Data, Graphs, and Functions

Many activities and processes used in today’s careers and lifestyles rely on data to analyze situations, make predictions, and take action.  Stacks of data alone would be too overwhelming to process efficiently and effectively.  So graphs and functions help visualize and analyze patterns, trends, and relationships within and between sets of data.  As with many technologies, there is no one perfect form that meets every need of the user, so we need to be adept at going between the forms of data, graphs, and functions to recognize and use the full breadth of information within a collection of measurements.

Data

Since data are so useful, humans have been collecting data for at least 20,000 years.  Data capture information or knowledge in a form that provides efficient processing and application.  The data we will focus on in this website are measured relative to a scale with exact differences between the values and may or may not have a “true zero.” For example, the Fahrenheit temperature scale has a zero value, but there are values below this, so 0ºF isn’t an absolute zero temperature value based on the Fahrenheit scale.  Air pressure does have a true zero value: there is nothing with less air pressure than 0 lbs/in².  The color intensities shown in the following screenshot from the game, Suzie the Artist, have a true zero value.

How We View Data

Sometimes, we are able to collect data through controlled experiments, where one variable changes in an ordered fashion, which helps us see trends in and relationships between other variables, and sometimes, we need to collect data about an ever-changing system, such as the atmosphere, oceans, ecosystems, financial and business systems, etc.  Viewing tabular sets of data from one perspective limits how well you can spot important patterns and details.

Graphs

Graphs are visual displays showing the relation between variable quantities and were created well after humans began collecting data. There are many types of graphs, such as pie charts, bar graphs, scatter plots, etc.  We will focus on scatter plots, which are typically two variables measured along one of a pair of axes at right angles.

Below are three scatter plots of the color intensity data shown above.  Notice that the order the data were picked cannot be discerned, so if this were important, you would need to change your graph, add additional information to your graph, or change how you viewed the data. 

What the graphs quickly show are that the color intensities are linearly related to each other.  These relationships will help us identify what Suzie’s remaining two colors she will use that day.

Functions

Functions are maps of how input values are converted to output values where each input has only one output.  Typically functions are written symbolically.  One common form is f(x) = x² where f is the symbol representing the function, x is the input to the function, and x² is the output.  Another form of this function is y = x² where y = output, x = input, and there isn’t a symbol of the function.

Each of the three graphs above contains an equation showing how one color intensity is mapped to another.  Using the Red vs Green graph, y = –x + 100 where x = the red intensity and y = the green intensity.  By substituting these into the equation, we get:

Green = -Red + 100 = 100 – Red

There are two remaining equations, but since all three colors are related to each other, we only need to use one more.  The most straightforward is y = 0.3333x – 4 = x/3 – 4 where x = the green intensity and y = the blue intensity, or:

Blue = Green / 3 – 4

Now, if we have a starting color intensity, we can calculate the other color intensities.  What if Green = 25?

Blue = Green / 3 – 4 = 25 / 3 – 4 = 4.3333

Does this fit the pattern observed in the data?  No. Looking more closely at the data, we see that all of the color intensities are integers.  So pick a green intensity that is divisible by 3, say 24.

Blue = 24 / 3 – 4 = 4

Does this fit the pattern observed in the data?  No, since all of the intensities are odd integers.  So pick a green intensity divisible by three, which yields an odd integer, say 27.

Blue = 27 / 3 – 4 = 5

And this fits the patterns in the data, so complete our calculation of the red intensity:

Red = 100 – Green = 100 – 27 = 73

Typing this into the game, we see this is correct.  After identifying a second correct color, notice that the equations the game used are not identical to what we calculated, but you can rewrite the ones we found to be the ones used in the game.

Combining the Strengths of Data, Graphs, and Functions

To complete the challenge illustrated above, you needed to study:

• the data to recognize that all of the color intensities were odd integers,
• the graphs to see that the color intensities were linearly related to each other, and
• the functions that quantified the relationships between the intensities to select two from the possible remaining colors that fit Suzie’s rules.

The following concept map is an attempt to show the connections between data, graphs, and functions.  To maximize the strength of each, you need to be able to create each form from another.