1. Exploring Assumptions
Chapter 5

2. Assumptions of Parametric Data
Parametric tests are based on the normal distributions and the following must be met:
Normally distributed data.
Homogeneity of variance:
Variances should be equal throughout the data
The groups should have equal variance
In corr: the variance in one variable (x) should be equal across all levels of the other variable (y).
Dependent variable should be interval or ratio.
Independence: for independent designs the data from different subjects should be independent; for repeated designs the time variable will be dependent, but different subjects should be independent.

3. Assumption of Normality
Many statistical tests (t, ANOVA) assume that the sampling distribution is normally distributed.
This is a problem, we don’t have access to the sampling distribution.
But, according to the central limit theorem if the sample data are approximately normal, then the sampling distribution will be normal.
Also from the central limit theorem, in large samples (n > 30) the sampling distribution tends to be normal, regardless of the shape of the data in our sample.

4. Explore Command In SPSS, use:
Analyze – Descriptives – Explore to get to the Explore Command.

5. The P-P Plots is a probability-probability plot.
From the Explore command it compares the variables cumulative probability to the cumulative probability of the normal distribution.
If the circles fall on the line the data are normal.
Day 2 & Day 3 are NOT normal. It is clear from the histograms that they are both positively skewed.

7. Interpretation of Deviations of Skew and Kurtosis based on Z Scores
Z > 1.95 is significant at p < .05
Z > 2.58 is significant at p < .01
Z > 3.29 is significant at p < .001
Day 1 is platykurtic (flat), Days 2 & 3 are leptokurtic (peaked)

8. Interpretation of Deviations of Skew and Kurtosis based on Z Scores
Z > 1.95 is significant at p < .05
Z > 2.58 is significant at p < .01
Z > 3.29 is significant at p < .001
Day 1 is platykurtic (flat), Days 2 & 3 are leptokurtic (peaked)

9. Interpretation of Deviations of Skew and Kurtosis based on Z Scores
Z > 1.95 is significant at p < .05
Z > 2.58 is significant at p < .01
Z > 3.29 is significant at p < .001
Day 1 is platykurtic (flat), Days 2 & 3 are leptokurtic (peaked)

10. Large Samples and Z Scores
Z > 1.95 is significant at p < .05
Z > 2.58 is significant at p < .01
Z > 3.29 is significant at p < .001
Significance tests of skew and kurtosis should not be used in large samples (because they are likely to be significant even when skew and kurtosis are not too different from normal.

11. With the combined groups (Duncetown & Sussex) Numeracy looks positively skewed.

12. You can split a file by groups (as in the next 2 slides) or just use existing groups and add the grouping variable to the factor list.

15. Just as in the p-p Plot, the Q-Q Plot the data should fall on the line if the data set is normal. Both of these variables have a problem with Normality.

16. The plot on the left shows unequal variance
The plot on the left shows unequal variance. This would be evident when looking at: sd’s, variance and the Levine’s Test of Homogeneity of Variance.

17. Numeracy from
SPSSExam.sav Variance
Duncetown University
Sussex University
N = 50

22. Outliers and Extreme Scores
KINE 5305 Applied Statistics in Kinesiology

23. SPSS – Explore BoxPlot
The top of the box is the upper fourth or 75th percentile.
The bottom of the box is the lower fourth or 25th percentile.
50 % of the scores fall within the box or interquartile range.
The horizontal line is the median.
The ends of the whiskers represent the largest and smallest values that are not outliers.
An outlier, O, is defined as a value that is smaller (or larger) than 1.5 box-lengths.
An extreme value, E , is defined as a value that is smaller (or larger) than 3 box-lengths.
Normally distributed scores typically have whiskers that are about the same length and the box is typically smaller than the whiskers.

24. Choosing a Z Score to Define Outliers
% Above
% +/- Above
3.0
0.0013
0.0026
3.1
0.0010
0.0020
3.2
0.0007
0.0014
3.3
0.0005
3.4
0.0003
0.0006

25. Decisions for Extremes and Outliers
Check your data to verify all numbers are entered correctly.
Verify your devices (data testing machines) are working within manufacturer specifications.
Use Non-parametric statistics, they don’t require a normal distribution.
Develop a criteria to use to label outliers and remove them from the data set. You must report these in your methods section.
If you remove outliers consider including a statistical analysis of the results with and without the outlier(s). In other words, report both, see Stevens (1990) Detecting outliers.
Do a log transformation.
If you data have negative numbers you must shift the numbers to the positive scale (eg. add 20 to each).
Try a natural log transformation first in SPSS use LN().
Try a log base 10 transformation, in SPSS use LG10().

26. Transform - Compute

27. Add 10 to the data. Then log transform.
Add 10 to each data point.
Try Natural Log.
Last option, use Log10.

28. Add 10 to each data point, since you can not take a log of a negative number.

29. First try a Natural Log Transformation

30. If Natural Log doesn’t work try Log10 Transformation.

31. Outlier Criteria: 1.5 * Interquartile Range from the Median
Milner CE, Ferber R, Pollard CD, Hamill J, Davis IS. Biomechanical Factors Associated with Tibial Stress Fracture in Female Runners. Med Sci Sport Exer. 38(2): , 2006.
Statistical analysis. Boxplots were used to identify outliers, defined as values >1.5 times the interquartile range away from the median. Identified outliers were removed from the data before statistical analysis of the differences between groups. A total of six data points fell outside this defined range and were removed as follows: two from the RTSF group for BALR, one from the CTRL group for ASTIF, one from each group for KSTIF, and one from the CTRL group for TIBAMI.

32. Outlier Criteria: 1.5 * Interquartile Range from the Median (using SPSS)

33. Outlier Criteria: 1.5 * Interquartile Range from the Median (using SPSS)
Outliers = 1.5 * 3.39 above and below

34. Outlier Criteria: ± 3 standard deviations from the mean
Tremblay MS, Barnes JD, Copeland JL, Esliger DW. Conquering Childhood Inactivity: Is the Answer in the Past? Med Sci Sport Exer. 37(7): , 2005.
Data analyses. The normality of the data was assessed by calculating skewness and kurtosis statistics. The data were considered within the limits of a normal distribution if the dividend of the skewness and kurtosis statistics and their respective standard errors did not exceed ± 2.0. If the data for a given variable were not normally distributed, one of two steps was taken: either a log transformation (base 10) was performed or the outliers were identified (± 3 standard deviations from the mean) and removed. Log transformations were performed for push-ups and minutes of vigorous physical activity per day. Outliers were removed from the data for the following variables: sitting height, body mass index (BMI), handgrip strength, and activity counts per minute.

35. Computing and Saving Z Scores
Check this box and SPSS creates and saves the z scores for all selected variables. The z scores in this case will be named zLight1…

36. Computing and Saving Z Scores
Now, you can identify and remove raw scores above and below 3 sds if you want to remove outliers.

37. Newcomb's measurements of the passage time of light
Comparison of Outlier Methods Median ± 1.5 * Interquartile Range 27 ± 1.5 * 7 gives a range of 16.5 – 37.5
TABLE 1.1
Newcomb's measurements of the passage time of light 28 22 36 26 24 32 30 27 33 21 31 25 34
-44 23 29 16 40 19 37 -2 20 39

38. Comparison of Outlier Methods ± 2 SDs (Notice the 16s are not removed)
TABLE 1.1
Newcomb's measurements of the passage time of light 28 22 36 26 24 32 30 27 33 21 31 25 34
-44 23 29 16 40 19 37 -2 20 39

39. Comparison of Outlier Methods ± 2 SDs (Notice the 16s are not removed)
TABLE 1.1
Newcomb's measurements of the passage time of light 28 22 36 26 24 32 30 27 33 21 31 25 34
-44 23 29 16 40 19 37 -2 20 39

40. Newcomb's measurements of the passage time of light
Comparison of Outlier Methods ± 3 SDs (Step 1, then run Z scores again)
TABLE 1.1
Newcomb's measurements of the passage time of light 28 22 36 26 24 32 30 27 33 21 31 25 34
-44 23 29 16 40 19 37 -2 20 39

41. Newcomb's measurements of the passage time of light
Comparison of Outlier Methods ± 3 SDs (Step 2, after already removing the -44, the -2 then has a SD of so it is removed)
TABLE 1.1
Newcomb's measurements of the passage time of light 28 22 36 26 24 32 30 27 33 21 31 25 34
-44 23 29 16 40 19 37 -2 20 39

42. Conclusions?
The median ± 1.5 * interquartile range appears to be too liberal.
± 2 SDs may also be too liberal and statisticians may not approve.
An iterative process where you remove points above and below 3 SDs and then re-check the distribution may be the most conservative and acceptable method.
Choosing 3.1, 3.2, or 3.3 as a SD increases the protection against removing a score that is potentially valid and should be retained.

43. Choosing a Z Score to Define Outliers
% Above
% +/- Above
3.0
0.0013
0.0026
3.1
0.0010
0.0020
3.2
0.0007
0.0014
3.3
0.0005
3.4
0.0003
0.0006

44. Decisions for Extremes and Outliers
Check your data to verify all numbers are entered correctly.
Verify your devices (data testing machines) are working within manufacturer specifications.
Use Non-parametric statistics, they don’t require a normal distribution.
Develop a criteria to use to label outliers and remove them from the data set. You must report these in your methods section.
If you remove outliers consider including a statistical analysis of the results with and without the outlier(s). In other words, report both, see Stevens (1990) Detecting outliers.
Do a log transformation.
If you data have negative numbers you must shift the numbers to the positive scale (eg. add 20 to each).
Try a natural log transformation first in SPSS use LN().
Try a log base 10 transformation, in SPSS use LG10().

45. Data Transformations and Their Uses
Can Correct For
Log Transformation (log(X))
Positive Skew, Unequal Variances
Square Root Transformation (sqrt(X))
Reciprocal Transformation (1/X)
Reverse Score Transformation – all of the above can correct for negative skew, but you must first reverse the scores. Just subtract each score from the highest score in the data set + 1.
Negative Skew