Imagine you have a list of numbers: **20**,** 21**, **22**, **25**, **32**, **33**, **34**, **36**, **37**, **39**, **41**, **43**, **43**, **44**, **47**. To organize these numbers, you might use a *stem-and-leaf plot.*

*Stem-and-leaf *plots separate a number into its tens place (the "stem"), and its ones place (the "leaf").

For example, all of the numbers in the 20s would have a stem of **2**, and all of the numbers in the 30s would have a stem of **3**.

We can see these numbers displayed in the stem-and-leaf plot below.

## Stem |
## Leaf |

2 |
0 1 2 5 |

3 |
2 3 4 6 7 9 |

4 |
1 3 3 4 7 |

Let's break down this stem-and-leaf plot. Since the stem represents the tens place and the leaf represents the ones place, we must place each leaf next to its respective stem, as shown below.

## Stem |
## Leaf |
## Value |

2 |
0 |
20 |

2 |
1 |
21 |

2 |
2 |
22 |

2 |
5 |
25 |

3 |
2 |
32 |

3 |
3 |
33 |

3 |
4 |
34 |

3 |
6 |
36 |

3 |
7 |
37 |

3 |
9 |
39 |

4 |
1 |
41 |

4 |
3 |
43 |

4 |
3 |
43 |

4 |
4 |
44 |

4 |
7 |
47 |

We can see that 2 | 0 =** 20**, 2 | 1 =** 21**, etc.

Stem-and-leaf plots are a way to keep lists of numbers more organized. You can visualize a stem with a bunch of leaves coming out of it, where the stem represents the tens digit and each leaf represents the ones digit of each number.

To create the numbers on the stem-and-leaf picture above, go from the stem (7) to each of the leaves. For example, the arrows point to 7 | 1 =** 71 **and 7 | 2 = **72**.

The numbers listed on the stem-and-leaf picture above are **71**, **72**, **74**, **76**, **77**, **77**, **77**, **78**, and **79**.

Now let's look at another stem-and-leaf plot. This plot shows the different grades received by members of a class on a recent math test.

## Stem |
## Leaf |

6 |
6 7 9 9 |

7 |
0 1 1 4 7 8 |

8 |
0 1 2 2 4 5 5 6 6 7 |

9 |
1 2 4 4 4 5 8 |

**What grade was the most commonly received grade?**

To find the most common grade, we can look at the leaves and see which number is most common.

We can see that in the last row (the **9** stem) the number **4** occurs 3 times, more than any other number in any of the other rows. That means that **94** is the most common grade on the test.

**How many students took the math test?**

We can find the number of students that took this test by adding up all of the data points.

We can use the stem and leaf plot to write out all of the data points, or we can simply count each of the "leaves" as one point.

For example, since the stem **6** creates the numbers **66**, **67**, **69**, **69**, just counting **6**, **7**, **9**, **9** as four data points would give you the same result.

Counting all of the leaves, we can see that** 27** people took the math test.

For each of the stem-and-leaf plots shown, we can construct a data set of all the possible values that the stem-and-leaf plot shows.

We may be asked questions involving the mean, mode, median and range of data sets.

To help illustrate these concepts, let's recall the first stem-and-leaf plot and the data set that corresponds with it.

## Stem |
## Leaf |

2 |
0 1 2 5 |

3 |
2 3 4 6 7 9 |

4 |
1 3 3 4 7 |

Data set: **{20**,** 21**, **22**, **25**, **32**, **33**, **34**, **36**, **37**, **39**, **41**, **43**, **43**, **44**, **47}**

The mean of a data set is also known as the average of the data set. To find the mean, add up all of the values in the data set and divide them by the number of data items in the data set.

In this case we have a total of **15** numbers, so we will add them up and then divide the result by **15**. The sum of the numbers in our data set is **517**.

This means that the mean or the average of our data set is **34.46**.

The mode of a data set is the number that appears the most in a data set. Data sets can have no modes, a single mode or multiple modes.

In this case, the number which appears the most in our data set is **43**. **{20**,** 21**, **22**, **25**, **32**, **33**, **34**, **36**, **37**, **39**, **41**, **43**, **43**, **44**, **47}**

This means that the mode of our data set is **43**.

Let's modify the data set. **{20**,** 21**, **22**, **25**, **32**, **33}
Does this data set have a mode?**

No, this data set does not have a mode since each of the numbers in the data set appear the same amount of times.

Let's modify the data set again.

Does this data set have a mode?

Yes, this data set actually has two modes since both

The median of a data set is the value which appears exactly in the middle of a data set when it is arranged in ascending order.

Let's find the median of our data set: **{20**,** 21**, **22**, **25**, **32**, **33**, **34**, **36**, **37**, **39**, **41**, **43**, **43**, **44**, **47}**

In this case, we have an odd number of entries in our data set. This means that the middle-most number will be our median.

Our median for this data set is **36**.

Let's modify our data set so that we have an even number of entries. **{20, 21, 22, 23, 24, 25}**

In this case, we won't be able to identify a number that is in the middle of the data set. To find the median of this data set, we need to find the average of the two middle values.

To find the average of the two middle values, add them up and divide them by two.

**22 + 23 = 45
45 ÷ 2 = 22.5**

The median of this data set is

The range of a data set is the difference between the greatest and smallest values in the data set.

To find the value of the range, subtract the smallest value from the largest value.

Recall our original data set: **{20**,** 21**, **22**, **25**, **32**, **33**, **34**, **36**, **37**, **39**, **41**, **43**, **43**, **44**, **47} **.

The greatest value of this data set is **47**, and the smallest is **20**.

To find the range, subtract **20** from **47**.

**47 − 20 = 27**

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