Covariation, Correlation, Linear Regression, Residuals

• Variance measures how much the data differs from the average value (mean)
• Variance measures identify whether most students have similar stress levels or if there are wide disparities from the mean within the group
• Co-variation is a statistical term that describes how two variables move together in relation
• If two variables both increase or decrease at the same time, they are said to co-vary.
• Co-variance predicts changes in one variable based on changes in another.
• Researcher finds that as stress levels increase, sleep quality tends to decrease, it would suggest a negative co-variation
• Correlation, not only observes the relationship between these two variables but also measures the strength and direction of this relationship.
• Co-variation would look at how two variables move together.
• The phrase "correlation does not imply causation" means that just because two variables appear to be related to each other, it does not mean that one variable actually causes the other to happen.
• Correlation is a statistical measure that describes the degree of strength and direction of a linear relationship between two variables.
• Goodness-of-fit refers to how well a statistical model fits observed data.
• Linear regression is about predictions for the future data
• Covariation and correlation measure a linear relationship between two variables
• correlation and co-variation cannot claim causation they are observable variables not manipulated
• what is the equation for a linear regression?


  Y= mX + c
• Covariance extends this concept of variance to consider two variables and their joint variability.
• Correlation further standardises covariance to provide a more interpretable measure of the relationship between two variables.
• Measures the degree and direction of the linear relationship between two variables, this describes what type of statistical test?

  Pearsons correlation
• Captures the relationship of a dependent variable as a function of independent variables, what type of statistical relationship is described here?

  Linear regression
• A negative co-variance relationship would imply that as one variable increases, the other tends to decrease because they are moving in opposite directions
• A positive co-variance relationship would imply that as one variable increases, the other tends to increase because they are moving in the same direction
• A positive co-variance relationship would imply that as one variable decreases, the other tends to decrease because they are moving in the same direction
• What does this linear regression plot represent?

  The sum of squares

  A) Sum of Squares

  
• A linear regression's aim is to try and find a line that best fits data to describe the relationship between two continuous variables
• How is the sum of squares calculated?

  By squaring the difference between our line's predicted values and the actual data points
• The goal of linear regression is to minimise the sum of squares to achieve the best-fit line for the data.
• Sum of squares measure of the total variability (residual left-overs)of the dataset explained by the regression model
• Analogy: A Seamstress (linear regression) wants to pin and tuck (minimise) that extra variant fabric (sum of squares) to make the dress (line) fit as sleekly as possible to the mannequin (data)

  
• Residuals on a linear regression plot represents the left over variability (extra fabric)
• The sum of squares calculation allows us to measure how well our linear regression fits the relationship between continuous variables within our data

  
• What are the two ways we are able to find if the linear regression fits our data?

  • Correlation equation r
  • Sum of squares (variance) R2