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.
