# Stem and Leaf Plot problems

*Directions*:

Use a Stem-and-Leaf Plotter accesable on the internet to explore the sets of problems and answer the questions.

## Set 1

Each person in a class used one hand to reach into a bowl of pennies and remove as many as they could and carry them across the room to another bowl. These are the results: 108, 122, 97, 99, 81, 78, 73, 95, 33, 97, 64, 100, 85, 83, 85, 88, 79, 81, 93, 86, 104, 83, and 71.

What are the mean, median, and mode of the data?

Describe the shape of the data set.

Is it symmetric or are there extreme values in the low or high numbers?

Given this information what would you guess is the students' age?

Explain.

## Set 2

A student babysits for several different families. In one month the student made: $12, $9, $15, $17, $20, $12, $9, $6, $13, $15, $11, $17, $18, $14, $15, $20, $25, $13, $12, $15.

What are the mean, median, and mode of the data?

Describe the shape of the data set.

Is it symmetric or are there extreme values in the low or high numbers?

What was the average daily income?

If they baby sat approximately 4 hours a day, what was the average daily income?

## Set 3

A sports team in a recreational league scored the following points for each game. If the most common score is determined what should be used (mean, median, or mode)? The scores were: 9, 5, 2, 10, 7, 6, 7, 2, 5, 7, 9, 10, 9, and 6.

What are the mean, median, and mode of the data?

Describe the shape of the data set.

Is it symmetric or are there extreme values in the low or high numbers?

What is your answer for the most common score.

## Set 4

At the grocery store one of your parents uses a whole lot of coupons in order to save money. You decide to find the mean, median, and mode of the coupons and you want to use a stem-and-leaf plot, but then you realize that the coupons contain decimals. How can you use the stem-and-leaf plot?

Here are the values of the coupons: $0.50, $0.75, $0.30, $1.00, $0.45, $0.30, $0.75, $0.25, $0.25, $1.50, $0.75, $0.45, $0.30, and $0.50.

Use the method that you described to find the mean, median, and mode of these values!

## Mean, Median, and Mode Discussion

Student: When do we use mean and when do we use median?

Mentor: It is up to the researcher to decide. The important thing is to make sure you tell which method you use. Unfortunately, too often people call mean, median and mode by the same name:average.

Student:What is mode?

Mentor: The easiest way to look at modes is on histograms. Let us imagine a histogram with the smallest possible class intervals (see also Increase or Decrease? Discussion).

Student: Then every different piece of data contributes to only one bin in the histogram.

Mentor: Now let us consider the value that repeats most often. It will look like the highest peak on our histogram. This value is called the mode. If there are several modes, data is called multimodal. Can you make an example of trimodal data?

Student: Data with three modes? Sure. Say, if somebody counted numbers of eggs in 20 tree creeper's nests, they could get these numbers: 4, 3, 1, 2, 6, 3, 4, 5, 2, 6, 4, 3, 3, 3, 6, 4, 6, 4, 2, 6. I can make a histogram:

Mentor: There are three values that appear most often: 3, 4, and 6, so all these values are modes. Modes are often used for so-called qualitative data, that is, data that describes qualities rather than quantities.

Student: What about median?

Mentor: Medianis simply the middle piece of data, after you have sorted data from the smallest to the largest. In your nest example, you sort the numbers first: 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6 eggs. There is an even number of values, so the middle (or median) is between the first and second 4. Because they are the same, we can easily say that the median is four, but if they were different, say if the median was between a 3 and a 4, we would do (3+4)/2=3.5.

Student: So, if there is an even number of values, the median is equal to the sum of the two middle values divided by two.

Mentor: If no birds had nests with only one egg, we would have values of 2, 3, 4, 5, and 6. In this case, the middle number or the median would be the second 4, and we would not need to add or divide because there were an odd number of values.

Student: The last type of averages I would like to know about is mean.

Mentor: Sometimes it is called arithmetic mean, because there are other things in math that are called mean. For example, there is a geometric mean and a harmonic mean. The arithmetic mean of a set of values is a sum of all values, divided by their number. In your nest example,

mean = (4+3+1+2+6+3+4+5+2+6+4+3+3+3+6+4+6+4+2+6)/20 = 3.65

Student: Which one is better: mean, median or mode?

Mentor: It depends on your goals. I can give you some examples to show you why. Consider a company that has nine employees with salaries of 35,000 a year, and their supervisor makes 150,000 a year. If you want to describe the typical salary in the company, which statistics will you use?

Student: I will use mode (35,000), because it tells what salary most people get.

Mentor: What if you are a recruiting officer for the company that wants to make a good impression on a prospective employee?

Student: The mean is (35,000*9 + 150,000)/10 = 46,500 I would probably say: "The average salary in our company is 46,500" using mean.

Mentor: In each case, you have to decide for yourself which statistics to use.

Student: It also helps to know which ones other people are using!

*Stem-and-Leaf Plots Discussion*

Mentor: I am going to show you how to make a stem-and-leaf plot. These plots are used by statisticians to organize the data that they have better. We draw a line that looks a bit like a "T." On the left of the line we put the stem, which is all except the last number of ach data point. On the right of the line we put the leaf, or the last number of the data point.

Student 1: So if I had a data point that is 57, I would put 5 on the left and 7 on the right?

Mentor: That's right. Who can tell me how I would put 257 on this plot?

Student 2: You would put 25 on the left and 7 on the right?

Mentor: Very good. Now, since I already have 57 on the chart, who can guess where I would put 53?

Student 3: You don't need a 5 on the left, since it's already there from the 57, but you need a 3 on the right. Could you put the 3 next to the 7?

Mentor: Wonderful thinking! It can be helpful to organize the numbers in order of magnitude so that we can see which ones occur more than once. If we added another 53, a 54 and two more 57's, it would look like this:

Student 4: Oh, I see, that way you can read the plot more easily when there is a lot of data on it!

Mentor: Right. As you can see, this chart could help us organize a large amount of information. I can see at a glance which number(s) would be in the middle of the data set or which ones occur most, and I can easily obtain the sum of the data points in order to calculate the mean, or average.