Unit 6 Associations In Data — Unit Plan

Consider the data collected from pulling back a toy car and then letting it go forward. In the first table, the data may not seem to have an obvious pattern. The second table has the same data and shows that both values are increasing together.

A scatter plot of the data makes the pattern clear enough that we can estimate how far the car will travel when it is pulled back 5 in.

Patterns in data can sometimes become more obvious when reorganized in a table or when represented in scatter plots or other diagrams. If a pattern is observed, it can sometimes be used to make predictions. This is a scatter plot for this scenario:

A scatterplot. Horizontal, from 0 to 12, by 2’s, labeled distance pulled back, inches. Vertical, from 0 to 40, by 10’s, labeled distance traveled, inches. 6 data points.  Trend linearly upward and right.

Here are two tables representing two different situations.

• A student runs errands for a neighbor every week. The table shows the pay he may receive, in dollars, in any given week.

  number of errands	pay in dollars	difference from previous week	factor from previous week
  0	10	-	-
  1	15	5	1.5
  2	20	5	1.33
  3	25	5	1.25
  4	30	5	1.2

• A student at a high school heard a rumor that a celebrity will be speaking at graduation. The table shows how the rumor is spreading over time, in days.

  day	people who have heard the rumor	difference from previous day	factor from previous day
  0	1	-	-
  1	5	4	5
  2	25	20	5
  3	125	100	5
  4	625	500	5

Once we recognize how these patterns change, we can describe them mathematically. This allows us to understand their behavior, extend the patterns, and make predictions.

Notice that in the situation with the student running errands, the difference is constant from week to week, while the factor changes. In the situation about a rumor spreading, the difference changes from day to day, but the factor is constant. This can give us clues to how we might write out the pattern in each situation.

Scatter plots show two measurements for each individual from a group. For example, this scatter plot shows the weight and height for each dog from a group of 25 dogs. 

A scatterplot. Horizontal, from 6 to 30, by 3’s, labeled dog height, inches. Vertical, from 0 to 112, by 16’s, labeled dog weight, pounds. 24 data points.  Trend upward and to right.

We can see that the tallest dogs are 27 inches, and that one of those tallest dogs weighs about 75 pounds while the other weighs about 110 pounds. This shows us that dog weight is not a function of dog height because there would be two different outputs for the same input. But we can see a general trend: taller dogs tend to weigh more than shorter dogs. There are exceptions. For example, there is a dog that is 18 inches tall and weighs over 50 pounds, and there is another dog that is 21 inches tall but weighs less than 30 pounds.

When we collect data by measuring attributes like height, weight, area, or volume, we call the data numerical data (or measurement data), and we say that height, weight, area, or volume is a numerical variable.

Here is a graph showing the luminescence of a glow-in-the-dark paint, measured in lumens, over a period of time, measured in hours. The luminescence of this glow-in-the-dark paint can be modeled by an exponential function.

Notice that the amounts are decreasing over time. The graph includes the point $(0, 12)$. This means that when the glow-in-the-dark paint started glowing, its glow measured 12 lumens. The point $(1, 6)$ tells us the glow measured 6 lumens 1 hour later. Between 3 and 4 hours after the glow-in-the-dark paint began to glow, the luminescence fell below 1 lumen.

We can use the graph to find out what fraction of luminescence stays each hour. Notice that $\frac{6}{12}=\frac{1}{2}$ and $\frac{3}{6}=\frac{1}{2}$. As each hour passes, the luminescence that stays is multiplied by a factor of $\frac{1}{2}$.

If $y$ is the luminescence, in lumens, and $t$ is time, in hours, then this situation is modeled by the equation:

$y=12 \boldcdot (\frac{1}{2})^t$

We can confirm that the data is changing exponentially because it is multiplied by the same value each time. When the growth factor is between 0 and 1, the quantity being multiplied decreases, the situation is sometimes called “exponential decay,” and the growth factor may be called a “decay factor.”

When a linear function fits data well, we say there is a “linear association” between the variables. For example, the relationship between height and weight for 25 dogs with the linear function whose graph is shown in the scatter plot.

We say there is a positive association between dog height and dog weight because knowledge about one variable helps predict the other variable, and when one variable increases, the other tends to increase as well.

What do you think the association between the weight of a car and its fuel efficiency is?

We say that there is a negative association between fuel efficiency and weight of a car because knowledge about one variable helps predict the other variable, and when one variable increases, the other tends to decrease.

Graphs are useful for comparing relationships. Here are two graphs representing the amount of caffeine in Person A and Person B, in milligrams, at different times, measured hourly, after an initial measurement.

Graph of an exponential function, origin O. Horizontal axis, time (hours), scale 0 to 10, by 1’s Vertical axis, caffeine (mg), scale 0 to 200, by 100’s. The function is discrete and has these approximate points: (0 comma 200), (1 comma 160), (2 comma 125), (3 comma 105), (4 comma 80), (5 comma 65), (6 comma 50), (7 comma 45), (8 comma 35), (9 comma 25), (10 comma 20), (11 comma 18) and (12 comma 12).

Graph of an exponential function, origin O. Horizontal axis, time (hours), scale 0 to 10, by 1’s. Vertical axis, caffeine (mg), scale 0 to 200, by 100’s. The function is discrete and has these approximate points: (0 comma 100), (1 comma 90), (2 comma 80), (3 comma 70 ), (4 comma 65), (5 comma 60), (6 comma 57), (7 comma 50), (8 comma 45), (9 comma 40), (10 comma 38), (11 comma 35) and (12 comma 30).

The graphs reveal interesting information about the caffeine in each person over time:

• At the initial measurement, Person A has more caffeine (200 milligrams) than Person B (100 milligrams).
• The caffeine in Person A's body decreases faster. It went from 200 to 160 milligrams in an hour. Because 160 is $\frac{8}{10}$ (or $\frac45$) of 200, the growth factor is $\frac45$.
• The caffeine in Person B's body went from 100 to about 90 milligrams, so that growth factor is about $\frac{9}{10}$. This means that after each hour, a larger fraction of caffeine stays in Person B than in Person A.
• Even though Person A started out with twice as much caffeine, because of the growth factor, Person A had less caffeine than Person B after 6 hours.

Equations are useful not only for representing relationships that change exponentially, but also for answering questions about these situations.

Suppose a bacteria population of 1,000,000 has been increasing by a factor of 2 every hour. What was the size of the population 5 hours ago? How many hours ago was the population less than 1,000?

We could go backward and calculate the population of bacteria 1 hour ago, 2 hours ago, and so on. For example, if the population doubled each hour and was 1,000,000 when first observed, an hour before then it must have been 500,000, and two hours before then it must have been 250,000, and so on.

Another way to reason through these questions is by representing the situation with an equation. If $t$ measures time in hours since the population was 1,000,000, then the bacteria population can be described by the equation:

$\displaystyle p = 1,000,000 \boldcdot 2^t$

The population is 1,000,000 when $t$ is 0, so 5 hours earlier, $t$ would be -5 and here is a way to calculate the population:

$\displaystyle \begin{aligned} 1,000,000 \boldcdot 2^{\text-5} &= 1,000,000 \boldcdot \frac{1}{2^5} \\ &= 1,000,000 \boldcdot \frac{1}{32} \\ &= 31,250 \end{aligned}$

Likewise, substituting -10 for $t$ gives us $1,000,000 \boldcdot 2^{\text-10}$ (or $1,000,000 \boldcdot \frac{1}{2^{10}}$), which is a little less than 1,000. This means that 10 hours before the initial measurement the bacteria population was less than 1,000.

People often collect data in two variables to investigate possible associations between two numerical variables and use the connections that they find to predict more values of the variables.

Data analysis usually follows these steps:

1. Collect data
2. Organize and represent the data, then look for an association
3. Identify any outliers and try to explain why these data points are exceptions to the trend that describes the association
4. Find a function that fits the data well

Although computational systems can help with data analysis by graphing the data, finding a function that might fit the data, and using that function to make predictions, it is important to understand the process and think about what is happening. A computational system may find a function that does not make sense or use a line when the situation suggests that a different model would be more appropriate.

When we collect data by counting things in various categories, like red, blue, or yellow, we call the data “categorical data,” and we say that color is a “categorical variable.”

We can use two-way tables to investigate possible connections between two categorical variables.

For example, this two-way table of frequencies shows the results of a study of meditation and state of mind of athletes before a track meet.

If we are interested in the question of whether there is an association between meditating and being calm, we might present the frequencies in a bar graph, grouping data about those who meditated and those who did not meditate, so we can compare the numbers of calm and agitated athletes in each group.

Double bar graph in blue and red. Horizontal labeled meditated and did not meditate. Vertical labeled 0 to 50, by 10's. Blue represents calm. Red represents agitated.

Notice that the number of athletes who did not meditate is small compared to the number who meditated (29 as compared to 68, as shown in the table).

If we want to know the proportions of calm meditators and calm non-meditators, we can make a two-way table of relative frequencies and present the relative frequencies in a segmented bar graph.

	meditated	did not meditate
calm	66%	28%
agitated	34%	72%
total	100%	100%

Stacked bar graph in blue and red.  Horizontal labeled meditated and did not meditate. Vertical labeled 0 to 100, by 25's. Blue represents calm. Red represents agitated.

In an earlier lesson, we looked at data on meditation and state of mind in athletes.

Double bar graph in blue and red. Horizontal labeled meditated and did not meditate. Vertical labeled 0 to 50, by 10's. Blue represents calm. Red represents agitated.

Stacked bar graph in blue and red.  Horizontal labeled meditated and did not meditate. Vertical labeled 0 to 100, by 25's. Blue represents calm. Red represents agitated.

Is there an association between meditation and state of mind?

The bar graph shows that more athletes were calm than agitated among the group that meditated, and more athletes were agitated than calm among the group that did not.

We can see the proportions of calm meditators and calm non-meditators from the segmented bar graph, which shows that about 66% of athletes who meditated were calm, whereas only about 27% of those who did not meditate were calm.

This does not necessarily mean that meditation causes calmness. It could be the other way around, where calm athletes are more inclined to meditate. However, it does suggest that there is an association between meditating and calmness.