Lesson Video: Stem-and-Leaf Plots Mathematics

In this video, we will learn how to use a stem-and-leaf plot to organize data sets.

17:43

Video Transcript

In this video, we’ll learn all about stem-and-leaf plots. We’ll learn how to create one, and we’ll learn how to interpret the information given in one.

A stem-and-leaf plot or a stem-and-leaf diagram is a special table where each data value is split into a stem and a leaf. Let’s take a look at this typical stem-and-leaf diagram here. It’s not immediately obvious what all of these digits represent, which brings us to the other important feature of a stem-and-leaf plot, a key. We need to have a key in order to tell us what the information actually represents. Here, we’re told that one and then the bar three equals 13. So that means in the first row, our data set will actually be 13, 16, and 16. In the second row, the digit zero would actually represent the value of 20.

Before we have a look at some questions, there’s a few things that we should know about stem-and-leaf plots. Firstly, the leaf is usually just one digit. That means then that the stem will consist of the first digit or digits of a number. So it could be just the tens as here, or it could be hundreds and tens with the leaf of the units. Next, we know that values can and should be repeated if the data values are repeated. For example, in our first row, we have two of the number 16. That’s because in the data set the value 16 occurred twice.

With stem-and-leaf diagrams, we do need to be very careful though. For instance, this value of six in the second row isn’t the same as the value six in the first row, as in the first row it represents 16 and in the second row it would represent 26. One excellent point about stem-and-leaf plots and the reason why we might choose to use them is that we could easily see any groupings with the highest frequency. In this stem-and-leaf plot, for example, we can see that a large number of the data values occurred in row three, which would represent the thirties. So now, let’s have a look at some questions. And in the first few questions, we’ll be creating our own stem-and-leaf plots.

Write the stems, in ascending order, for the following data set and, hence, make a stem-and-leaf plot from the data: 14, 42, 21, 33, 36, 27, 29.

We can remember that a stem-and-leaf plot is a special type of table where each data value is split into a stem part and a leaf part. When we create a stem-and-leaf plot, the leaves are usually single-digit values. As all of the data values here are two-digit numbers, then the stems will be the numbers in the tens column and the leaf will be the part that’s in the units column. We can set up the structure of our stem-and-leaf plot, but we must remember to include one important thing. And that’s a key. Here, for example, we could say that a stem of one and a leaf of four would indicate the value of 14.

We’re asked to begin by writing the stems in ascending order from smallest to greatest. And that’s actually the way any stem-and-leaf plot would be. The lowest value here is 14. So the smallest stem we have would be one, which would be the same as 10. The largest value in the data set is 42. So our stem must go up to four. And there we have the stems in ascending order: one, two, three, and four. We can now take each value in turn and fill it in to the stem-and-leaf plot. The first value of 14, as we’ve already demonstrated with the key, would be written in the row with the stem of one, and we fill in the value four for the leaf.

If we have a large data set, then it’s worthwhile crossing through or taking each value as we go. The next value is 42 so that we’ll have a stem of four and a leaf of two. We can continue crossing off the values as we go. When we have two values with the same stem, for example, 33 and 36, then we must put a comma between the leaves. And there is our completed stem-and-leaf plot. It’s always worth checking that we have the same number of data values as we do in the stem-and-leaf plot. For example, we were given seven data values to start with. And if we count the leaves, we can see that there are seven different leaves. So there we have our complete answer for the stem-and-leaf plot and not forgetting the key.

Just one final point, when we filled in the data, the leaves are actually in ascending order. For example, the leaves with a stem of two go from one, seven, and nine. If, for example, we had the value of 29, then 21, then 27, we’d need to reorder these so that they occurred one, then seven, then nine. This will be true of every stem. The leaves must be given in ascending order.

Let’s have a look at our second question.

Write the stems, in ascending order, of a stem-and-leaf plot for the following data set: 67, 81, 74, 10, and 36. Then, complete the stem-and-leaf plot.

In order to complete a stem-and-leaf plot, we first need to remember that it’s a special table where the data values are split into stems and leaves. As all of our data values here are two-digit numbers, then our leaf, which is usually the digit at the end, will be the value that’s in the units or ones column and the stem will be the digit in the tens column. We’re asked to write the stems in ascending order. That’s from smallest to greatest. So let’s start by identifying that our stems will be six, eight, seven, one, and three. In ascending order then, we could say that the stems are one, three, six, seven, and eight. And that’s the answer for the first part of this question.

Next, we need to create the stem-and-leaf plot. We can set up our stem-and-leaf plot in the following way. Even though we have a very small data set with just five values, we don’t just want to write the stems that we have, but we should include the stems in between these values. For example, we could start with a stem of zero and go up to stem of nine. This would allow us, if in the case of further data values, that we could continue to fill them in to the stem-and-leaf plot.

We can now fill in the values. So let’s look at the first value of 67. This will have a stem of six and a leaf of seven. The next value of 81 has a stem of eight and a leaf of one. 74 is next. But then, when we come to the value of 10, we need to be careful. We still need to record a leaf of zero as, don’t forget, it doesn’t mean a value of zero; it means a value of 10. Finally, we have the value of 36 with a stem of three and a leaf of six. In a stem-and-leaf plot, we must always make sure that the leaves are also given in ascending order. For example, if we also have the value of 35, then when we wrote the leaves of six and five, we’d have to order these so that the leaf of five came before the leaf of six.

But in this case, these five stems only have one single leaf. But there’s still one thing that we’re missing in our stem-and-leaf plot. We need to include a key. We can choose any value to represent how to interpret our stem-and-leaf diagram, but it’s common just to select the first value. Here, we can write that one and the bar zero equals 10 to indicate that a stem of one and a leaf of zero would mean 10. And we can give our answer. Here is the fully complete stem and leaf plot, complete with key.

In the next question, we’ll see a larger data set where we’ll need to order the leaves as well as the stems.

A class of students were asked to count and record the number of bugs that they could find in their yards in 30 minutes. The results were as follows: 21, 35, 42, 11, 13, 77, 14, 17, 31, 27, 15, 53, 12, 25, 31, 18, 22, 16, 72, 14, 91, 34, 19, 23. Write the stems, in ascending order, of a stem-and-leaf plot for the data. Display the data in a stem-and-leaf plot. Use your stem-and-leaf plot to find out how many students counted bugs in their yards.

We can see the list of values here, which begins with the value of 21 that would indicate that one student had found 21 different bugs in their yard. What we’re asked to do is to take this list of numerical values and put it into a stem-and-leaf plot. A stem-and-leaf plot is a special type of table where the leaves of each number will represent the final digit of each number. As all of the values here are two-digit numbers, then the leaves will all be the values in the ones column and the stem will be the value in the tens column.

For example, the stem of the first number would be two, the second stem would be three, the third stem would be four, and so on. Notice that a number of these stems will have duplicates. For example, there’s a number 11 and a number 13, which indicates that there’ll be two values that have a stem of one. The best way perhaps to write the stems in ascending order is to start by creating our stem-and-leaf plot. The smallest value that we have is 11, and the largest value is 91. Therefore, our stems will need to go from one to nine.

So here we have the stems written. We can include the stem of zero if we wish, although it’s not essential. As we go through to creating the stem-and-leaf plot, we’ll also need to include a key. We can choose any value to represent the key. Here, I’ve chosen a value with a stem of one and a leaf of two to indicate that that means 12 bugs. So that would be the first part of the question answered. The stems are written in ascending order.

We’ll now begin to answer the second question, which is where we fill in the data values into the stem-and-leaf plot. Because we have a large data set here, we can begin by filling in the values so that the leaves are given in an unordered way. Once we have all of the values into the stem-and-leaf plot, then we’ll rewrite it so that the leaves are in order. So let’s take the first value of 21. This will have a stem of two and a leaf of one. The second value of 35 has a stem of three and a leaf of five.

We can input the next two values of 42 and 11. But notice that when it comes to the value of 13, we need to put a comma and then the leaf of three. We can continue until all the values are in the stem-and-leaf plot. We can notice, for example, that the value of 31 occurred twice, but we must still write it into our stem-and-leaf plot twice. We’ll now put the leaves in each stem into ascending order. Having the leaves in order also allows us to interpret the data and draw any conclusions we may want to. And so that’s our answer for the second part of the question. Here’s a fully complete and ordered stem-and-leaf diagram along with the key.

The third part of this question asks us to find out how many students counted bugs in their yards from the stem-and-leaf plot. In order to do this, we need to count the total number of leaves. In the first row, we have 10 different leaves, then five leaves, four, one, one, two, and one. Adding these all together would give us the value of 24. So our answer would be that 24 students counted bugs in their yards. It’s also a very good check as we’re completing the stem-and-leaf diagram to make sure that we’ve got all of the values in. If we’d counted each individual data value that we were originally given, we would also find that there were 24 different data values.

In the next question, we’ll see an example where the values of the stem and leaves in a stem-and-leaf plot aren’t given as the tens and the ones values.

The monthly rainfall was measured in inches in Minneapolis, Minnesota, over one year, with the following results. Illustrate the rainfall data in a stem-and-leaf plot.

Let’s remember that a stem-and-leaf plot is a special type of table where we have different parts of the number that represent the stem and the remaining part representing the leaf. If we take a look at all the values of the rainfall, we can see that all except for one consist of a value in the ones column and a value in the tens column. So therefore, if we took a value, for example, the value of 1.8, we could say that a stem of one and a leaf of eight would represent 1.8 inches of rainfall. Looking at the data, the smallest value is 0.8 and the largest value is 4.3, which actually occurs twice.

Therefore, our stems will need to go from zero to four, written in that order. We can put the data into the stem-and-leaf table in one of two ways. For example, we could say which values would have a stem of zero, and notice that we have a 0.9 and 0.8. Alternatively, we could take each data value in turn and fill it into the stem-and-leaf plot. This is particularly useful when there’s a large data set. If we did it this way, then we would say our first value of 0.9 would have a stem of zero and a leaf of nine. 0.8 would be next, representing a stem of zero and a leaf of eight. Notice that we’ve written these two leaves in the form of a list with a comma in between.

At the minute, these leaves aren’t written in order, but we can put them in order once we’ve completed our values. We can fill in the next few values. But notice when it comes to the value of four, this doesn’t actually have a leaf. However, a value of four would, of course, represent the equivalent value of 4.0. So we can fill this in with a stem of four and a leaf of zero. Once we’ve got all these values into the stem-and-leaf diagram, we need to make sure that the leaves are ordered within each stem row.

The leaves of nine and eight should be ordered as eight and nine. Next, the leaves of nine, eight, and two should be written as two, eight, and nine. And we can continue ordering the rest of the stem rows. Finally, this description of how to interpret the stem-and-leaf diagram can be given as the key. And there we have the answer. We have our stem-and-leaf plot with the leaves given in order. And we also have a key to indicate how to interpret the stem-and-leaf plot.

We’ll now summarize some of the key points of this video. We saw, firstly, how a stem-and-leaf plot is a special table where each data value is split into a stem and a leaf. We can write the stems and leaves in different ways. For example, we can have the stem as the tens and the leaf as the units, the stem as the hundreds and tens and the leaf as the units, or even a stem which is the ones value and the leaf is the tenths value. Each stem should have the leaves written in ascending order. If the data set is large, consider creating a stem-and-leaf plot with unordered leaves first and then ordering them afterwards.

We must make sure that we include every value in the data set in the stem-and-leaf plot, even and especially if it’s a duplicate. After all, if a value is given more than once in the data set, then it should be given the same number of times in the stem-and-leaf plot. We must make sure that we always include a key. That will allow whoever’s looking at the data to be able to interpret it correctly. Finally, the good thing about stem-and-leaf plots and the reason that we use them is that they show us the spread of data very nicely and allow us to also see each individual value in the original data set.

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