SHOWTIME!
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 (S2)
• STANDARD DEVIATION = S = 4
• VARIANCE = S2 = 42 = 16
QUESTIONS OR COMMENTS??
THANK YOU!!