# Probability and Statistics

### Topics

## Introduction to :

A stem is something from which stuff grows. We can think of a digit like a stem: if we write down a single digit, each choice for the digit we write next "grows'' us a new number. If it doesn't seem to be growing, you can try watering it, but don't go crazy. We don't want to drown the poor things.

For example, we could use the digit 3 as a stem from which to "grow" either the number 31 or the number 35:

A **stem and leaf plot** is a fancy-shmancy sort of table that captures this idea of ''growing'' different numbers out of the same stem.

Let's forge ahead and build a couple. Since we're building numbers instead of actual plants, we're not playing God so much as we're, um, playing Euclid.

In the simplest case, our data consists of two-digit numbers.

### Sample Problem

Build a stem-and-leaf plot for the list

32, 34, 35, 36, 37, 45, 46, 47, 48, 55, 56, 57.

The stems of the numbers in the list are the digits that occur in the tens place: 3, 4, and 5.

**3**2, **3**4, **3**5, **3**6, **3**7, **4**5, **4**6, **4**7, **4**8, **5**5, **5**7.

We start our stem-and-leaf plot by putting these numbers into a table:

Now, for each stem we write down its leaves; that is, the digits that showed up in the ones place with that stem. For the stem 3, the leaves are 2, 4, 5, 6, and 7:

3**2**, 3**4**, 3**5**, 3**6**, 3**7**, 45, 46, 47, 48, 55, 56, 57.

We put these into the table by the appropriate stem:

Similarly, we find the leaves for the stem 4:

32, 34, 35, 36, 37, 4**5**, 4**6**, 4**7**, 4**8**, 55, 56, 57

and enter them in the table:

Finally, we find the leaves for the stem 5:

32, 34, 35, 36, 37, 45, 46, 47, 48, 5**5**, 5**6**, 5**7**

and enter those in the table:

The only thing left is to write a down a note at the bottom (formally called a **key**) that explains how we use the stem and the leaf to "grow'' a number. This note is not the appropriate place for you to write something like "Mark likes Mandy" or "I love your shoes." Let's keep it professional, people.

Our data could also consist of numbers with decimal points. Oh joy, right?

### Sample Problem

Build a stem-and-leaf plot for the following list:

0.4, 1.2, 3.4, 3.6, 3.7, 0.5, 2.3.

Very first thing, let's put the list in order:

0.4, 0.5, 1.2, 2.3, 3.4, 3.6, 3.7.

Now we find the stems:

**0**.4, **0**.5, **1**.2, **2**.3, **3**.4, **3**.6, **3**.7.

We use the stems to start our stem-and-leaf plot:

Now we find the leaves for each stem. First we find the leaves for 0:

0.**4**, 0.**5**, 1.2, 2.3, 3.4, 3.6, 3.7

and we put them in the plot:

We do the same for the other stem, filling in the rest of our table-like thing:

Finally, we make a key that explains how the numbers grow from a stem and a leaf:

**Be careful:** Remember to make a key for your stem and leaf plot. Otherwise, we won't know if 1 | 5 means 1.5 or 15. This distinction makes a huge difference, especially if we're cooking. We suspect our dough does not require 15 cups of flour.

One other fine point: in the table, it's nice to write the leaves in numerical order. If we start by ordering the list of numbers, we don't need to think too hard about it. Then our mind can be free to wander and think about other things, like whether or not we left the air conditioning running. Shoot!

There's a fun, interactive stem-and-leaf tutorial here. If you ever get bored of it, head back on over here and finish reading this unit. In the meantime, we'll miss you.

#### Exercise 1

Make a stem-and-leaf plot for the following list of numbers:

3.1, 3.4, 4.5, 2.7, 2.9.

#### Exercise 2

Make a stem-and-leaf plot for the following list of numbers:

2.1, 2.0, 3.2, 3.7, 1.4, 1.5, 2.6.

#### Exercise 3

Make a stem-and-leaf plot for the following list of numbers:

23, 19, 28, 45, 36, 47, 48, 50, 51, 13.