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

Descriptive Statistics IREVIEW

• 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?

• Measures of central tendency• Mean, median mode• 3, 4, 5, 5, 8

• Distribution shapes

Variability

• RangeHi – 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 statsThe square root of the variance

Most commonly used measure of variability

True Variance

Totalvariance

Error

2SS

Variance (Table 3.2)

The didactic formula

The calculating formula

1

2

2

n

MXS

1

2

2

2

nn

XX

S

4+1+0+1+4=10 10 = 2.5 5-1=4 4

55 - 225 = 55-45=10 = 2.5 5 4 4

4

Standard Deviation

The square root of the variance

Nearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations

M + S100 + 10

2SS

The Normal Distribution

M + 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)-1301-30

S= √20 5

S= √4

S=2

N

MXS

2

(X-M)2

19019

20

Coefficient of Variation (V)

Relative variability around the mean ORDetermines 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

Descriptive Statistics II

• What is the “muddiest” thing you learned today?

Descriptive Statistics IIREVIEW

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+10What does this tell you?

Standard Scores

S

MXZ

•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?

Standard Scores

• Z-scoreM=0, s=1

• T-scoreT = 50 + 10 * (Z)

M=50, s=10

S

MXZ

S

MXT

1050

Variability

• RangeHi – 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 statsThe square root of the variance

Most commonly used measure of variability

True Variance

Totalvariance

Error

2SS

Variance (Table 3.2)

The didactic formula

The calculating formula

1

2

2

n

MXS

1

2

2

2

nn

XX

S

4+1+0+1+4=10 10 = 2.5 5-1=4 4

55 - 225 = 55-45=10 = 2.5 5 4 4

4

Standard Deviation

The square root of the variance

Nearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations

M + S100 + 10

2SS

The Normal Distribution

M + 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)-1301-30

S= √20 5

S= √4

S=2

N

MXS

2

(X-M)2

19019

20

Coefficient of Variation (V)

Relative variability around the mean ORDetermines 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

Descriptive Statistics II


Descriptive Statistics IIREVIEW

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




Standard Scores

S

MXZ

•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?

Standard Scores

• Z-scoreM=0, s=1

• T-scoreT = 50 + 10 * (Z)

M=50, s=10

S

MXZ

S

MXT

1050

Conversion to Standard Scores

Raw scores37451

• Mean: 4• St. Dev: 2

S

MXZ

X-M-1 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?

Normal distribution of scores Figure 3.7

99.9

Descriptive Statistics II Accelerated REVIEW

Standard Scores• Converting scores to Z scores expresses a score’s

distance from its own mean in sd units• Value?




Between what values do 95% of the scores in this data set fall?

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• 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

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?

• Assume M=700, S=100

Percentile T score z score Raw score

64 53.7 .37 737

43

–1.23

618

17

68

68

835

.57

Descriptive Statistics III

• Explain one thing that you learned today to a classmate
