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Chapter 5
Individual measures
Raw scores (Xi)
Differential scores (xi)
Standard scores (Zi)
Fundamental statistical
characteristics
Group indexes




Central tendency
Variability (Dispersion)
Bias (Asymmetry)
Skewness (Kurtosis)
Individual indexes




Position
 Centiles (Ci)
 Percentiles (Pi)
 Quartiles (Qi)
Raw scores (Xi)
Differentials scores (xi)
Standard scores (Zi)
2
1) X: it’s the score given to each subject to be
subjected to any test.
Xi
2) x: it’s the raw score minus the mean.
xi  X i  X
3) Z: it’s the differential score divided by the
standard deviation.
Xi  X
Zi 
S
3
Scores comparison
1) The comparison of raw scores (Xi) can
lead us to misleading conclusions.
2) The solution based on the distances to
the mean (xi) is not an entirely
satisfactory solution.
3) The solution is to use the Typical scores,
Typed or standardized.
4
Raw scores interpretation: Xi
5
CASE 1: a subject obtains a score on
an IQ test:

1 subject = 1 variable

How can I interpret the score of X = 25?
A raw score such as 25 does not give us more than
a number. But, 25 is too much? Is it enough?
The answer is: IT DEPENDS.
What is the factor on which this score depend?
* group mean and
* their variability
So, raw scores are not ENOUGH.




X=25
6
CASE 2: 1 subject  2 variables
(not comparable)
John weighs 75 kg and is 1.80
m. tall

His weight, is it more or less than his height?

The answer is that they are not directly
comparable.
7
CASO 3: 1 subject  2 variables
(supposedly comparable)
A student X has recently been examined
in Motivation and Methods.

If the scores have been respectively 30 and
15:

Can we say that the student has done
better in Motivation than in Methods?
8
Differential scores
interpretation: xi
9
Scores of
Deviation
Diversions
Errors or bias
Symbolically
xi = X i - X
10
Differentials scores: xi

Tell us if a raw score is HIGHER,
SMALLER than the mean or THE SAME
as the mean.

This information is inferred from the sign
of the differential score.

There exist differential scores positive
and negative.
11
Raw and differential scores:
Xi = 7 – 9 – 10 – 11 - 13
Xi
7
Xi2
49
(Xi - X )
7-10 = -3
(Xi - X )2
9
9
81
9-10 = -1
1
10
100
10-10 = 0
0
11
121
11-10 = 1
1
13
169
13-10 = 3
9
50
520
0
20
12
3
3
1
1
X
7
9
10
11
13
x
-3
-1
3
0
1
1
1
3
3
13
Conclusions
• RAW
X  10
S 4
2
X
SX  2



DIFFERENTIALS
x 0
S 4
2
x
Sx  2
14
Example
Two groups (A and B) were measured in
their intellectual level:

A:
97
102

B:
92
97
107
102
112
117
107
112
117
122
15
X A  107

X B  107
Supposing that 2 students, one
belonging to group A and another to the
group B, received the same intellectual
level of 117 points.
x  X - X  117  107   10
16
This equality in differential scores may be
masking very different realities.

• Group A
97
• Group B
102
107
112
117
     
92
97
102
107
112
117
122
17
Conclusions

As we can see, group A’s scores are much
more homogeneous (less scattered, less
variable) than group B’s scores.

This means that, while the score 117 in the
group A represents an extreme value –
because it’s the maximum score-, the same
score in group B is not as extreme, there is
another score above it (122).
18
Standards scores: Zi

One solution to this situation is:
Xi  X
Zi 
SX
19
Typing or Standardization

The process of obtaining the typical scores is
known as:
Typing or Standardization.

Typical scores are also known as scores:
Typified or Estandardized.
20
DEFINITION

“The standard score, typified or
estandardized indicates the number of
standard deviations that a particular raw
score is separated from its mean."
21
Standard scores calculation: Zi
Xi
Zi
Zi2
7
-3/2 = -1.5
2.25
9
-1/2 = -0.5
0.25
10
0
0
11
1/2 = 0.5
0.25
13
3/2 = 1.5
2.25
Tot. 50
0
n=5
22
Graphically there is a shift to the left side of the x-axis
when we convert raw scores in differential:
xi
[-3

-1
0
Xi
1
3]
[7
9
10
11 13]
And there is a concentration of scores when they
become standards.
Zi
-1.5 –0.5 0 0.5 1.5
23
Zs properties
1
The sum of standard scores is 0:
Zi = 0
24
Demonstration
 Xi  X 
 0
 Zi  0   
 S 
1
   ( X i  X)  0
S
25
2
The mean of standard scores is 0
Z=0
26
Demonstration
Zi
Z
0
n
as Zi  0
0
Z 0
n
27
3
The sum of the squares of standard
scores is equal to n:
Zi = n
2
28
Demonstration
2

(
X

X
)
1
i
2
2
 Zi 



(
X

X
)

2
2
i
S
S
1
n
2
2


(
X

X
)



(
X

X
)
n
i
i
2
2
 ( X i  X)
 ( X i  X)
n
29
4
The standard deviation and variance of
standard scores is equal to 1.
 Zi
n
2
=
-Z = =1
n
n
2
SZ
2
30
Demonstration
2
Z
S 
Z
n
but, as Z  0 y Z2  n
n
2
SZ   1 y S Z  1  1
n
2
Z
2
i
31
Standard scores properties
summary
1. The sum of typical scores is 0:
Z=0
2. The mean of standard scores is 0:
Z=0
3. The sum of the squares of standard scores is equal
to n:
Zi =n
2
4. The standard deviation and variance of standard
2
scores is equal to 1:
2

Z
n
2
i
SZ =
-Z = =1
n
n
Example

The following scores:
4,85 1,68 1,12 0,56 0 -0,56 -1,12 -1,68 -4,85
Answer justification:
 A) Can they be the difference scores of a
data set?

B) Can they be the standard scores of a data
set?
33
Typical scores and normal curve
34

6
5
4
The limit of the bar chart: to increase indefinitely
the number of bars, reduces its size and thin bars
are more and more. In the limit, the broken line
is identified with the curve (Normal or Gaussian).
20
18
16
14
12
3
2
1
0
10
8
6
4
2
0
35
Among all probability distributions from absolutely
continuous random variables, this is the most
important.
These distributions are knowing as Laplace-Gauss,
but there were had already been used by De
Moivre.
36
Normal curve
1) The sample size grows.
2) The bar size decreases.
For each pair of values of  and  we’ll have one
normal curve.
37
Normal curve characteristics
38
1ª
  0
- 
  
+ 
This distribution depends on 2 parameters :  and 
39
2ª
Mo = Mdn =

The normal curve has a single maximum.
40
3ª
 
  1
  1
The normal curve has 2 inflection points
41
4ª
The normal curve is asymptotic to the abscissa.
42
5ª
g1 = 0
It is a symmetrical distribution(g1 = 0)
43
6ª
g2 = 0
It is a mesokurtic distribution (g2 = 0).
44
X  N ( ;  )


The parameter  gives the distribution center and the parameter
variability , verifying the following equalities :

, the
45
-3
-2
-
68 %

2
3
95,5 %
99,7 %
46
Case 1
A  B  C
A  B  C
47
 A   B  C
A

A
B
A

C
B
48
Applications to the typified normal
curve
49

The characteristics of the normal
distribution have many distributions, each
with their means and standard deviations.

Implies the existence of infinite normal
distributions.
Solution: convert raw scores into typical.
 Implies to transform the normal
distribution in a standardized normal
distribution.


It has always mean 0 and standard deviation 1.

Generally, standardized distances in
themselves are not usually of our interest,
but as an intermediate product between
raw scores and proportions.

Translation of raw scores into proportions.

Translation of proportion to raw scores.

Translation: use the Tables

¿Qué distancia estandarizada representa
el 33.4% de los datos, inmediatamente
por encima de la media?
Attending to raw scores and table use,
we’ll going to use one of the following 2
choices:
1. Translations since raw scores to
proportions:




Translation since raw scores to standard
scores.
Look for the area in the table.
We work with the areas if it’s necessary to
obtain the result.

Ex. In a normal distribution with mean
100 and standard deviation 15, how much
data proportion do the values between 70
and 130 have?
2. Translations since proportions to raw
scores:



We work with the area.
Translation the area (to the mean) in typical
scores.
Translation this score in a raw score.

Ex. In a normal distribution with mean
100 and standard deviation 15, what score
defines the top 10% of the data?