2.1 Introducing Statistics: Are Variables Related?

• Correlation is a statistical measure that describes the linear relationship between two variables
• A correlation coefficient between 0 and 1 indicates a positive correlation, which becomes stronger as $r$ approaches 1
• A correlation coefficient of r = -1</latex> indicates a perfect negative correlation, where variables move in opposite directions and $r$ is at its strongest
• Match the example with its explanation:

  Increased exercise ↔️ Improves health

  Higher education ↔️ Higher income

  Ice cream sales and crime rates ↔️ Confounding factors
• Correlation is a statistical measure that describes the linear relationship between two variables
• Causation occurs when one variable directly causes a change in another
• What is an example of causation mentioned in the study material?

  Exercise improves health
• Match the correlation type with its visual pattern in a scatter plot:

  Positive correlation ↔️ Dots trend upwards

  Negative correlation ↔️ Dots trend downwards

  No correlation ↔️ Dots are scattered randomly
• Causation requires rigorous evidence to prove.


  True
• The X-axis of a scatter plot represents the independent variable.
• What is an example of a positive correlation in a scatter plot?

  Height vs Weight
• Scatter plots imply causation between variables.

  False
• What does a correlation coefficient of $r =$$0.979$ indicate?


  Strong positive correlation
• Outliers are data points that lie far from the general pattern of a dataset.
• A correlation coefficient of r = 1</latex> indicates a perfect positive correlation.


  True
• A correlation coefficient of $- 1 < r < 0$ indicates a negative correlation.


  True
• Correlation measures the linear relationship between variables but does not necessarily indicate causation
• Order the following examples based on whether they indicate causation or correlation:

  1️⃣ Increased exercise improves health (causation)

  2️⃣ Higher education leads to higher income (causation)

  3️⃣ Ice cream sales and crime rates increase in summer (correlation)

  4️⃣ Number of firefighters at a fire increases with fire size (correlation)
• Match the r value with its interpretation:

  r = 1 ↔️ Perfect positive correlation

  r = 0 ↔️ No linear correlation

  r = -1 ↔️ Perfect negative correlation
• Order the key aspects of causation and correlation based on their relationship strength:

  1️⃣ Causation: Direct cause-and-effect

  2️⃣ Correlation: Statistical association
• The X-axis of a scatter plot represents the independent variable.
• Causation implies a direct cause-and-effect relationship
• What is a scatter plot used to visualize?

  Relationship between two variables
• Match the correlation with its visual pattern:

  Positive ↔️ Dots trend upwards from left to right

  Negative ↔️ Dots trend downwards from left to right

  No Correlation ↔️ Dots are scattered randomly
• What does a scatter plot with no correlation look like?

  Dots are scattered randomly
• Steps to calculate the correlation coefficient ($r$)


  1️⃣ Calculate the mean of x ($\bar{x}$) and y ($\bar{y}$)

  2️⃣ Find the deviations by subtracting each mean from its corresponding value

  3️⃣ Multiply each pair of deviations and sum them up

  4️⃣ Square each deviation, then sum each series separately

  5️⃣ Apply the correlation coefficient formula
• What determines the strength of a correlation?

  Absolute value of $r$
• Why is it important to identify and address outliers in statistical analysis?

  For accurate interpretation
• What is the range of values for the correlation coefficient $r$?


  -1 to 1
• What does a correlation coefficient of r = 0</latex> mean?

  No linear correlation
• What determines the strength of a correlation in the correlation coefficient $r$?


  Absolute value of $r$
• Correlation between ice cream sales and crime rates in summer indicates that ice cream sales cause higher crime rates.

  False
• The correlation coefficient r ranges from -1 to 1.

  True
• Correlation implies causation.

  False
• Scatter plots are used to visualize the relationship between two quantitative variables.

  True
• What is an example of a positive correlation in a scatter plot?

  Height vs Weight
• What is an example of correlation without causation?

  Ice cream sales and crime rates
• The Y-axis of a scatter plot represents the dependent variable.


  True
• What is an example of a negative correlation in a scatter plot?

  Temperature vs Ice Cream Sales
• To calculate the correlation coefficient, the first step is to calculate the mean of x and y.