SHOWTIME STATISTICAL TOOLS IN EVALUATION DESCRIPTIVE VALUES MEASURES

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STATISTICAL TOOLS IN EVALUATION DESCRIPTIVE VALUES MEASURES OF VARIABILITY

MEASURES OF CENTRAL TENDENCY • 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 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 = 7 - 2 = 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 = 42 = 16

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