Correlation is a statistical technique that describes the degree of relationship between two variables when you have bivariate data.

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• Correlation is a statistical technique that
  describes the degree of relationship between two
  variables when you have bivariate data.
• A bivariate distribution is required for
  correlation and regression technique.
• A bivariate distribution is a joint distribution
  of two variables, the individual scores of which
  are paired in some logical way.
• A bivariate distribution may show positive
  correlation, negative correlation, or zero
  correlation.

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• In a case of a positive correlation between two
  variables, high measurements on one variable tend
  to be associated with high measurements on the
  other variable, and low measurements on one
  variable with low measurements on the other.
• Table 1 shows a graph of relationship of tall
  fathers tend to have sons who grew up to be tall
  men.
• Short fathers tend to have sons who grow up to be
  short men.
• If such were the case (which, of course, is
  ridiculous), then it would be possible to predict
  without error the adult height of an unborn son
  simply by measuring his father.

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• In Figure 1, each point represents a pair of
  scores, the height of a father and the height of
  his son.
• Such an array of points is called a scatterplot.
• The line that runs through the points is called a
  regression line.
• It is a line of best fit.
• When there is perfect correlation (r 1.00), all
  points fall exactly on the regression line.

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• When a correlation is negative, increases in one
  variable are accompanied by decreases in the
  other variable (an inverse relationship).
• With negative correlation, the regression line
  goes from the upper left corner of the graph to
  the lower right corner.
• As you may recall, such lines have a negative
  slope.
• Although some correlation coefficients are
  positive and some are negative, one is not more
  valuable than the other.

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• The algebraic sign simply tells you the direction
  of the relationship (which is important when you
  are describing how the variables are related).
• The absolute size of r, however tells you the
  degree of the relationship.
• A strong relationship (either positive or
  negative) is usually more valuable than a weaker
  one.
• A zero correlation means there is no linear
  relationship between two variables.

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• High and low scores on the two variables are not
  associated in any predictable manner.
• Figure 2 shows a scatterplot that produces a zero
  correlation coefficient.
• When r 0, the regression line is horizontal at
  a height of Y.
• This make sense if r 0, then your best
  estimate of Y for any value is Y.

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• A correlation coefficient provides a quantitative
  way to express the degree of relationship that
  exists between two variables.
• The definition formula is
• where r Pearson product-moment correlation
  coefficient
• zx a z score for variable X
• zy the corresponding z score for variable Y
• N number of pairs of X and Y values

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• Because researchers often use means and standard
  deviations when telling the story of the data,
  this formula (Blanched formula) is used by many
• Where x y paired observations
• xy product of each x value multiplied
  
• by its paired y value
• mean of variable x
• mean of variable y
• standard deviation of variable x
• standard deviation of variable y
• N number of pairs of observations

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• With the raw score formula, you calculate r from
  the raw scores without computing means and
  standard deviations.
• The formula is
• Remember that N is the number of pairs of values.
  

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• The basic simple interpretation of r is probably
  familiar to you at this point.
• A correlation coefficient measures the degree of
  linear relationship between two variables of a
  bivariate distribution.
• What is qualifies as a large correlation
  coefficient? What is small?
• Jacob Cohen proposed that the question be
  answered by calculating an effect size index (d)
  and that d values of .20, .50, and .80 were
  designated as small, medium, and large,
  respectively.

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• In a similar way, Cohen addressed the question of
  calculating an effect size index for correlation
  coefficients.
• Small r .10
• Medium r .30
• Large r .50
• The correlation coefficient is also the basis of
  the coefficient of determination, which tells the
  proportion of variance that two variables in a
  bivariate distribution have in common.

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• The coefficient of determination is calculated by
  squaring r it is always a positive value between
  0 and 1.
• coefficient of determination r2

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• Linear regression is a technique that uses the
  data to write an equation for a straight line
  then to make predictions.
• Often predictions are based on an assumption that
  the relationship between two variables is linear.
  
• Formula for a straight line is
• Y mX b
• Where - Y and X are variables representing
  scores on
• the Y and X axes
• - m slope of the line
• - b intercept of the line with the Y axis

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• Least square method is
• Consider this data
• Father height 64 in
• First daughter height 66 in
• Second daughter height 64 in
• Draw a regression line using least square method

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• There is an error for each person on the
  scatterplot.
• The least square method creates a straight line
  such that the sum of the squares of the errors is
  a minimum.
• The least square method produces numerical values
  for the slope and the intercept write the
  equation for a straight line this line is the
  one that best fits the data.
• In statistics, the regression equation is

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• Where Y Y value predicted from a particular X
• value
• a point at which the regression line
• intersects the s the axis
• b slope of the regression line
• X X value for which you wish to predict
  a Y
• value
• Note
• In correlation problem, the symbol Y can be
  assigned to either variable, but in regression
  equation, Y is assigned to the variable you wish
  to predict.

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• You need the values for a and b, which are called
  regression coefficients can be calculated from
  any bivariate set of data.
• To calculate b,
• To calculate a,

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• Let use our data from earlier correlation
  problem, predict math from oral test in SPM.

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• Entering these regression coefficient values into
  the regression equation produces a formula that
  predicts math from oral test

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