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Ways of Describing Data Emily H. Wughalter, Ed.D. Measurement and Evaluation in Kinesiology Summer 2010 Symbols • • • • • X = sum of scores X2 = sum of the squared scores (X) 2 = sum of the scores squared n = sample N = population Describing Data • Grouping information into meaningful categories or themes to make sense of it Frequency Distributions • A frequency distribution is a method of describing scores X (midterm scores) f 95 1 88 3 86 5 85 2 80 2 Frequency Distributions X (midterm scores) f cf X=the scores 95 1 13 f=frequency of the score 88 3 12 86 5 9 85 2 4 80 2 2 cf=cumulative frequency n=number of participants Frequency Distributions X (midterm scores) f cf xf x2 95 1 13 95 9025 88 3 12 264 23232 86 5 9 430 36980 85 2 4 170 14450 80 2 2 160 12800 Types of Curves • Normal Curve is a perfectly bisymmetrical curve. In a normal curve 50 % falls on one side and 50 % of the other side of the curve. • A Skewed Curve occurs when fewer scores fall to the left of the curve or to the right of the curve. A negatively skewed curve has fewer scores in the negative direction. • A positively skewed curve has fewer scores in the positive direction. • A Leptokurtic curve occurs when many people scored the same score. • A Platykurtic curve occurs when lots of scores have the same frequency of occurrence. Measures of Central Tendency • Points around which the scores tend to cluster 3 Measures of Central Tendency • Mode • Median • Mean Mode • The mode is the most frequently occurring score. • The mode is affected by a single change to the scores in the distribution. Median • The median is the 50th percentile score. It is that score at which 50% of the group falls below or above. • A problem with the median is that it can be affected by a change in a score; also, it is based upon the number of scores not the value of the scores. Mean • The mean is the average score. It is the best measure of central tendency. • The mean is prized by researchers and evaluators because of its use and because the mean or average is well understood. Measures of Variability • Measures of variability indicate information about the spread of the scores 3 Measures of Variability • Range • Semi-Interquartile Deviation (SID) • Standard Deviation 3 Measures of Variability • Range associated and reported with the mode • Semi-Interquartile Deviation (SID) associated and reported with the median • Standard Deviation associated and reported with the mean Example of Between and Within Subject Variability 120 100 80 Exam 1 60 Exam 2 Exam 3 40 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Range • The range is the measure of variability reported with the mode. • It provides a description of the total spread of the scores. • The range can be calculated by Range = (X high - X low) + 1 Problems with the Range • A change in one score in the distribution can change the range of scores. Semi-Interquartile Deviation • The semi-interquartile deviation is reported with the median. • It provides a measure of variability for the middle 50% of the scores. Problems with the SemiInterquartile Deviation • The semi-interquartile deviation measures the variability of only the middle 50% of the scores. • If heterogeneity of the scores or homogeneity of the scores exists in the extremes and this is different than the middle then this will not be reflected in the semi-interquartile deviation. Standard Deviation • The standard deviation measures variability and is reported with the mean. • Theoretically, the standard deviation represents the mean of the differences of all of the scores from the mean. • The standard deviation represents the deviation of an entire set of scores.