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Descriptive Statistics and the Normal Distribution HPHE 3150 Dr. Ayers

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Introduction Review Terminology Reliability Validity Objectivity Formative vs Summative evaluation Norm- vs Criterion-referenced standards

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Scales of Measurement Nominal name or classify Major, gender, yr in college Ordinal order or rank Sports rankings Continuous Interval equal units, arbitrary zero Temperature, SAT/ACT score Ratio equal units, absolute zero (total absence of characteristic) Height, weight

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Summation Notation is read as "the sum of" X is an observed score N = the number of observations Complete ( ) operations first Exponents then * and / then + and -

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Operations Orders 65 26 -5 2-34

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Summation Notation Practice: Mastery Item 3.2 Scores: 3, 1, 2, 2, 4, 5, 1, 4, 3, 5 Determine: ∑ X (∑ X) 2 ∑ X 2 30 900 110

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Percentile The percent of observations that fall at or below a given point Range from 0% to 100% Allows normative performance comparisons If I am @ the 90 th percentile, how many folks did better than me?

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Test Score Frequency Distribution Figure 3.1 (p.42 explanation) ValidFrequencyPercentValid PercentCumulative Percent 4111.5 4334.6 6.2 4434.6 10.8 4557.7 18.5 4657.7 26.2 47710.8 36.9 481116.9 53.8 49812.3 66.2 50710.8 76.9 5169.2 86.2 5234.6 90.8 5334.6 95.4 5423.1 98.5 5511.5 100.0 Total65100.0

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Central Tendency Mean sum scores / # scores Median (P 50 ) exact middle of ordered scores Mode most frequent score Where do the scores tend to center?

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Mean Median (P 50 ) Mode Raw scores 2 7 5 1 Rank order 1 2 5 5 7 Mean:4 (20/5) Median:5 Mode:5

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Distribution Shapes Figure 3.2 So what?OUTLIERS Direction of tail = +/-

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Kampert, MSSE, Suppl. 2004, p. S135

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Histogram of Skinfold Data

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Three Symmetrical Curves Figure 3.3 The difference here is the variability; Fully normal More heterogeneous More homogeneous

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Descriptive Statistics I What is the most important thing you learned today? What do you feel most confident explaining to a classmate?

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Descriptive Statistics I REVIEW Measurement scales Nominal, Ordinal, Continuous (interval, ratio) Summation Notation: 3, 4, 5, 5, 8Determine: ∑ X, (∑ X) 2, ∑X 2 9+16+25+25+64 25 625 139 Percentiles: so what?

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Measures of central tendency 3, 4, 5, 5, 8 Mean (?), median (?), mode (?) Distribution shapes

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Variability Range Hi – Low scores only (least reliable measure; 2 scores only) Variance (s 2 ) inferential stats Spread of scores based on the squared deviation of each score from mean Most stable measure of variability Standard Deviation (S) descriptive stats Square root of the variance Most commonly used measure of variability True Var- iance Total variance Error

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Variance (Table 3.2) The didactic formula The calculating formula 4+1+0+1+4=1010 = 2.5 5-1=4 4 55 - 225 = 55-45=10 = 2.5 5 44 4

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Standard Deviation The square root of the variance Nearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations M + S 100 + 10

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The Normal Distribution M + 1s = 68.26% of observations M + 2s = 95.44% of observations M + 3s = 99.74% of observations

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Calculating Standard Deviation Raw scores 3 7 4 5 1 ∑ 20 Mean: 4 (X-M) 3 0 1 -3 0 S= √20 5 S= √4 S=2 (X-M) 2 1 9 0 1 9 20

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Coefficient of Variation (V) Relative variability Relative variability around the mean OR determine homogeneity of two data sets with different units S / M Relative variability accounted for by the mean when units of measure are different (ht, hr, running speed, etc.) Helps more fully describe different data sets that have a common std deviation (S) but unique means (M) Lower V=mean accounts for most variability in scores.1 -.2=homogeneous>.5=heterogeneous

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Descriptive Statistics II What is the “muddiest” thing you learned today?

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Descriptive Statistics II REVIEW Variability Range Variance: Spread of scores based on the squared deviation of each score from meanMost stable measure Standard deviation Most commonly used measure Coefficient of variation Relative variability around the mean (homogeneity of scores) Helps more fully describe relative variability of different data sets 50+10 What does this tell you?

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Standard Scores Z or t Set of observations standardized around a given M and standard deviation Score transformed based on its magnitude relative to other scores in the group Converting scores to Z scores expresses a score’s distance from its own mean in sd units Use of standard scores: determine composite scores from different measures (bball: shoot, dribble); weight?

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Standard Scores Z-score M=0, s=1 T-score T = 50 + 10 * (Z) M=50, s=10 Percentile p = 50 + Z (%ile)

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Conversion to Standard Scores Raw scores 3 7 4 5 1 Mean: 4 St. Dev: 2 X-M 3 0 1 -3 Z -.5 1.5 0.5 -1.5 Allows the comparison of scores using different scales to compare “apples to apples” SO WHAT? You have a Z score but what do you do with it? What does it tell you?

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Normal distribution of scores Figure 3.6 99.9

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Descriptive Statistics II REVIEW Standard Scores Converting scores to Z scores expresses a score’s distance from its own mean in sd units Value? Coefficient of variation Relative variability around the mean (homogeneity of scores) Helps more fully describe relative variability of different data sets 100+20 What does this tell you? Between what values do 95% of the scores in this data set fall?

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Normal-curve Areas Table 3.4 Z scores are on the left and across the top Z=1.64: 1.6 on left,.04 on top=44.95 Since 1.64 is +, add 44.95 to 50 (mean) for 95 th percentile Values in the body of the table are percentage between the mean and a given standard deviation distance ½ scores below mean, so + 50 if Z is +/- The "reference point" is the mean +Z=better than the mean -Z=worse than the mean

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p. 51

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Area of normal curve between 1 and 1.5 std dev above the mean Figure 3.7

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Normal curve practice Z score Z = (X-M)/S T score T = 50 + 10 * (Z) Percentile P = 50 + Z percentile (+: add to 50, -: subtract from 50) Raw scores Hints Draw a picture What is the z score? Can the z table help?

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Assume M=700, S=100 PercentileT scorez scoreRaw score 6453.7.37737 43 –1.23 618 17 68 835.57

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Descriptive Statistics III Explain one thing that you learned today to a classmate What is the “muddiest” thing you learned today?

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