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220109P - VARIABLES and STATISTICAL DISTRIBUTIONS

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Presented in the Research Camp of AlMaarefa University on Sunday, 9 January, 2022 at 12.30-1.00. By: Professor Omar Hasan Kasule Sr. MB ChB (MUK), MPH (Harvard), DrPH (Harvard), Professor of Epidemiology and Bioethics

 

CONSTANTS and VARIABLES

  • A constant has only one unvarying value under all circumstances for example p and c = speed of light.
  • A random variable can be qualitative (descriptive with no intrinsic numerical value) or quantitative (with intrinsic numerical value).
  • A random quantitative variable results when numerical values are assigned to results of measurement or counting. It is called a discrete random variable if the assignment is based on counting. It is called a continuous random variable if the numerical assignment is based on measurement.
  • The numerical continuous random variable can be expressed as fractions and decimals. The numerical discrete random variable can only be expressed as whole numbers.


VARIABLES and STATISTICAL DISTRIBUTIONS

  • Statistical distributions are graphical representations of mathematical functions of random variables.
  • Each random variable has a corresponding statistical distribution.
  • Each statistical distribution is associated with a specific statistical analytic technique.
  • Choice of the technique of statistical analysis depends on the type of statistical distribution. The statistical distribution depends on the type of random variable.


QUALITATIVE (NON NUMERICAL) RANDOM VARIABLES

  • Qualitative variables (nominal, ordinal, and ranked) are attribute or categorical with no intrinsic numerical value.
  • The nominal has no ordering.
  • The ordinal has ordering.
  • The rank has observations arrayed in ascending or descending orders of magnitude.


QUANTITATIVE (NUMERICAL) DISCRETE RANDOM VARIABLES

  • The discrete random variables are the Bernoulli, the binomial, the multinomial, the negative binomial, the Poisson, the geometric, the hypergeometric, and the uniform. Each random variable is associated with a statistical distribution.
  • The Binomial is the commonest. is the number of successes in more than 2 consecutive trials each with a dichotomous outcome?



QUANTITATIVE (NUMERICAL) CONTINUOUS RANDOM VARIABLES

The continuous random variables can be natural such as the normal, the exponential, and the uniform, or artificial such as chi-square, t, and F variables.

  • The Normal represents the result of a measurement on the continuous numerical scale such as height and weight.



QUANTITATIVE (NUMERICAL) CONTINUOUS RANDOM VARIABLES, con’t.

  • The Exponential is the time until the first occurrence of the event of interest. [Line graph of an exponential distribution]
  • The uniform represents the results of a measurement and takes on the same value at repeated trials.



SCALES OF QUANTITATIVE (NUMERIC-AL) CONTINUOUS RANDOM VARIABLES

  • The continuous R.V can be measured on either the interval or the ratio scales. Only 2 measurements are made on the interval scale, the calendar, and the thermometer. The rest of the measurements are on the ratio scale.
  • Properties of the interval scale: Zero is arbitrary with no biological meaning, and both negative and positive values are allowed.
  • Properties of the ratio scale: Zero has a biological significance, values can only be positive.


SIX PROPERTIES OF A RANDOM VARIABLE (DISCRETE OR CONTINUOUS)

  • The expectation of a random variable is a central value around which it hovers most of the time for example the mean (average).
  • The variations of the random variable around the expectation are measured by its variance.
  • Covariance measures the co-variability of the two random variables.


SIX PROPERTIES OF A RANDOM VARIABLE (DISCRETE OR CONTINUOUS), cont’. - 1

  • Correlation measures the linear relation between two random variables.



SIX PROPERTIES OF A RANDOM VARIABLE (DISCRETE OR CONTINUOUS), cont’. - 2

  • Skewness measures the bias of the distribution of the random variable from the center.



SIX PROPERTIES OF A RANDOM VARIABLE (DISCRETE OR CONTINUOUS), cont’. - 3

  • Kurtosis measures the peakedness of the random variable is at the point of its expectation.



TRANSFORMATIONS OF RANDOM VARIABLES

  • Quantitative variables can be transformed into qualitative ones.
  • Qualitative variables can be transformed into quantitative ones but this is less desirable.
  • The continuous variable can be transformed into a discrete variable.
  • Transformation of the discrete into the continuous may be misleading.