Biostatistics course Part 6 Normal distribution

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Biostatistics coursePart 6Normal distribution

• Dr. en C. Nicolás Padilla Raygoza
• Facultad de Enfermería y Obstetricia de Celaya
• Universidad de Guanajuato México

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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.
  
• padillawarm_at_gmail.com

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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.

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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.

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Quantitative variables

• They can take many values and are negative or
  positive.

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Quantitative variables
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Quantitative variables

• What happen if the sample size is more big?
• Changed the histogram?

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Quantitative variables

• The distributions of many variables are
  symmetrical, specially when the sample size is
  big.

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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.

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Normal distribution

• We can use the Normal distribution, to answer
  questions as
• What is the probability of a adult man has a
  glycemia level lt or to 50 mg/100 ml?
• We can answer, taken the percentage of observed
  men, with glycemia levels lt 150 mg/100 ml.

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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.

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Normal standard distribution

• When Z0.00 is 0.5
• When Z 1.00 is 0.159 or 0.841

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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.

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
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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.

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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

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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!

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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.

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Non-Normal distribution

• Not all quantitative variables have a Normal
  distribution.
• We measured levels of glycemia in 10 personas
  the distribution are skewed.
• Do we can use the properties of Normal
  distribution?

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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).

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Bibliography

• 1.- Last JM. A dictionary of epidemiology. New
  York, 4ª ed. Oxford University Press, 2001173.
• 2.- Kirkwood BR. Essentials of medical
  ststistics. Oxford, Blackwell Science, 1988 1-4.
• 3.- Altman DG. Practical statistics for medical
  research. Boca Ratón, Chapman Hall/ CRC 1991
  1-9.