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STATISTICAL TOOLS IN EVALUATION DESCRIPTIVE VALUES MEASURES OF VARIABILITY

WHEN THE GRAPH OF THE SCORES IS A NORMAL CURVE, THE MODE, MEDIAN, AND MEAN ARE EQUAL THE MEAN IS THE MOST COMMON MEASURE OF CENTRAL TENDENCY WHEN THE SCORES ARE QUITE SKEWED OR THE DATA IS ORDINAL LACKING A COMMON INTERVAL, THE MEDIAN IS A BETTER MEASURE OF CENTRAL TENDENCY THE MODE IS USED ONLY WHEN THE MEAN OR MEDIAN CANNOT BE CALCULATED (E.G., NOMINAL DATA) OR WHEN THE ONLY INFORMATION WANTED IS THE MOT FREQUENT SCORE (E.G., MOST UNIFORM SIZE OR INJURY SITE) MEASURES OF CENTRAL TENDENCY

MEASURES OF VARIABILITY DESCRIBES THE SET OF SCORES IN TERMS OF THEIR SPREAD, OR HETEROGENEITY CONSIDER TWO GROUPS OF SCORES GROUP 1 = 9, 5, 1; GROUP 2 = 5, 6, 4 BOTH HAVE A MEAN AND MEDIAN OF 5 BUT GROUP 2 HAS MUCH MORE HOMOGENEOUS OR SIMILAR SCORES THAN GROUP 1

MEASURES OF VARIABILITY RANGE STANDARD DEVIATION VARIANCE

RANGE EASIEST MEASURE OF VARIABILITY TO CALCULATE USED WHEN THE MEASURE OF CENTRAL TENDENCY IS THE MODE (NOMINAL DATA OR WHEN THE MOST FREQUENT SCORE IS OF INTEREST) OR MEDIAN (ORDINAL DATA OR SKEWED DATA) SIMPLY THE DIFFERENCE BETWEEN THE HIGHEST AND LOWEST SCORES

WHAT IS THE RANGE IN THE SET OF SCORES BELOW? SET OF SCORES: 7, 2, 7, 6, 5, 6, 2 RANGE = HIGHEST SCORE MINUS LOWEST SCORE = = 5

STANDARD DEVIATION (S) MEASURE OF VARIABILITY USED WITH THE MEAN (NORMALLY DISTRIBUTED INTERVAL OR RATIO DATA) INDICATES THE AMOUNT THAT ALL SCORES DIFFER OR DEVIATE FROM THE MEAN THE MORE THE SCORES DIFFER FROM THE MEAN, THE HIGHER THE STANDARD DEVIATION (S) SUM OF THE DEVIATIONS OF SCORES FROM THE MEAN IS ALWAYS IS 0

DEFINITIONAL FORMULA FOR STANDARD DEVIATION FORMULA 2.1 SHOULD BE USED IF THE GROUP TESTED IS VIEWED AS THE GROUP OF INTEREST; CONSIDERED THEN THE POPULATION (E.G., CALCULATING STANDARD DEVIATION OF THE TEST SCORES ON EXAM #1 IN THIS CLASS) X = SCORES BAR X = MEAN OF SCORES N = NUMBER OF SCORES MANY CALCULATORS USE THIS FORMULA

DEFINITIONAL FORMULA FOR STANDARD DEVIATION FORMULA 2.2 SHOULD BE USED IF THE GROUP TESTED IS VIEWED AS A REPRESETATIVE PART OF THE POPULATION; CONSIDERED THEN A SAMPLE STANDARD DEVIATION CALCULATED ON THE SAMPLE IS USED AS AN ESTIMATE OF THE POPULATION STANDARD DEVIATION (E.G., CALCULATION OF THE STANDARD DEVIATION OF THE PERCENT BODY FAT OF COLLEGE RUNNERS THAT IS USED AS AN ESTIMATION OF THE STANDARD DEVIATION OF ALL COLLEGE RUNNERS) X = SCORES BAR X = MEAN OF SCORES N = NUMBER OF SCORES MANY CALCULATORS AND MOST COMPUTER PROGRAMS USE THIS FORMULA

SAMPLE CALCULATION OF THE STANDARD DEVIATION USING FORMULA 2.1 AND 2.2 AND THE FOLLOWING TESTS SCORES: 7, 2, 7, 6, 5, 6, 2

CALCULATIONAL FORMULA FOR STANDARD DEVIATION FORMULA 2.3 SHOULD BE USED IF THE GROUP TESTED IS VIEWED AS THE GROUP OF INTEREST; CONSIDERED THEN THE POPULATION (E.G., CALCULATING STANDARD DEVIATION OF THE 50-M SWIM TIMES AT A SWIM MEET ) X = SCORES N = NUMBER OF SCORES FORMULA TYPICALLY USED FOR HAND CALCULATION

CALCULATIONAL FORMULA FOR STANDARD DEVIATION FORMULA 2.4 SHOULD BE USED IF THE GROUP TESTED IS VIEWED AS A REPRESETATIVE PART OF THE POPULATION; CONSIDERED THEN A SAMPLE STANDARD DEVIATION CALCULATED ON THE SAMPLE IS USED AS AN ESTIMATE OF THE POPULATION STANDARD DEVIATION (E.G., CALCULATION OF THE STANDARD DEVIATION OF THE 40-YARD TIME OF COLLEGE WIDE RECEIVERS THAT IS USED AS AN ESTIMATION OF THE STANDARD DEVIATION OF ALL COLLEGE WIDE RECEIVERS) X = SCORES N = NUMBER OF SCORES FORUMULA TYPICALLY USED FOR HAND CALCULATION

SAMPLE CALCULATION OF THE STANDARD DEVIATION USING FORMULA 2.3 AND 2.4 AND THE FOLLOWING TESTS SCORES: 7, 2, 7, 6, 5, 6, 2

VARIANCE USEFUL STATISTIC IN CERTAIN HIGH LEVEL STATISTICAL PROCEDURES LIKE REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE (ANOVA) CALCULATED BY SQUARING THE STANDARD DEVIATION (S 2 ) STANDARD DEVIATION = S = 4 VARIANCE = S 2 = 4 2 = 16

QUESTIONS OR COMMENTS?? THANK YOU!!