How Does Theã¢â‚¬â€¹ Stem-and-leaf Plot Show The Distribution Of Theseã¢â‚¬â€¹ Data?

Stem and leaf plots

Archived Content

Information identified as archived is provided for reference, research or recordkeeping purposes. It is not subject to the Government of Canada Web Standards and has not been altered or updated since it was archived. Delight contact us to request a format other than those bachelor.

Elements of a good stem and leaf plot
Tips on how to describe a stem and leaf plot
- Example 1 – Making a stem and foliage plot
The primary advantage of a stem and leaf plot
- Instance 2 – Making a stem and leaf plot
- Example iii – Making an ordered stalk and leaf plot
Splitting the stems
- Example 4 – Splitting the stems
- Example five – Splitting stems using decimal values
Outliers
Features of distributions
Using stem and leaf plots as graphs
- Case 6 – Using stem and foliage plots as graph

A stalk and leaf plot, or stem plot, is a technique used to allocate either discrete or continuous variables. A stalk and foliage plot is used to organize information as they are collected.

A stem and foliage plot looks something like a bar graph. Each number in the data is broken down into a stalk and a leaf, thus the proper noun. The stem of the number includes all but the terminal digit. The leaf of the number will always be a single digit.

Elements of a good stem and leafage plot

A good stem and foliage plot

shows the first digits of the number (thousands, hundreds or tens) as the stem and shows the last digit (ones) as the leaf.
commonly uses whole numbers. Anything that has a decimal signal is rounded to the nearest whole number. For case, test results, speeds, heights, weights, etc.
looks like a bar graph when information technology is turned on its side.
shows how the data are spread—that is, highest number, lowest number, almost common number and outliers (a number that lies exterior the main grouping of numbers).

Summit of Folio

Tips on how to depict a stalk and leaf plot

In one case you have decided that a stem and leaf plot is the best way to show your data, draw information technology as follows:

On the left hand side of the folio, write down the thousands, hundreds or tens (all digits but the terminal one). These volition be your stems.
Depict a line to the right of these stems.
On the other side of the line, write down the ones (the last digit of a number). These volition be your leaves.

For case, if the observed value is 25, then the stem is two and the leaf is the 5. If the observed value is 369, so the stalk is 36 and the leaf is 9. Where observations are authentic to one or more decimal places, such as 23.vii, the stem is 23 and the leaf is 7. If the range of values is too great, the number 23.7 can be rounded up to 24 to limit the number of stems.

In stem and foliage plots, tally marks are not required because the bodily data are used.

Not quite getting information technology? Try some exercises.

Case 1 – Making a stem and leaf plot

Each morning time, a teacher quizzed his class with xx geography questions. The course marked them together and everyone kept a record of their personal scores. As the year passed, each student tried to improve his or her quiz marks. Every day, Elliot recorded his quiz marks on a stalk and leafage plot. This is what his marks looked like plotted out:

Table ane. Elliot'south scores on the basic facts quiz final yr
Stem	Leaf
0	3 6 5
i	0 1 4 3 5 6 five half-dozen 8 9 vii 9
2	0 0 0 0

Analyse Elliot'due south stem and leaf plot. What is his most common score on the geography quizzes? What is his highest score? His everyman score? Rotate the stalk and leafage plot onto its side so that information technology looks like a bar graph. Are most of Elliot's scores in the 10s, 20s or under 10? Information technology is difficult to know from the plot whether Elliot has improved or not considering we exercise non know the lodge of those scores.

Endeavour making your own stem and leafage plot. Use the marks from something similar all of your examination results last year or the points your sports squad accumulated this flavor.

Tiptop of Page

The main advantage of a stem and foliage plot

The main advantage of a stem and leafage plot is that the data are grouped and all the original data are shown, too. In Example 3 on battery life in the Frequency distribution tables section, the table shows that ii observations occurred in the interval from 360 to 369 minutes. However, the table does not tell you what those actual observations are. A stem and leaf plot would show that information. Without a stem and leafage plot, the two values (363 and 369) can only exist plant by searching through all the original data—a dull task when you have lots of data!

When looking at a information set, each observation may be considered as consisting of ii parts—a stalk and a leaf. To make a stalk and leaf plot, each observed value must start be separated into its two parts:

The stem is the starting time digit or digits;
The leaf is the final digit of a value;
Each stem can consist of whatsoever number of digits; merely
Each leafage can have only a single digit.

Elevation of Folio

Example ii – Making a stem and leaf plot

A teacher asked 10 of her students how many books they had read in the last 12 months. Their answers were every bit follows:

12, 23, 19, six, 10, 7, xv, 25, 21, 12

Ready a stalk and leaf plot for these data.

Tip: The number 6 can be written equally 06, which ways that it has a stem of 0 and a leaf of half-dozen.

The stalk and foliage plot should look like this:

Table 2. Books read in a year by 10 students
Stem	Leaf
0	6 seven
1	2 9 0 v 2
2	3 five 1

In Table ii:

stem 0 represents the class interval 0 to 9;
stem 1 represents the class interval 10 to xix; and
stem 2 represents the class interval 20 to 29.

Usually, a stalk and leaf plot is ordered, which simply means that the leaves are bundled in ascending gild from left to correct. Also, there is no need to split up the leaves (digits) with punctuation marks (commas or periods) since each foliage is always a single digit.

Using the data from Tabular array 2, we fabricated the ordered stem and leaf plot shown beneath:

Table 3. Books read in a year by 10 students
Stem	Leaf
0	6 7
one	0 2 two 5 9
2	1 3 5

Peak of Page

Case 3 – Making an ordered stalk and leaf plot

Xv people were asked how often they drove to work over x working days. The number of times each person drove was as follows:

5, 7, 9, 9, iii, 5, 1, 0, 0, 4, 3, vii, 2, 9, viii

Make an ordered stem and leafage plot for this table.

Information technology should be drawn as follows:

Tabular array four. Number of drives to piece of work in 10 days
Stem	Leafage
0	0 0 i 2 three 3 iv 5 v 7 7 eight ix 9 9

Splitting the stems

The system of this stalk and leaf plot does non requite much data about the data. With only i stem, the leaves are overcrowded. If the leaves go also crowded, and then it might be useful to split each stem into two or more components. Thus, an interval 0–9 can be split into two intervals of 0–4 and v–ix. Similarly, a 0–nine stalk could be split into five intervals: 0–ane, ii–3, 4–five, half-dozen–seven and 8–9.

The stem and leafage plot should then look similar this:

Table five. Number of drives to work in 10 days
Stem	Leaf
0⁽⁰⁾	0 0 one 2 3 iii 4
0^(five)	5 5 7 7 8 ix 9 9

Annotation: The stalk 0⁽⁰⁾ ways all the information inside the interval 0–four. The stem 0^(v) means all the data within the interval five–nine.

Top of Folio

Example 4 – Splitting the stems

Britney is a swimmer training for a competition. The number of 50-metre laps she swam each day for 30 days are as follows:

22, 21, 24, 19, 27, 28, 24, 25, 29, 28, 26, 31, 28, 27, 22, 39, 20, ten, 26, 24, 27, 28, 26, 28, 18, 32, 29, 25, 31, 27

Prepare an ordered stem and foliage plot. Brand a brief comment on what information technology shows.
Redraw the stalk and leafage plot by splitting the stems into five-unit intervals. Make a brief annotate on what the new plot shows.

Answers

The observations range in value from 10 to 39, so the stem and leaf plot should have stems of 1, ii and 3. The ordered stem and foliage plot is shown below:

Table 6. Laps swum by Britney in 30 days
Stalk	Foliage
i	0 8 ix
ii	0 1 2 2 four four 4 5 5 6 6 6 seven 7 seven 7 8 8 8 viii 8 9 9
three	ane one 2 9

The stem and leafage plot shows that Britney usually swims betwixt 20 and 29 laps in grooming each 24-hour interval.

Splitting the stems into 5-unit intervals gives the following stem and foliage plot:

Table 7. Laps swum by Britney in xxx days
Stalk	Leaf
1⁽⁰⁾	0
1⁽⁵⁾	8 ix
2⁽⁰⁾	0 1 ii ii 4 4 4
two⁽⁵⁾	5 5 half dozen 6 six 7 7 7 7 8 8 8 eight 8 nine nine
3⁽⁰⁾	1 1 2
3⁽⁵⁾	ix

Annotation: The stem 1⁽⁰⁾ ways all data between 10 and fourteen, 1⁽⁵⁾ means all data between fifteen and 19, and so on.

The revised stem and leaf plot shows that Britney usually swims betwixt 25 and 29 laps in training each day. The values one⁽⁰⁾ 0 = ten and 3^(five) nine = 39 could exist considered outliers—a concept that will be described in the next section.

Elevation of Page

Instance 5 – Splitting stems using decimal values

The weights (to the nearest 10th of a kilogram) of thirty students were measured and recorded every bit follows:

59.2, 61.5, 62.3, 61.four, 60.9, 59.eight, 60.5, 59.0, 61.ane, 60.7, 61.vi, 56.three, 61.nine, 65.vii, 60.4, 58.9, 59.0, 61.2, 62.1, 61.4, 58.4, sixty.8, threescore.two, 62.7, 60.0, 59.three, 61.9, 61.7, 58.4, 62.2

Prepare an ordered stem and leafage plot for the data. Briefly comment on what the analysis shows.

Answer

In this case, the stems volition be the whole number values and the leaves volition be the decimal values. The data range from 56.three to 65.vii, and then the stems should showtime at 56 and terminate at 65.

Table viii. Weights of xxx students
Stem	Leaf
56	3
57
58	four 4 ix
59	0 0 2 three viii
60	0 2 4 five vii viii 9
61	1 two 4 iv 5 6 seven 9 9
62	ane 2 3 7
63
64
65	7

In this example, it was non necessary to split stems considering the leaves are not crowded on besides few stems; nor was it necessary to round the values, since the range of values is not large. This stalk and leaf plot reveals that the group with the highest number of observations recorded is the 61.0 to 61.9 group.

Top of Folio

Outliers

An outlier is an extreme value of the information. It is an ascertainment value that is significantly unlike from the rest of the data. At that place may be more than one outlier in a prepare of data.

Sometimes, outliers are significant pieces of information and should not exist ignored. Other times, they occur because of an error or misinformation and should exist ignored.

In the previous example, 56.3 and 65.vii could be considered outliers, since these two values are quite different from the other values.

By ignoring these two outliers, the previous instance'due south stem and leaf plot could be redrawn as beneath:

Table ix. Weights of 30 students except for outliers
Stem	Leaf
58	4 4 9
59	0 0 two iii 8
60	0 2 4 5 7 8 9
61	1 2 4 4 v 6 vii 9 9
62	1 ii three 7

When using a stem and leafage plot, spotting an outlier is often a thing of judgment. This is because, except when using box plots (explained in the section on box and whisker plots), at that place is no strict rule on how far removed a value must exist from the rest of a data set to authorize as an outlier.

Height of Folio

Features of distributions

When yous assess the overall pattern of any distribution (which is the pattern formed past all values of a particular variable), look for these features:

number of peaks
general shape (skewed or symmetric)
centre
spread

Number of peaks

Line graphs are useful because they readily reveal some characteristic of the data. (Run into the section on line graphs for details on this blazon of graph.)

The beginning characteristic that can be readily seen from a line graph is the number of high points or peaks the distribution has.

While nigh distributions that occur in statistical information have simply one principal peak (unimodal), other distributions may take 2 peaks (bimodal) or more than two peaks (multimodal).

Examples of unimodal, bimodal and multimodal line graphs are shown below:

Examples of unimodal, bimodal and multimodal line graphs.

Full general shape

The second main characteristic of a distribution is the extent to which it is symmetric.

A perfectly symmetric curve is one in which both sides of the distribution would exactly match the other if the figure were folded over its central point. An example is shown beneath:

Example of a perfectly symmetric curve.

A symmetric, unimodal, bell-shaped distribution—a relatively common occurrence—is chosen a normal distribution.

If the distribution is lop-sided, information technology is said to exist skewed.

A distribution is said to be skewed to the right, or positively skewed, when well-nigh of the data are concentrated on the left of the distribution. Distributions with positive skews are more common than distributions with negative skews.

Income provides ane instance of a positively skewed distribution. Almost people make nether $xl,000 a year, only some brand quite a bit more, with a smaller number making many millions of dollars a twelvemonth. Therefore, the positive (right) tail on the line graph for income extends out quite a long way, whereas the negative (left) skew tail stops at zero. The correct tail conspicuously extends farther from the distribution's centre than the left tail, as shown below:

Example of a positively skewed distribution.

A distribution is said to exist skewed to the left, or negatively skewed, if near of the data are concentrated on the correct of the distribution. The left tail clearly extends farther from the distribution'southward centre than the correct tail, as shown below:

Example of a negatively skewed distribution.

Centre and spread

Locating the centre (median) of a distribution can exist done by counting half the observations up from the smallest. Obviously, this method is impracticable for very large sets of data. A stem and leaf plot makes this like shooting fish in a barrel, however, considering the data are arranged in ascending order. The mean is another measure of primal tendency. (See the chapter on central tendency for more detail.)

The amount of distribution spread and any large deviations from the general blueprint (outliers) can exist quickly spotted on a graph.

Using stem and foliage plots as graphs

A stem and leaf plot is a elementary kind of graph that is made out of the numbers themselves. Information technology is a means of displaying the chief features of a distribution. If a stalk and leafage plot is turned on its side, it will resemble a bar graph or histogram and provide similar visual information.

Top of Page

Example 6 – Using stalk and leaf plots as graph

The results of 41 students' math tests (with a best possible score of seventy) are recorded below:

31, 49, 19, 62, fifty, 24, 45, 23, 51, 32, 48, 55, 60, 40, 35, 54, 26, 57, 37, 43, 65, 50, 55, 18, 53, 41, fifty, 34, 67, 56, 44, four, 54, 57, 39, 52, 45, 35, 51, 63, 42

Is the variable discrete or continuous? Explain.
Set up an ordered stalk and leaf plot for the data and briefly describe what it shows.
Are there whatsoever outliers? If so, which scores?
Look at the stem and leaf plot from the side. Draw the distribution's main features such as:
1. number of peaks
2. symmetry
3. value at the center of the distribution

Answers

A test score is a discrete variable. For example, it is not possible to have a test score of 35.74542341....

The lowest value is 4 and the highest is 67. Therefore, the stem and leaf plot that covers this range of values looks like this:

Table x. Math scores of 41 students
Stalk	Leaf
0	iv
1	eight 9
two	3 iv half dozen
iii	1 two 4 5 five 7 ix
iv	0 ane 2 iii 4 5 v 8 ix
5	0 0 0 ane 1 2 3 4 4 five 5 6 7 7
6	0 ii three 5 seven

Annotation: The notation 2|4 represents stem two and leaf iv.

The stem and leaf plot reveals that about students scored in the interval between 50 and 59. The large number of students who obtained high results could hateful that the test was too easy, that most students knew the textile well, or a combination of both.

The result of 4 could be an outlier, since in that location is a big gap between this and the next result, xviii.
If the stalk and leaf plot is turned on its side, it will look similar the following:

The distribution has a single height within the l–59 interval.

Although there are simply 41 observations, the distribution shows that most data are clustered at the right. The left tail extends farther from the data heart than the correct tail. Therefore, the distribution is skewed to the left or negatively skewed.

Since there are 41 observations, the distribution centre (the median value) will occur at the 21st observation. Counting 21 observations up from the smallest, the centre is 48. (Note that the aforementioned value would accept been obtained if 21 observations were counted down from the highest observation.)