My intern just started a unit on statistics with my favourite starter question of ** all time**.

(First blogged near the end of this post in 2011…)

The question is simple: floor is very low, and ceiling is very high.

**Create a data set with the following characteristics:**

**Mean = 3**

**Mode = 3**

**Median = 3**

During the teacher rotation between groups, I picked up on some lines of reasoning. (Not being directly responsible for the teaching of the lesson, allows me to sit back, be inspired, and follow the lines of inspiration).

Student justifications for their data sets were very interesting. It inspired me to pen some burning questions I may have asked groups if I were “in charge”.

- If the mean of a data set is 3, all data points must be multiples of 3.
- You cannot have a mode of 3 without at least 2 threes.
- Adding a data point lower than the mean will always lower the mean.
- Adding a negative and positive data point will never change the median of a data set.
- If the mean and mode of a data set match, so must the median.
- The median of a data set must appear as a data point in the set.
- The mode of a data set will always be closer to the mean than the median.
- Adding a data point lower than the median will always change the median.
- Adding a group of points with the same mean as a set of data will not change the mean.
- If we keep adding threes to a data set, the mean will eventually become 3.

This type of extreme *agree* or *disagree* questions are great for discussion. I believe some teachers have coined them as “talking points”. (If you have a link to a great description of “talking point” comment below.)

Asking students to *prove* or *disprove* these statements (ranging in difficulty) would allow them to surpass the calculation of the statistics and work directly with the mechanism of central tendency. That is a huge win for me.

If you have more questions, comment them!

NatBanting

## 3 replies on “Central Tendency: 10 Burning Questions”

One solution to the original question is "3". That is, a single data point. One may avoid that triviality by asserting that one of the data points is "5". Of course, trivialities can be valuable, too.

Yes!! That is exactly the conversation I want. Usually, several groups write a single "3" and act like they cheated the problem. Then I ask them to add more. They add many more 3's and give me the same look. I ask, "Why"?

Eventually, we add a non-three data point, and work on balance.

Great anticipation of student thought.

Talking Points HOW-TO: http://twittermathcamp.pbworks.com/f/READ+ME+-+the+Talking+Points+Activity+Structure.pdf

Talking Point Resources: http://twittermathcamp.pbworks.com/w/search?q=talking%20points

I love this list, and it's perfect for what we're about to cover in Math 1! ~Carolyn (Learning with Tape)