Descriptive Statistics I REVIEW Measurement scales Nominal, Ordinal, Continuous (interval, ratio) Summation Notation: 3, 4, 5, 5, 8 Determine: ∑ X, (∑ X)2, ∑X2 9+16+25+25+64 25 625 139 Percentiles: so what? Chapter 3

Measures of central tendency Mean, median mode 3, 4, 5, 5, 8 Distribution shapes Chapter 3

Variability Range Hi – Low scores only (least reliable measure; 2 scores only) Variance (S2) Spread of scores based on the squared deviation of each score from mean Most stable measure Standard Deviation The square root of the variance Most commonly used measure of variability Error Total variance True Variance These are measures of variability. The range is the most unreliable measure because it depends only two scores. The standard deviation is the square root of the variance. Chapter 3

Variance (Table 3.2) The didactic formula 4+1+0+1+4=10 10 = 2.5 4+1+0+1+4=10 10 = 2.5 5-1=4 4 The calculating formula Introduce the variance from the didactic equation (i.e., the average squared deviation from the mean) and then the calculating formula. Use Table 3-2 on page 40 to illustrate that you get the same answer with both formulae. Then illustrate that you obtain the standard deviation by simply taking the square root of the variance. 55 - 225 = 55-45=10 = 2.5 5 4 4 4 Chapter 3

Standard Deviation The square root of the variance Nearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations Indicate the Mean plus and minus 3 standard deviations captures nearly 100% of the scores in a normal distribution. Chapter 3

The Normal Distribution Indicate the Mean plus and minus 3 standard deviations captures nearly 100% of the scores in a normal distribution. M + 1s = 68.26% of observations M + 2s = 95.44% of observations M + 3s = 99.74% of observations Chapter 3

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

Coefficient of Variation (V) Relative variability around the mean OR Determines homogeneity of scores S M 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 Chapter 3

Descriptive Statistics II What is the “muddiest” thing you learned today? Chapter 3

Descriptive Statistics II REVIEW Variability Range Variance: Spread of scores based on the squared deviation of each score from mean Most stable measure Standard deviation Most commonly used measure Coefficient of variation Relative variability around the mean (homogeneity of scores) Helps more fully describe different data sets that have a common std deviation (S) but unique means (M) 50+10 What does this tell you? Chapter 3

Standard Scores 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? Chapter 3

Standard Scores Z-score M=0, s=1 T-score T = 50 + 10 * (Z) M=50, s=10 Provide an example of the use of standard scores For example, determining a composite score for basketball from shooting (high is better) and dribbling (low is better). You might want to weight shooting 2 or 3 times that of dribbling and show how scores change. Chapter 3

Conversion to Standard Scores Raw scores 3 7 4 5 1 Mean: 4 St. Dev: 2 X-M -1 3 1 -3 Z -.5 1.5 .5 -1.5 SO WHAT? You have a Z score but what do you do with it? What does it tell you? Provide an example of the use of standard scores For example, determining a composite score for basketball from shooting (high is better) and dribbling (low is better). You might want to weight shooting 2 or 3 times that of dribbling and show how scores change. Allows the comparison of scores using different scales to compare “apples to apples” Chapter 3

Normal distribution of scores Figure 3.7 Provide illustrations of use interpretation of the normal distribution. Point out that the only numbers that ever change on the figure are for the specific test that one is using. Here it is VO2 but it could be body fat or written test score, or a scale from the affective domain 99.9 Chapter 3

Normal-curve Areas Table 3-3 Z scores are on the left and across the top Z=1.64: 1.6 on left , .04 on top=44.95 Values in the body of the table are percentage between the mean and a given standard deviation distance The "reference point" is the mean Students need to clearly see that the reference point is the mean and that the values in the BODY of the table are percentages of observations between the MEAN and the given standard deviation units away from the mean. Use many of the homework problems for students to get practice working with the Z-table Chapter 3

Area of normal curve between 1 and 1.5 std dev above the mean Figure 3.9

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? There a numerous homework problems on the WWW about converting between z, T, percentile and raw scores The animation shows that when z = 1, T = 60, Percentile = 84, and VO2 = 65 Chapter 3

Assume M=700, S=100 Percentile T score z score Raw score 64 53.7 .37 737 43 –1.23 618 17 68 835 .57 There a numerous homework problems on the WWW about converting between z, T, percentile and raw scores Chapter 3

Descriptive Statistics III Explain one thing that you learned today to a classmate What is the “muddiest” thing you learned today? Chapter 3