Graphs

Graphs

Graphs are visual representations of data. They help show the relationship between the independent and dependent variables. 

• Linear Relationships - A change in the independent variable causes a constant change in the dependent variable. The relationship can be positive (an increase in the independent causes an increase in the dependent) or negative (an increase in the independent causes a decrease in the dependent). If the relationship goes through the origin of the graph, the relationship is said to be directly proportional.

• Non-Linear Relationships - A change in the independent variable does not cause a constant change in the dependent variable. Often, non-linear data is manipulated during the Analysis phase of a procedure to generate a linear relationship.

Graphs can be presented two different ways.

• Sketched - Sketched graphs simply show the relationship between the independent and dependent variable. They do not require data points to be plotted accurately. To properly sketch a graph, both the x and y axes must be labeled with units to state how the variables relate to one another.

• Plotted - Plotted graphs accurately place data based on values of the independent and dependent variables. Once plotted, trend lines are added to better show the relationship between the variables. Labels for the x and y axes are also included.

Lines of Best Fit

Lines of best fit represent the general trend of the relationship between the variables while providing a mathematical model to allow predictions to be made. The line can be linear or non-linear in nature. 

When sketching lines of best fit, the line should have roughly the same number of points above and below the line. If the line of best fit is linear, a straightedge should be used to generate a straight line.

Linear lines of best fit provide a coefficient of determination, R2, a quantitative method to state how successful the data is at predicting the changes in the dependent variable as the independent variable changes. The closer the value is to 1.0, the closer the data points are located to the line of best fit and the more successful the data is at predicting the dependent variable. If an R2 is greater than 0.81, it can be assumed that the linear trend line is a reliable model.

Deriving Quantities from Graphs

The line of best fit of a linear relationship can be described by the equation y = mx + b. Once this equation is known, quantities can be derived from the graph.

• Gradients - The gradient, or slope, of a range of data provides a quantitative value of the relationship between the variables. Steep gradients state that the dependent variable changes quickly when the independent variable changes while less steep gradients state that the changes occurs slowly.

Knowing the gradient of a line allows the value of either the independent or dependent variable to be determined if the other variable is known. To do this:

Gradient = y variable / x variable

Interpolated Values - Interpolated values are values that are within the range of collected data.

Extrapolated Values - Extrapolated values are values that are outside the range of collected data.

• Intercepts - The intercepts, both x and y, provide a value for a variable when the other variable = 0.