Download Standard Scores

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?
• Measures of central tendency
• Mean, median mode
• 3, 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 mean
Most stable measure
Error
True
Varia
nce
Total
variance
• Standard Deviation (S) descriptive stats
The square root of the variance
Most commonly used measure of variability
S 
S2
Variance (Table 3.2)
The didactic formula
S
2

 X
 M
2
n  1
4+1+0+1+4=10
5-1=4
10 = 2.5
4
The calculating formula
 X 
2
S2 
X
2

n 1
n
55 - 225 = 55-45=10 = 2.5
5
4
4
4
Standard Deviation
The square root of the variance
S 
S
2
Nearly 100% scores in a normal distribution are captured
by the mean + 3 standard deviations
M+S
100 + 10
The Normal Distribution
M + 1s = 68.26% of observations
M + 2s = 95.44% of observations
M + 3s = 99.74% of observations
Calculating Standard Deviation
Raw scores (X-M)
3
-1
7
3
4
0
5
1
1
-3
∑ 20
0
Mean: 4
(X-M)2
1
9
0
1
9
20
S 

S= √20
5
S= √4
S=2
X  M 
2
N
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
Descriptive Statistics II
• What is the “muddiest” thing you learned
today?
Descriptive Statistics II
REVIEW
Variability
• Range
• Variance: Spread of scores based on the squared deviation of
each score from mean
• Standard deviation
Most stable measure
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?
Standard Scores
•Set of observations standardized around a given M and
standard deviation
X M
Z

•Score transformed based on its magnitude relative
S 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-score
M=0, s=1
• T-score
T = 50 + 10 * (Z)
M=50, s=10
X M
Z 
S
10  X  M 
T  50 
S
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 mean
Most stable measure
Error
True
Varia
nce
Total
variance
• Standard Deviation (S) descriptive stats
The square root of the variance
Most commonly used measure of variability
S 
S2
Variance (Table 3.2)
The didactic formula
S
2

 X
 M
2
n  1
4+1+0+1+4=10
5-1=4
10 = 2.5
4
The calculating formula
 X 
2
S2 
X
2

n 1
n
55 - 225 = 55-45=10 = 2.5
5
4
4
4
Standard Deviation
The square root of the variance
S 
S
2
Nearly 100% scores in a normal distribution are captured
by the mean + 3 standard deviations
M+S
100 + 10
The Normal Distribution
M + 1s = 68.26% of observations
M + 2s = 95.44% of observations
M + 3s = 99.74% of observations
Calculating Standard Deviation
Raw scores (X-M)
3
-1
7
3
4
0
5
1
1
-3
∑ 20
0
Mean: 4
(X-M)2
1
9
0
1
9
20
S 

S= √20
5
S= √4
S=2
X  M 
2
N
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
Descriptive Statistics II
• What is the “muddiest” thing you learned
today?
Descriptive Statistics II
REVIEW
Variability
• Range
• Variance: Spread of scores based on the squared deviation of
each score from mean
• Standard deviation
Most stable measure
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?
Standard Scores
•Set of observations standardized around a given M and
standard deviation
X M
Z

•Score transformed based on its magnitude relative
S 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-score
M=0, s=1
• T-score
T = 50 + 10 * (Z)
M=50, s=10
X M
Z 
S
10  X  M 
T  50 
S
Conversion to Standard Scores
Raw scores
3
7
4
5
1
• Mean: 4
• St. Dev: 2
X-M
-1
3
0
1
-3
Z
-.5
1.5
0
.5
-1.5
X M
Z 
S
SO WHAT?
You have a Z score but what
do you do with it? What
does it tell you?
Allows the comparison of
scores using different
scales to compare “apples
to apples”
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?
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)
100+20
What does this tell you?
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
• What is the “muddiest” thing you learned
today?