Review of basic statistics

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Review of basic statistics

• Variables and Random Variables
• A variable is a quantity (such as height, income,
  the inflation rate, GDP, etc.) that takes on
  different values across individuals, families,
  nations, months, quarters, etc. A constant, on
  the other hand, does not vary--e.g., the number
  of heads on a person.
• A random variable is a type of variable which has
  its value determined at least in part by the
  element of chance

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• Measures of Central Tendency
• The mode, median, and mean are measures of the
  central tendency of a random variable such as the
  height of males. If the statement is made with
  respect to this variable that the mode is
  5'10", it means that most common height (or the
  height which occurs with the greatest frequency)
  among males is 5'10".
• The median is the value of the random variable
  such that half the observations are above it and
  half below it. To say that median family income
  in the U.S. is 38,450" is to say that half of
  U.S. households have an income below that figure
  and half above it.

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• The population mean (symbolized by the Greek
  letter µ) is the average value of the variable
  for the population. Let m denote the number of
  observations (corresponding to the size of the
  population). Thus, we have

• Suppose we want to know the average height of
  adult males in the U.S. The practical approach
  would be to measure a representative sample
  (meaning, for example, that basketball players
  would not be disproportionately represented in
  the sample) of the population rather than the
  entire population. That is, we estimate the
  population mean by calculating a sample mean (?
  ). Let n be the number of observations in our
  sample. Thus we have

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• Measures of Dispersion
• Often we are interested in looking at the degree
  of dispersion of a random variable about its mean
  value. That is, are our observations of adult
  male height all bunched up around the mean or do
  we have wide dispersion about the mean? The
  population variance (? 2) is a measure of the
  dispersion of a random variable . The variance of
  random variable X is defined as

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• If we observe only a representative sample of the
  population, then (1) µ is unknown and (2) all
  the Xi s are not known. Thus, we estimate ?2
  by substituting ? for µ and summing across our
  sample observations of X This is called a sample
  variance (s2)

• Note that we must divide through by n - 1 to
  obtain an unbaised estimate of ?2 --that is s2 is
  an unbaised estimator of ?2 if E(s2 ) ?2

• The population standard deviation (?) is given by
  the square root of the population variance ( 2).
  You can think of as the average deviation from
  the mean. In the case of male adult height, one
  would like to see that measure expressed in
  inches--hence we take the square root of the
  variance.
• Similarly, the sample standard deviation (s) is
  given by the square root of the sample variance
  (s2).

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• Probability Distributions
• The probability density function of variable X is
  constructed such that, for any interval (a, b),
  the probability that X takes on a value in that
  interval is the total area under the curve
  between a and b. Expressed in terms of integral
  calculus, we have

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You should be familiar with this diagram
P(X)
Area under curve represents probability
a
b
X
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The standard normal distribution

• The normal distribution is probability density
  function which is symmetric about the mean--i.e.,
  the left-hand side of the distribution is a
  mirror image of the right-hand side. The formula
  for the normal probability density function is
  given by

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The normal distribution
68.27
95.45
?
?
2?
-2?
- ?
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• A random variable Z is said to be standard normal
  if it is normally distributed with mean of zero
  or and a variance of 1. If X is normally
  distributed with mean µ and variance ?2, we
  abbreviate with the expression
• X N(?, ?2)

• Thus, the expression used to indicate that the
  distribution of Z is standard normal is
• Z N(0, 1)

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The standard normal distribution

• For example
• If a 1.93, then Pr(Z ? a ) 0.1093
• And Pr(Z ? a ) 1 - 0.1093 0.8907

P(Z)
a
0
Pr(Z gt a) when Z N(0, 1)
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• Correlation of Random Variables
• ?To say that random variables X and Y are
  correlated is to say that changes in X are
  associated with changes in Y in the probabilistic
  or statistical sense. However, this does not
  necessarily mean that a change in X was the cause
  of a change in Y, or vice-versa. That is,
  correlation does not imply causality.
• Technically speaking, the statement X and Y are
  positively correlated means that the covariance
  between random variables X and Y is positive (or
  greater than zero).

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1, X gt E(X) and Y gt E(Y) 2, X lt E(X) and Y gt E(Y)
3, X lt E(X) and Y lt E(Y) 4, X gt E(X) and Y lt E(Y)
Y
2
1
E(Y)
3
4
0
E(X)
X
X and Y are positively correlated random variables
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• The sample covariance between X and Y (i.e., our
  estimate of the covariance when we do not observe
  the entire populations of Xs or Ys) is given by
  the following formula (the hat indicates an
  estimate)

• The covariance is positive if above average
  values of X tend to be paired with above average
  values of Y, and vice versa. The covariance is
  negative (and hence the variables are negatively
  correlated) if below average values of X tend to
  be paired with above average values of Y, and
  vice-versa. The magnitude of the covariance
  depends partly on the unit of measurement. Hence,
  we cannot depend on the size of the covariance to
  give an accurate measure of the strength of the
  relationship

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• The correlation coefficient (? ) is a unit-free
  measure of correlation. The sample correlation
  coefficient is given by

• It will always be the case that -1? ? ? 1.
• If ? 1, there is a perfect positive ( linear)
  correlation between X and Y. If ? -1, there
  is a perfect negative (linear) correlate between
  X and Y.