1. Dr.Rehab F.M. Gwada

2. Measures of Central Tendency the average or a typical, middle observed value of a variable in a data set. There are three commonly used kinds of averages: Mean Median Mode

3. Mean 1.Measure of Central Tendency 2. the average score 3.Most Common Measure 4.Affected by Extreme Values 5. Formula (Sample Mean)

4. Finding the Mean X = (Σ X) / N If X = {3, 5, 10, 4, 3} X = (3 + 5 + 10 + 4 + 3) / 5 = 25 / 5 = 5

5. Median 1.Measure of Central Tendency 2.Middle Value In Ordered Sequence If Odd n, Middle Value of Sequence If Even n, Average of 2 Middle Values 3. Position of Median in Sequence 4. Not Affected by Extreme Values PositioningPoint n 1 2

6. Median Example Odd-Sized Sample Raw Data:24.122.621.523.722.6 Positioning Point Median = 22.6  n +1 2 5 +1 2 3 Ordered:21.522.622.623.724.1 Position:12345

7. Median Example Even-Sized Sample Raw Data:10.34.98.911.76.37.7 Positioning Point Point Median   n +1 2 6 +1 2 35 7.7 + 8.9 2 8.3. Ordered:4.9 6.3 7.7 8.9 10.3 11.7 Position: 1 2 3 4 5 6

8. Mode 1.Measure of Central Tendency 2.Value That Occurs Most Often 3.Not Affected by Extreme Values 4.May Be No Mode or Several Modes 5.May Be Used for Numerical & Categorical Data

9. Mode Example No Mode Raw Data:10.34.98.911.76.37.7 One Mode Raw Data:6.34.98.9 6.3 4.94.9 More Than 1 Mode Raw Data:212828414343

10. Summary of Central Tendency Measures MeasureEquationDescription Mean  X i /n Balance Point Median(n+1) Position Position 2 Middle Value When Ordered Modenone Most Frequent

11. Measures of variation Range Variance standard deviation

12. Range 1.Measure of Dispersion 2.Difference Between Largest & Smallest Observations 3.Ignores how data are distributed,it does not tell us much about the shape of the distribution and how much the scores vary from the mean.

13. Variance & Standard Deviation 1. Measures of Dispersion 2.Most Common Measures 3.Consider How Data Are Distributed 4.Show Variation About Mean 5. It is useful when we need to compare groups using the same scale

14. Calculating the Variance and/or Standard Deviation Formulae: Variance: Examples Follow... Standard Deviation: SD

15. Calculating a Mean and a Standard Deviation

16. The Normal Distribution Curve It is bell-shaped and symmetrical about the mean The mean, median and mode are equal Mean, Median, Mode It is a function of the mean and the standard deviation Most of the cases will fall in the center portion of the curve

18. Appropriate Measures of Central Tendency Nominal variables Mode Ordinal variables Median Interval level variables Mean

19. Types of Data 1-Nominal (The lowest measurement level ) A nominal scale, is classify items into categories that have no particular ordered relationship to one another. Examples in Therapy and Rehabilitation Gender; (Male=1 and Female= 0) Marital Status (Married =1,= 0, Unmarried =2) Hand dominance (Left =2, Right =1). Smoking (Smoking =1, Ex-smoker =2, Non-smoker= 0) Answer to a questioner (YES=1, NO=2) Stroke classification according to side (right=0, left=1, both=2)

20. Types of Data 2-Ordinal An ordinal scale is next up the list in terms of power of measurement. There is no objective distance between any two points on your subjective scale. we can not assume equal interval. The simplest ordinal scale is a ranking.

21. 2-Ordinal Examples in Therapy and Rehabilitation Ordinal scales are also used for the measurement of pain and pain relief (McQuay, 2004), for example: 4= complete relief 3= good relief 2= moderate relief 1= slight relief 0= no relief Pain scale mild, moderate, sever These consecutive grades are not linear, however. A gain from 0 to 1 is not the equivalent of gain from 1 to 2..

22. 2-Ordinal Examples in Therapy and Rehabilitation Functional assessment scales where 0 = Means dependent, 1 = Physical assistance is needed, 2=independence Statistics Ordinal data would use non-parametric statistics. Median Median Mode Mode Rank Rank Correlation Correlation

23. 3-Interval An interval scale is the level where the scale is truly quantitative. Intervals between adjacent scale values are equal with respect to the attribute being measured. Zero point on an interval scale is arbitrary.

24. 3-Interval Example Temperature is interval scale: For example, 38°C is not twice as hot as 19°C, although it is 19°C warmer. The Centigrade scale has a zero point but it is an arbitrary one. Statistics Interval scale data would use parametric statistical techniques:  Mean & standard deviation  Correlation  Analysis of variance  Factor analysis  Regression analysis Remember that you can use non-parametric techniques with interval and ratio data. But non-parametric techniques are less powerful than the parametric ones.

25. 4-Ratio A ratio scale has the properties of order, equal distance between units and a fixed origin or absolute zero point. Parametric statistics can be used to analyze ratio scales Examples in Therapy and Rehabilitation The length (walking distance in meter ) Age, Height, and Weight. Temperature in Kelvin Speed,volume. ROM Isokeintic Statistics :The same as for Interval data

26. VariablesProperties RatioIntervalOrdinal Nomina l √√√√ Classify objects √√√ Rank objects √√ Equal interval √ True zero point

27. Task 1 VariablesMeasurement RatioIntervalOrdinalNominal Speed ROM VAS Sex Temperature in centigrade Smoking Weight

28. P Values The probability that any observation is due to chance alone assuming that the null hypothesis is true Typically, an estimate that has a p value of 0.05 or less is considered to be “statistically significant” or unlikely to occur due to chance alone. The P value used is an arbitrary value P value of 0.05 equals 1 in 20 chance P value of 0.01 equals 1 in 100 chance P value of 0.001 equals 1 in 1000 chance.

29. Parametric versus Nonparametric Parametric – Characteristic is normally distributed in the population; Sample was randomly selected; Data is interval or ratio Sample size >20 Nonparametric Use when you have a specialized population, you’ve not randomly selected, Data is ranked or nominal Sample size <20

30. Parametric Statistics T-Test for mean differences (≤ 2 groups of participants) T for one (paired t test) T for two (unpaired t test) Analysis of Variance when >2 groups of participants

31. Paired T Tests T for one (paired t test) Uses the change before and after intervention in a single individual (group) T for two (unpaired t test) Uses the change before and after intervention in two groups. Reduces the degree of variability between the groups Given the same number of patients, has greater power to detect a difference between groups Unpaired T Tests

32. Analysis of Variance (ANOVA) Usually used for 3 or more measurements or/ samples. If there is significant among group used post hock test to determine differences between group.

33. Nonparametric Nonparametric Techniques for Quantitative Data Wilcoxon signed ranks—for one The Mann-Whitney U Test—for two The Kruskal-Wallis One Way Analysis of Variance— for ANOVA Nonparametric Technique for Categorical Data Chi-Squared test of frequencies Is there a relationship between eye and hair color?

34. Correlation Assesses the linear relationship between two variables Example: height and weight Strength of the association is described by a correlation coefficient- r r =.2 -.4low, possible importance r =.4 -.6moderate correlation r =.6 -.8high correlation r =.8 - 1very high correlation Can be positive or negative Pearson’s, Spearman correlation coefficient Tells nothing about causation

35. Correlation Source: Harris and Taylor. Medical Statistics Made Easy

36. Correlation Perfect Correlation Source: Altman. Practical Statistics for Medical Research

37. Correlation Source: Altman. Practical Statistics for Medical Research Correlation Coefficient -.5 Correlation Coefficient.7

38. Summary Understanding basic statistical concepts is central to understanding the medical literature. Not important to understand the basis of the tests or the underlying math. Need to know when a test should be used and how to interpret its results