1996, pp. 138-142). Standard error was calculated using a formula derived by

Tabachnick & Fidell (1996):

!"# = 24 !

To find out whether a variable lies within the acceptable range of +/- 2 SEK, the kurtosis statistic from the SPSS output was used in the calculations. An approximate estimate of the SEK for this set of data is:

!"# = 24 ! =

24

By multiplying the standard error of the kurtosis by 2, a value of 0.44 is achieved and can be taken as the range that the kurtosis statistic should fall between (range =/- 0.44). Therefore comparing the kurtosis statistic for variable 1 which is 1.072 and greater than 0.44, the assumption can be made that the distribution has a significant kurtosis issue. Since the sign of the kurtosis statistic is positive, the distribution would be flat with heavy tails. Alternatively, item 2 has a kurtosis statistic of -0.576 and is greater than - 0.44 indicating a negative kurtosis issue. Item 3 however has a kurtosis statistic of 0.14 and falls within the range between - 0.44 and + 0.44, in which case, the assumption is that the kurtosis was within the expected range of chance fluctuations in that statistic, which would further indicate a distribution with no significant kurtosis problem. Items 3, 6,12,16,18, 19, 22, 23, 25, 26, 27 and 39 produced scores that were within the acceptable range and follows a normal distribution shape similar to figure 5.

Fig. 5. Example of Acceptable Level of Kurtosis.

Items 2, 33, 34, 35 and 40 produced scores that are deemed negative and out with the range of +/- 0.44. The profile is flat with heavy tails when compared to the normal distribution curve and would be similar to figure 6 below.

Fig. 6. Example of Negative Kurtosis Fig. 7. Example of Positive Kurtosis

The remaining items 1, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 20, 21, 28, 29, 30, 31, 32, 36, 37 and 38 are deemed positive and out with the range of +/- 0.44. Figure 7. above illustrates the positive profile with a sharp curve and lighter tails as compared to the normal distribution shape.