12.2 Scatter Plots
4 min read•june 25, 2024
Scatter plots are powerful tools for visualizing relationships between two variables. They show how one variable changes as another varies, revealing patterns, trends, and in data. Understanding scatter plots is crucial for interpreting data and making informed decisions.
Constructing and interpreting scatter plots involves plotting data points, identifying relationships, and assessing their strength and direction. This skill is essential for analyzing data in various fields, from economics to science, and forms the foundation for more advanced statistical techniques like regression analysis.
Scatter Plots
Scatter plot construction
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- Graphical representation of the relationship between two quantitative variables
- () plotted on the (time, age)
- () plotted on the (height, weight)
- Each point represents a single observation or
- Coordinates determined by the values of the independent and dependent variables for that observation (x = age, y = height)
- Steps to create a :
- Determine the appropriate scale for each axis based on the range of values for each variable
- Plot each data point on the graph using its corresponding x and y values
- Label the axes with the variable names and units, if applicable (x-axis: Age (years), y-axis: Height (cm))
- Provide a descriptive title for the scatter plot (Relationship between Age and Height)
Interpreting scatter plot relationships
- Direction of the relationship:
- Positive relationship: dependent variable increases as the independent variable increases
- Points move from the bottom left to the top right (height increases with age)
- Negative relationship: dependent variable decreases as the independent variable increases
- Points move from the top left to the bottom right (test scores decrease with increasing distractions)
- No relationship: no clear pattern between the dependent and independent variables
- Points appear randomly scattered with no discernible trend (shoe size and IQ)
- Positive relationship: dependent variable increases as the independent variable increases
- Strength of the relationship:
- Strong relationship: points closely follow a clear linear or nonlinear pattern (weight and calorie intake)
- Weak relationship: points are more scattered and deviate from any apparent pattern (hours of sleep and grades)
- Relationship strength described as strong, moderate, or weak
- Quantified by the , which measures the strength and direction of the
- Pattern of the relationship:
- Linear relationship: points roughly follow a straight line (income and years of education)
- : points follow a curved pattern (age and running speed)
- Outliers: data points that deviate significantly from the overall pattern
- Can have a substantial impact on the interpretation of the relationship (a few extremely high income individuals in a sample)
Appropriateness of regression lines
- ( or ) is appropriate when:
- Linear relationship exists between the dependent and independent variables
- , where changes in the independent variable cause changes in the dependent variable (studying time and test scores)
- Goal is to make predictions about the dependent variable based on values of the independent variable (predicting sales based on advertising expenditure)
- Regression line is not appropriate when:
- Nonlinear (curved) relationship between the variables (age and physical strength)
- No clear relationship between the variables with randomly scattered points (favorite color and math ability)
- Non-causal relationship with no logical basis for using one variable to predict the other (number of cars owned and GPA)
- Significant outliers that unduly influence the regression line (a few extremely large companies in a study of business profits)
- Considerations before calculating a regression line:
- Context of the data and the research question being addressed
- Assumptions of linear regression:
- : the relationship between variables is linear
- Independence of observations: data points are independent of each other
- Constant variance of : variability of the residuals is consistent across the range of the independent variable
Advanced scatter plot analysis
- : measures how closely the data points follow a pattern
- Can be visually assessed by the tightness of the points around a trend
- Formally quantified using statistical measures like the coefficient
- : groups of data points that are close together in the scatter plot
- May indicate subgroups or patterns within the data
- Can provide insights into different categories or behaviors in the dataset
- : a grid of scatter plots showing relationships between multiple variables
- Useful for exploring relationships in datasets with more than two variables
- Helps identify potential correlations and patterns across multiple dimensions
Key Terms to Review (28)
Bivariate Data: Bivariate data refers to the collection and analysis of two variables or characteristics for each individual or observation in a dataset. It involves studying the relationship and interdependence between two variables, allowing for a deeper understanding of patterns and trends within the data.
Causal Relationship: A causal relationship is a relationship between two variables where a change in one variable directly causes a change in the other variable. It establishes a direct connection between a cause and an effect, allowing for predictions and explanations about the relationship between the two variables.
Clustering: Clustering is a data analysis technique that groups similar data points together based on their characteristics or features. It is a way of identifying patterns and structure within a dataset by partitioning the data into meaningful groups or clusters.
Correlation: Correlation is a statistical measure that describes the strength and direction of the linear relationship between two variables. It quantifies the degree to which changes in one variable are associated with changes in another variable.
Correlation Coefficient: The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with -1 indicating a perfect negative correlation, 0 indicating no correlation, and 1 indicating a perfect positive correlation.
Data Point: A data point is a single observation or measurement within a set of data. It represents a specific value or piece of information that is collected and analyzed as part of a dataset.
Dependent Variable: The dependent variable is the outcome or response variable in a study or experiment. It is the variable that is measured or observed to determine the effect of the independent variable. The dependent variable depends on or is influenced by the independent variable.
Explanatory variable: An explanatory variable is a type of independent variable used in experiments to explain variations in the response variable. It is manipulated by researchers to observe its effect on the dependent variable.
Explanatory Variable: An explanatory variable, also known as an independent variable, is a variable that is manipulated or controlled in a study to determine its effect on the dependent or response variable. It is the variable that is believed to influence or cause changes in the outcome or dependent variable.
Independent Variable: The independent variable is a variable that is manipulated or changed by the researcher in an experiment to observe its effect on the dependent variable. It is the variable that the researcher has control over and intentionally varies to measure its impact on the outcome.
Least-squares regression line: A least-squares regression line is a straight line that best fits the data points on a scatter plot by minimizing the sum of the squares of the vertical distances (residuals) between observed values and the line.
Line of Best Fit: The line of best fit, also known as the regression line, is a straight line that best represents the relationship between two variables in a scatter plot. It is used to make predictions and estimate the value of one variable based on the value of the other variable.
Linear Relationship: A linear relationship is a mathematical relationship between two variables where the change in one variable is proportional to the change in the other variable. This type of relationship is often depicted visually through a scatter plot and can be further analyzed using regression techniques.
Linearity: Linearity refers to the property of a relationship between two variables where the change in one variable is directly proportional to the change in the other variable. This linear relationship can be represented by a straight line on a scatter plot.
Negative Correlation: Negative correlation refers to a relationship between two variables where as one variable increases, the other variable decreases, and vice versa. It indicates an inverse or opposing relationship between the variables.
Nonlinear Relationship: A nonlinear relationship is a type of relationship between two variables where the change in one variable is not proportional to the change in the other variable. This means the relationship between the variables does not follow a straight line pattern, but rather a curved or more complex pattern.
Outliers: Outliers are data points that significantly differ from the rest of the data in a dataset. They can skew the results and lead to misleading interpretations, affecting measures of central tendency, variability, and visual representations.
Positive Correlation: Positive correlation refers to a relationship between two variables where an increase in one variable is associated with an increase in the other variable. It indicates a direct, linear relationship between the variables, with both moving in the same direction.
Regression Line: The regression line is a best-fit line that represents the linear relationship between two variables in a scatter plot. It is used to predict the value of one variable based on the value of the other variable.
Residuals: Residuals, in the context of statistical analysis, refer to the differences between the observed values and the predicted values from a regression model. They represent the unexplained or unaccounted-for portion of the variability in the dependent variable, providing insights into the quality and fit of the regression model.
Response variable: A response variable is the outcome or dependent variable that researchers measure in an experiment to determine the effect of treatments. It is what changes as a result of variations in the independent variable.
Response Variable: The response variable, also known as the dependent variable, is the variable that is measured or observed in an experiment or study. It is the outcome or the characteristic of interest that may be influenced or predicted by the independent variable(s).
Scatter plot: A scatter plot is a graphical representation that uses dots to show the relationship between two quantitative variables. Each point on the plot corresponds to an observation from a dataset, where the position of the dot represents the values of the variables being compared. This visualization helps to identify patterns, trends, and correlations in the data, serving as a fundamental tool in descriptive statistics, linear equations, and prediction analysis.
Scatterplot Matrix: A scatterplot matrix, also known as a correlation matrix, is a visual representation of the relationships between multiple variables in a dataset. It displays a grid of individual scatterplots, each showing the relationship between two variables, providing a comprehensive overview of the multivariate relationships within the data.
Strength of Association: The strength of association refers to the degree or magnitude of the relationship between two variables in a statistical analysis. It quantifies how strongly the variables are related or correlated with one another, providing insight into the nature and importance of the relationship.
Trend Line: A trend line is a line on a scatter plot that represents the overall direction or pattern of the data points. It is used to identify and visualize the relationship between two variables.
X-Axis: The x-axis is the horizontal axis on a graph or chart, typically used to represent the independent variable or the variable being measured. It serves as a reference point for displaying and interpreting data along the horizontal dimension.
Y-axis: The y-axis is the vertical line in a two-dimensional graph that represents the values of the dependent variable. It is essential for visually interpreting data, as it allows for the comparison of different sets of values and highlights relationships among them. The position and scale of the y-axis can significantly affect how data is perceived and understood.