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Bivariate Data
Consider the below results table, for the variables x and y.
5 10 15 20 25 30 35 40 45
40 44 106 91 175 138 169 175 187
Data that consists of pairs of values of two random variables, like the above table, is called Bivariate data.
The explanatory variable is usually plotted as x, and, in the context of an experiment, is the variable that
is changed. The response variable changes as a result, and is dependent on the explanatory variable.…read more

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A line of best fit, called a least squares regression line in the Statistics world, will be a linear line. Linear
lines have the following formula:
The gradient, b, needs to be calculated, as well as the y-intercept, a. If the data follows a negative
relationship, then the gradient of the least squares regression line will be negative, and if the data follows a
positive relationship, the gradient of the least squares regression line will be positive.…read more

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So what is the significance of this equation for the least squares regression line?
250 The straight line shown on the diagram to
the left is the plotted version of the equation:
200
150
y Fitting a straight line to a set of Bivariate
100 data is called fitting a linear regression
50 model.
0 We can use the least squares regression
0 10 20 30 40 50 60 line to predict values of x which are outside
the range of the original data.…read more

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Linear Regression Example Question 2
A travelling salesman wants to be able to predict his journey time to clients from the distance he travels. He
records the distances he travels to 11 clients and his journey times:
Distance
65 137 257 84 105 346 124 157 201 98 132
(miles)
Time (hours) 2 2.5 6 2.25 3 7.75 3.25 3.75 4.5 2.75 3.5
The distance is the explanatory variable, whilst the time is the response variable.…read more

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Residuals
Presented below is a Bivariate data table and its plotted scatter diagram, with the least squares regression
line given by
5 10 15 20 25 30 35 40 45
40 44 106 91 175 138 169 175 187
250
As you can plainly see, none of the
200
actual points lie on the least squares
regression line. If we were to predict the
150
y value of x=5 using the least squares
100 regression line, we would get a value of
y at 48.08.…read more

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Covariance
Covariance measures the extent to which the values of x and y are associated, high with high and low with
low. It is given by the formula:
Covariance is denoted by the expression:
When a set of data follows a generally positive gradient, the covariance is positive. The two variables are
said to be positively correlated.
When a set of data follows a generally negative gradient, the covariance is negative. The two variables are
said to be negatively correlated.…read more

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Correlation Example Question 1
A family records for a number of days the midday temperature outside in degrees C, and the number of
units of electricity they use in that one day.
From this Bivariate data, the following values were calculated:
Calculate the product moment correlation coefficient of this data, and make a conclusion about what the
p.m.c.c informs us in the context of the question.…read more

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In Summary:
The explanatory variable is usually plotted as x, and, in the context of an experiment, is the
variable that is changed. The response variable changes as a result, and is dependent on the
explanatory variable.
A line of best fit, called a least squares regression line in the Statistics world, will be a linear line.
Linear lines have the following formula:
The gradient, b, needs to be calculated, as well as the y-intercept, a.…read more

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