# Common Core Standards: Math

### Statistics and Probability 8.SP.A.4

4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

Sometimes, we can look at a situation and conclude—incorrectly—that one thing causes another. For example, a child might look at the moon each night, listen to the sound of the crickets outside, and think that the moon caused the sound. In fact, he might be able to give a hundred examples of the moon and the sound coexisting. But having two things happen at the same time doesn't necessarily mean that one causes the other.

Statistics can be funny like that. They can sometimes be confusing, misleading, and even downright wrong. Of course, there might also be a legitimate tie between two seemingly unrelated pieces of information. Statisticians can experiment enough to find that connection, even if they might not be able to explain it.

Students should know that statistical data can be shown a number of ways, from frequency tables to bar graphs to scatter plots to make the information easier to understand. This depends on the different types of data collected and how it's meant to be interpreted—in totals or in frequencies. Students should know how to construct these tables and how to read them.

Students should also know that depending on the data, tables, charts, or scatter plots could make the connections easier to see. But before coming to any conclusions about how these factors interact, students should know to consider any and all factors that might affect the data.

For instance, a study shows that kids who have earlier curfews usually do more chores around the house. While the two might be linked, students should know that having more chores doesn't cause earlier curfews. Is it because of strict parents? Rowdy kids being punished? Or maybe they like mowing the lawn and doing the dishes. (Yeah, right!)

While students can come up with different ways to explain the trends they see in statistical data, they should know that they cannot conclusively determine that one causes another without examining all other factors involved.

#### Drills

1. Kenny wants to prove to his parents that he deserves a raise in his allowance. He has taken a survey of all the other eighth graders in his school and has determined that he makes \$1 per week less than average. Is this a valid argument for a raise?

No, because of (B) and (C), among other things

Before Kenny can reach any kind of conclusion about the data, he needs to consider all the different factors that affect it. Among those factors are (B) and (C), but there are other factors too. For instance, having siblings or behaving well might affect whether or not Kenny gets a raise or not. While Kenny has done some of the work and is headed in the right direction, he needs to consider as many factors as he can before coming to a solid conclusion.

2. What is the difference between correlation and causation?

Correlation means there's a relationship, while causation means one happens as a result of the other

"Correlation" has the word "relation" inside it. It simply means that the two things are connected in some way, but it doesn't really explain how or why. Like height and weight—they're related somehow, but being tall doesn't make you either heavy or skinny. "Causation" is when one thing makes another thing happen, like how throwing a piano off a balcony causes it to break into a million pieces. There's a difference, but the two aren't opposites. Only (C) gives the right explanation.

3. Kyle decided that he needed exercise, so he spent 4 hours a day doing water aerobics in his pool. Two weeks later, Kyle had green hair. What might Kyle infer only from this information?

All of the above

Really, Kyle doesn't have enough information to prove any kind of causation, so all of the above are possible. Swimming, exercising, or sweating may have all caused his hair to turn green because the only correlation we can draw is between swimming in pool water for 4 hours a day and hair turning green. Any of (A), (B), or (C) could have caused it, for all we know. (Actually, there may have been other sources, too!) To really be sure, Kyle would need to conduct a few at-home experiments with exercise, swimming, and sweating. Sounds a bit gross—but it's for the good of statisticians everywhere!

4. Kyle decided it was probably the chlorine in the pool water that turned his hair green. He decided to steal his sister's dolls and put them into the pool to test his theory. What's wrong with his method?

The dolls don't have human hair

To show a correlation, Kyle should test the same population—or as close to it as possible. While dolls are humanlike (well, some of them, anyway), their hair is synthetic and might react to the chlorine differently than human hair would. Kyle would have been better off performing the experiment not on the dolls, but on his sister instead. Kyle's sister, on the other hand…

5. In the magical land of Forfleugenhagen, there exist different types of dragons. Some can fly and some can breathe fire. Which of the following is true about the dragons of Forfleugenhagen?

 Can Breathe Fire Can't Breathe Fire Total Can Fly 2% 65% 67% Can't Fly 22% 11% 33% Total 24% 76% 100%

Most of the dragons that can fly can't breathe fire

Out of 100% of the dragons in Forfleugenhagen, we can see that only a third of them can't fly, which means most of them can, so (A) isn't right. Out of the 24% of dragons that can breathe fire, only 2% can fly also, which means (C) is wrong. Only 11% of the dragons in Forfleugenhagen can't breathe fire or fly (so…aren't they just big lizards?), which means (D) is wrong too. Of the dragons in the "Can Fly" row, most are in the "Can't Breathe Fire" column—so (B) is the answer we're looking for.

6. If there are a total of 850 dragons in Forfleugenhagen, how many of them can both fly and breathe fire?

17

We can use the percentages in the table and the total number of dragons in Forfleugenhagen to find the exact number of dragons of each kind. If 850 corresponds to 100%, we can set up a proportion between those two numbers and see which number corresponds to the number of dragons who can fly and breathe fire—or just multiply 850 by 2% (or 0.02). Either way, we should get 17 as our answer. The other answers give different percentages, since 2 is less than a fourth of a percent (0.25%) of 850, 46 is 5.4%, and 94 is 11%.

7. Chester has always wanted a pet alligator, and wondered how many of his classmates felt the same. He decided ask his classmates whether they'd want an alligator for a pet and made a table of the results. Which of the following is true about the data Chester collected?

 Boys Girls Total Want Alligator 12 5 17 Don't Want Alligator 4 9 13 Total 16 14 30

One sixth of Chester's classmates are girls who want an alligator for a pet

We can see that out of all the 16 boys in Chester's class, 4 of them wouldn't want an alligator for a pet. This means (C) is wrong because there are some boys who wouldn't want a pet alligator. In comparing 16 boys to 14 girls, we can see that (D) is wrong—especially if we aren't counting Chester. If we look at the 14 girls in Chester's class, 9 of them wouldn't want an alligator and 5 would. Comparing these values, , which is definitely more than ½, so (B) is wrong, too. Comparing the girls who want an alligator (5) to all of Chester's classmates (30), we can see that , which means (A) is the answer we're looking for.

8. Which of the following represents the percentage of boys who wouldn't want a pet alligator compared to all of Chester's classmates?

13%

If we're interpreting the table correctly, we should be comparing the number in the "Boy" and "Don't Want Alligator" column (4) to the total number of classmates Chester interviewed (30). Dividing 4 by 30 gives us 0.13, which gives us about 13% when multiplied by 100%. All other values result from dividing 4 by 16, 14, or 12, and are wrong.

9. Of all of Chester's classmates who don't want an alligator for a pet, what percentage are girls?

69%

The first part tells us we're looking at the row of "Don't Want Alligator," and comparing the "Girls" column to the "Total" column. Basically, compare 9 to 14, and you should get (C) as your answer. All other percentages come from dividing the wrong numbers, which is something you should generally try to avoid.

10. Does Chester's data suggest a correlation between the appeal of alligators to boys and to girls?

Probably, since the numbers are skewed