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Biostatistics course
Part 6
Normal distribution
Dr. en C. Nicolás Padilla Raygoza
Facultad de Enfermería y Obstetricia de Celaya
Universidad de Guanajuato México
Presentación
 Médico Cirujano por la Universidad Autónoma de Guadalajara.
 Pediatra por el Consejo Mexicano de Certificación en Pediatría.
 Diplomado en Epidemiología, Escuela de Higiene y Medicina




Tropical de Londres, Universidad de Londres.
Master en Ciencias con enfoque en Epidemiología, Atlantic
International University.
Doctorado en Ciencias con enfoque en Epidemiología, Atlantic
International University.
Profesor Asociado B, Facultad de Enfermería y Obstetricia de
Celaya, Universidad de Guanajuato.
[email protected]
Competencies
 The reader will define what is Normal
distribution and standard Normal distribution.
 He (she) will know how are the Normal and
standard Normal distribution.
 He (she) will apply the properties of standard
Normal distribution.
 He (she) will know how standardize values to
change in a standard Normal distribution.
Introduction
 We know how calculate probabilities and to
find binomial distribution.
 But, there are other variables that they can
take more values that only two.


If they have a limited number of categories,
are categorical variables.
If they can take many different values, are
numeric variables.
Quantitative variables
 They can take many values and are
negative or positive.
Distribution of glycemia in 500 persons
N° of persons
120
100
80
60
40
20
0
<40
40-69
70-99
100-129
130-159
160-189
Level of glycemia mg/100 ml
190-219
220+
Quantitative variables
Percentage (%)
Distribution of glucose in blood levels in 500
persons
25
20
15
10
5
0
<40
40-69
70-99
100-129
130-159
160-189
Level glycemia mg/100 ml
190-219
220+
Quantitative variables
 What happen if the sample size is more big?
 Changed the histogram?
Distribution of glycemia in 3,500 persons
N° of persons
1200
1000
800
600
400
200
0
<40
40-69
70-99
100-129
130-159
160-189
Level of glycemia mg/100 ml
190-219
220+
Quantitative variables
 The distributions of many variables are
symmetrical, specially when the sample
size is big.
Stature in meters. n=1000
5
4
3
2
1
0
N° of persons
N° of persons
Stature in meters. n=10
1.5
1.6
1.7
Stature (m t)
1.8
400
300
200
100
0
1.5
1.6
1.7
Stature (m t)
1.8
Normal distribution
 It is used to represent the distribution of values that
they should observe, if we include all population. It
show the value distribution, if we repeat many times
the measure in a great population.
 Because of this, Y axis of Normal distribution is called
probability.
 An histogram show the value distribution observed in
a sample.
 An Normal plot show value distribution that it is
thinking that they can occur in the population of which
the sample was obtained.
Normal distribution
 We can use the Normal distribution, to
answer questions as:


What is the probability of a adult man has a
glycemia level < or = to 50 mg/100 ml?
We can answer, taken the percentage of
observed men, with glycemia levels < 150
mg/100 ml.
Normal standard distribution
 Normal distribution is defined by a
complicated mathematical formulae, but we
have published tables that define area under
the Normal curve: Normal standard
distribution.
 In this the mean is 0 and standard deviation
is ±1.
 These tables are in statistic textbooks.
Normal standard distribution
 When Z=0.00 is 0.5
 When Z = 1.00 is 0.159 or 0.841
-1
0
+1
Normal standard distribution
 Many times, we want the range out of area of curve.
 Area out of range is complementary to the range in of
area of the curve.
-1
0
+1
Range
Area in the
range
Area out the
range
-1, + 1
68.3%
31.7%
-2, +2
95.4%
0.6%
-3, +3
99.7%
0.3%
-4, +4
99.99%
0.01%
Normal standard distribution
 Area out of range is that we shall use with
more frequency.
 There are tables published with these values
and they are the table with two tails.
Standardized values
 Any Normal distribution can be changed in a
Normal standard distribution.
 To standardize value, we subtract of each
value its mean and it is divided by standard
deviation.



Example
Mean of stature is 1.58 mt with s = 0.12
Standardized value for stature of 1.7 is 1.7 1.58/0.12 = 0.12/0.12 = 1.00
Standardized values
 How we do apply the lessons learned?
 What is the probability of one person in the
population, has less than 1.6 mts of stature?




We know that should calculate what is the area under
curve to the left of 1.6 mts, under a Normal curve with
a mean of 1.58 and s of 0.12.
1.6 -1.58/0.12 = 0.167
Using the tables of Normal standard distribution p
lower value (to the left of mean) for 0.167 is 0.5675 =
56.75%.
We can answer that the probability of an individual of
this population has less than 1.6 mts is 56.75%
 It is a population with low stature!
Standardized values
 Cautions:

Sample size


We have used a sample of 1000 measures, if the
sample size is less, the results are different.
Supposition:


The results depend of the supposition that
statures are distributed Normally with the same
mean and standard deviation found in the
sample.
If the supposition is incorrect, the results are
wrong.
Non-Normal distribution
 Not all quantitative variables have a Normal distribution.
Distribution of levels of glucose in
blood n=10
10
010
9
80
-8
9
60
-6
9
4
3
2
1
0
<5
0

We measured levels of glycemia in 10 personas: the
distribution are skewed.
Do we can use the properties of Normal distribution?
Number of
patient

Glucose in blood (m g/100 m l)
Non-Normal distribution
 If, it is skewed to the right, we can apply logarithmic
transformations.
 If, it is skewed to the left, we squared each value.
 Original values is transformed in natural logarithmic
values (button “ln” in scientific calculators).
Distribution of log values of
glucose in blood n=10
Glucose in blood (m g/100 m l)
-5
.1
4.
9
4.
3
-4
.5
4
3
2
1
0
<4
.0
Number of
patients
10
010
9
80
-8
9
60
-6
9
4
3
2
1
0
<5
0
Number of
patients
Distribution of values of glucose in
blood n=10
log glucose in blood (m g/100 m l)
Bibliography
 1.- Last JM. A dictionary of epidemiology.
New York, 4ª ed. Oxford University Press,
2001:173.
 2.- Kirkwood BR. Essentials of medical
ststistics. Oxford, Blackwell Science, 1988: 14.
 3.- Altman DG. Practical statistics for medical
research. Boca Ratón, Chapman & Hall/
CRC; 1991: 1-9.