# Descriptive Statistics I REVIEW Measurement scales Nominal, Ordinal, Continuous (interval, ratio) Summation Notation: 3, 4, 5, 5, 8Determine: ∑ X, (∑ X)

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• Descriptive Statistics IREVIEW

Measurement scalesNominal, Ordinal, Continuous (interval, ratio)

Summation Notation:3, 4, 5, 5, 8Determine: X, ( X)2, X29+16+25+25+64 25 625 139

Percentiles: so what?

• Measures of central tendencyMean, median mode3, 4, 5, 5, 8

Distribution shapes

• Variability

Range Hi Low scores only (least reliable measure; 2 scores only)

Variance (S2) inferential stats Spread of scores based on the squared deviation of each score from meanMost stable measure

Standard Deviation (S) descriptive stats The square root of the varianceMost commonly used measure of variabilityTrue VarianceError

• Variance (Table 3.2)The didactic formulaThe calculating formula4+1+0+1+4=1010 = 2.5 5-1=4 455 - 225 = 55-45=10 = 2.5 5 444

• Standard Deviation

The square root of the varianceNearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations

M + S100 + 10

• The Normal DistributionM + 1s = 68.26% of observationsM + 2s = 95.44% of observationsM + 3s = 99.74% of observations

• Calculating Standard Deviation

Raw scores37451 20

Mean: 4 (X-M)-1301-30S= 20 5

S= 4

S=2

(X-M)21901920

• Coefficient of Variation (V)

Relative variability around the mean ORDetermines homogeneity of scoresSM

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

• Descriptive Statistics II

What is the muddiest thing you learned today?

• Descriptive Statistics IIREVIEW

VariabilityRangeVariance: Spread of scores based on the squared deviation of each score from meanMost stable measureStandard deviation Most commonly used measureCoefficient of variationRelative 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+10What does this tell you?

• 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 scores distance from its own mean in sd units

Use of standard scores: determine composite scores from different measures (bball: shoot, dribble); weight?

• Standard Scores

Z-scoreM=0, s=1

T-score T = 50 + 10 * (Z)M=50, s=10

• Variability

Range Hi Low scores only (least reliable measure; 2 scores only)

Variance (S2) inferential stats Spread of scores based on the squared deviation of each score from meanMost stable measure

Standard Deviation (S) descriptive stats The square root of the varianceMost commonly used measure of variabilityTrue VarianceError

• Variance (Table 3.2)The didactic formulaThe calculating formula4+1+0+1+4=1010 = 2.5 5-1=4 455 - 225 = 55-45=10 = 2.5 5 444

• Standard Deviation

The square root of the varianceNearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations

M + S100 + 10

• The Normal DistributionM + 1s = 68.26% of observationsM + 2s = 95.44% of observationsM + 3s = 99.74% of observations

• Calculating Standard Deviation

Raw scores37451 20

Mean: 4 (X-M)-1301-30S= 20 5

S= 4

S=2

(X-M)21901920

• Coefficient of Variation (V)

Relative variability around the mean ORDetermines homogeneity of scoresSM

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

• Descriptive Statistics II

What is the muddiest thing you learned today?

• Descriptive Statistics IIREVIEW

VariabilityRangeVariance: Spread of scores based on the squared deviation of each score from meanMost stable measureStandard deviation Most commonly used measureCoefficient of variationRelative 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+10What does this tell you?

• 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 scores distance from its own mean in sd units

Use of standard scores: determine composite scores from different measures (bball: shoot, dribble); weight?

• Standard Scores

Z-scoreM=0, s=1

T-score T = 50 + 10 * (Z)M=50, s=10

• Conversion to Standard Scores

Raw scores37451

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

• Normal distribution of scores Figure 3.7

99.9

• Descriptive Statistics II Accelerated REVIEW

Standard ScoresConverting scores to Z scores expresses a scores distance from its own mean in sd unitsValue? Coefficient of variationRelative 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)100+20What does this tell you?Between what values do 95% of the scores in this data set fall?

• Normal-curve Areas Table 3-3Z scores are on the left and across the topZ=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 scores below mean, so + 50 if Z is +/-

The "reference point" is the mean+Z=better than the mean-Z=worse than the mean

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

• Normal curve practiceZ score Z = (X-M)/ST score T = 50 + 10 * (Z)Percentile P = 50 + Z percentile (+: add to 50, -: subtract from 50)Raw scores

HintsDraw a pictureWhat is the z score?Can the z table help?

• Assume M=700, S=100

PercentileT scorez scoreRaw score64 53.7 .37 737 43 1.23618 176868835.57

• Descriptive Statistics III

Explain one thing that you learned today to a classmate

What is the muddiest thing you learned today?

Chapter 3*Chapter 3*Chapter 3*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*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.Chapter 3*Indicate the Mean plus and minus 3 standard deviations captures nearly 100% of the scores in a normal distribution.Chapter 3*Indicate the Mean plus and minus 3 standard deviations captures nearly 100% of the scores in a normal distribution.Chapter 3*Chapter 3*Chapter 3*Chapter 3*Chapter 3*Chapter 3*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*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*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.Chapter 3*Indicate the Mean plus and minus 3 standard deviations captures nearly 100% of the scores in a normal distribution.Chapter 3*Indicate the Mean plus and minus 3 standard deviations captures nearly 100% of the scores in a normal distribution.Chapter 3*Chapter 3*Chapter 3*Chapter 3*Chapter 3*Chapter 3*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*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*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 domainChapter 3*Chapter 3*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-tableChapter 3*There a numerous homework problems on the WWW about converting between z, T, percentile and raw scoresThe animation shows that when z = 1, T = 60, Percentile = 84, and VO2 = 65Chapter 3*There a numerous homework problems on the WWW about converting between z, T, percentile and raw scoresChapter 3* ##### Section 2.3 Properties of 2 âˆ’3. gx x (âˆ’=âˆ’) 2 ( ) 2 âˆ’3 = âˆ’=2 3 x fx. 2. Even function symmetric
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