Understanding how different types of graphs are used is crucial for presenting data effectively. The question asks for the most suitable diagrammatic presentation to show the relationship between two sets of data: population size and number of services.

We need a type of graph that can display individual data points for two related variables and show if there is a pattern or correlation between them. Let's look at the given options:

Analyzing Data Visualization Options

Let's examine each option to determine its suitability for showing the relationship between population size and number of services.

• Pie chart: A pie chart is used to show the proportion of different categories that make up a whole. It represents parts of a whole as slices of a pie. This is not suitable for showing how two different, potentially continuous variables like population size and number of services relate to each other.
• Scatter graph: A scatter graph (also called a scatter plot) uses Cartesian coordinates to display values for typically two variables for a set of data. Each point on the scatter graph represents an observation, with the value of one variable determining the position on the horizontal axis (x-axis) and the value of the other variable determining the position on the vertical axis (y-axis). This type of graph is specifically designed to visualize the relationship or correlation between two quantitative variables. For example, we could plot population size on the x-axis and number of services on the y-axis. The pattern of the points would show if areas with larger populations tend to have more services, fewer services, or no clear relationship.
• Bar chart: A bar chart uses rectangular bars of varying heights or lengths to represent and compare discrete categories or show changes over time for a single variable. While you could show population size or number of services for different locations using bars, a standard bar chart doesn't effectively display the relationship between these two variables simultaneously for multiple data points in the way a scatter plot does.
• Triangular graph: A triangular graph (also known as a ternary plot) is used to plot the proportions of three variables that add up to a constant sum (usually 100%). This is typically used in fields like geology (e.g., plotting percentages of sand, silt, and clay in soil). It is not suitable for plotting two independent variables like population size and number of services.