How to Draw Stem and Leaf Diagram

Stem and leaf plots

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• Elements of a skillful stalk and leaf plot
• Tips on how to draw a stem and foliage plot
  • Instance one – Making a stem and foliage plot
• The principal advantage of a stalk and leaf plot
  • Case 2 – Making a stem and leaf plot
  • Example 3 – Making an ordered stem and leaf plot
• Splitting the stems
  • Example iv – Splitting the stems
  • Example 5 – Splitting stems using decimal values
• Outliers
• Features of distributions
• Using stem and leaf plots equally graphs
  • Case half dozen – Using stem and leaf plots equally graph

A stem and leaf plot, or stem plot, is a technique used to allocate either discrete or continuous variables. A stem and leaf plot is used to organize data as they are nerveless.

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

Elements of a good stem and leaf plot

A practiced stalk and leaf plot

• shows the showtime digits of the number (thousands, hundreds or tens) as the stem and shows the last digit (ones) equally the leaf.
  
• normally 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 information are spread—that is, highest number, everyman number, almost mutual number and outliers (a number that lies exterior the main grouping of numbers).

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Tips on how to draw a stem and leaf plot

One time you have decided that a stem and foliage plot is the all-time manner to show your information, draw it as follows:

• On the left paw side of the page, write down the thousands, hundreds or tens (all digits merely the last one). These will be your stems.
• Describe a line to the correct of these stems.
• On the other side of the line, write downward the ones (the last digit of a number). These volition be your leaves.

For instance, if the observed value is 25, then the stem is 2 and the leaf is the 5. If the observed value is 369, then the stalk is 36 and the leaf is ix. Where observations are accurate to one or more decimal places, such as 23.7, the stem is 23 and the leaf is seven. If the range of values is also bang-up, the number 23.7 can be rounded upward to 24 to limit the number of stems.

In stem and leaf plots, tally marks are not required because the actual data are used.

Not quite getting it? Try some exercises.

Example 1 – Making a stem and leaf plot

Each morning, a teacher quizzed his class with 20 geography questions. The form marked them together and everyone kept a record of their personal scores. As the twelvemonth passed, each student tried to amend his or her quiz marks. Every day, Elliot recorded his quiz marks on a stem and leaf plot. This is what his marks looked like plotted out:

Table 1. Elliot's scores on the bones facts quiz last year

Stalk	Leafage
0	3 vi v
1	0 1 iv 3 5 six 5 half dozen eight 9 7 9
2	0 0 0 0

Analyse Elliot's stem and leaf plot. What is his most common score on the geography quizzes? What is his highest score? His lowest score? Rotate the stem and leaf plot onto its side and so that it looks like a bar graph. Are most of Elliot's scores in the 10s, 20s or under ten? It is difficult to know from the plot whether Elliot has improved or not because we exercise not know the gild of those scores.

Try making your own stem and foliage plot. Use the marks from something like all of your exam results last year or the points your sports team accumulated this season.

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The main reward of a stem and leaf plot

The main reward of a stalk and foliage 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 tabular array shows that two observations occurred in the interval from 360 to 369 minutes. However, the tabular array does not tell you what those actual observations are. A stalk and leaf plot would bear witness that data. Without a stem and leaf plot, the two values (363 and 369) can only be establish past searching through all the original information—a tedious task when you have lots of information!

When looking at a data set, each ascertainment may exist considered as consisting of two parts—a stem and a foliage. To make a stem and leaf plot, each observed value must first exist separated into its two parts:

• The stem is the start digit or digits;
• The leafage is the terminal digit of a value;
• Each stem tin consist of whatsoever number of digits; just
• Each leafage can take but a single digit.

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Example 2 – Making a stem and leaf plot

A teacher asked ten of her students how many books they had read in the final 12 months. Their answers were equally follows:

12, 23, 19, 6, x, 7, 15, 25, 21, 12

Prepare a stem and leaf plot for these information.

Tip: The number half dozen tin be written every bit 06, which means that information technology has a stalk of 0 and a leaf of half dozen.

The stem and leaf plot should expect like this:

Table 2. Books read in a year by 10 students

Stem	Foliage
0	half-dozen seven
1	ii 9 0 5 2
two	3 five 1

In Table 2:

• stem 0 represents the class interval 0 to 9;
• stem 1 represents the class interval 10 to 19; and
• stem 2 represents the class interval twenty to 29.

Usually, a stem and leaf plot is ordered, which merely means that the leaves are arranged in ascending lodge from left to right. Likewise, in that location is no need to carve up the leaves (digits) with punctuation marks (commas or periods) since each leaf is always a single digit.

Using the data from Table 2, we made the ordered stalk and leaf plot shown below:

Tabular array 3. Books read in a year by ten students

Stem	Leaf
0	6 seven
1	0 two 2 5 9
2	1 3 5

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Example 3 – Making an ordered stem and leaf plot

Fifteen people were asked how often they collection to piece of work over 10 working days. The number of times each person drove was as follows:

5, seven, ix, 9, three, five, i, 0, 0, 4, three, 7, 2, 9, 8

Make an ordered stem and leaf plot for this table.

It should be drawn as follows:

Table 4. Number of drives to work in 10 days

Stem	Leafage
0	0 0 one two iii 3 four 5 v 7 vii 8 ix ix 9

Splitting the stems

The organisation of this stem and leaf plot does non give much information about the data. With merely one stalk, the leaves are overcrowded. If the leaves become too crowded, then it might be useful to divide each stalk into two or more than components. Thus, an interval 0–ix can be split into ii intervals of 0–4 and 5–ix. Similarly, a 0–9 stem could exist split into v intervals: 0–one, ii–iii, four–5, 6–7 and 8–nine.

The stalk and leaf plot should and then look like this:

Table v. Number of drives to work in x days

Stem	Leafage
0(0)	0 0 1 two 3 iii 4
0(5)	v 5 7 7 8 ix 9 ix

Note: The stalk 0⁽⁰⁾ means all the information within the interval 0–iv. The stem 0^(v) means all the information within the interval 5–9.

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Example 4 – Splitting the stems

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

22, 21, 24, 19, 27, 28, 24, 25, 29, 28, 26, 31, 28, 27, 22, 39, xx, 10, 26, 24, 27, 28, 26, 28, 18, 32, 29, 25, 31, 27

1. Prepare an ordered stem and leaf plot. Make a cursory comment on what information technology shows.
2. Redraw the stem and leaf plot by splitting the stems into five-unit of measurement intervals. Make a brief comment on what the new plot shows.

Answers

1. The observations range in value from 10 to 39, so the stem and leaf plot should have stems of 1, 2 and 3. The ordered stalk and foliage plot is shown below:

   Table 6. Laps swum by Britney in 30 days

   Stem	Foliage
   1	0 8 9
   two	0 1 2 2 4 four iv 5 v 6 half dozen 6 seven vii 7 7 8 8 8 8 8 ix ix
   three	ane i 2 9

   The stalk and leaf plot shows that Britney usually swims betwixt xx and 29 laps in training each day.
2. Splitting the stems into five-unit intervals gives the post-obit stem and leaf plot:

   Table seven. Laps swum by Britney in xxx days

   Stalk	Leafage
   one(0)	0
   1(five)	eight ix
   2(0)	0 i 2 ii 4 4 4
   2(5)	5 5 6 6 6 vii 7 7 7 8 8 eight eight 8 ix 9
   3(0)	ane i ii
   3(5)	9

   Note: The stem i⁽⁰⁾ means all data between ten and 14, 1⁽⁵⁾ ways all data betwixt 15 and 19, and so on.

   The revised stem and leaf plot shows that Britney usually swims between 25 and 29 laps in training each twenty-four hours. The values ane⁽⁰⁾ 0 = ten and 3⁽⁵⁾ ix = 39 could be considered outliers—a concept that volition be described in the next department.

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Example 5 – Splitting stems using decimal values

The weights (to the nearest tenth of a kilogram) of xxx students were measured and recorded as follows:

59.2, 61.5, 62.iii, 61.iv, sixty.nine, 59.eight, 60.5, 59.0, 61.i, 60.7, 61.6, 56.3, 61.9, 65.7, 60.4, 58.9, 59.0, 61.2, 62.1, 61.4, 58.iv, sixty.8, sixty.2, 62.7, lx.0, 59.iii, 61.9, 61.7, 58.4, 62.2

Prepare an ordered stem and leaf plot for the data. Briefly annotate on what the analysis shows.

Answer

In this case, the stems will be the whole number values and the leaves volition be the decimal values. The information range from 56.3 to 65.7, so the stems should starting time at 56 and stop at 65.

Table viii. Weights of thirty students

Stem	Foliage
56	three
57
58	four 4 nine
59	0 0 ii 3 viii
threescore	0 two iv five 7 8 9
61	one 2 iv 4 five six seven 9 9
62	1 2 3 7
63
64
65	7

In this example, information technology was not necessary to split stems because the leaves are not crowded on too few stems; nor was information technology necessary to round the values, since the range of values is not big. This stem and leafage plot reveals that the grouping with the highest number of observations recorded is the 61.0 to 61.9 group.

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Outliers

An outlier is an extreme value of the data. It is an observation value that is significantly different from the residuum of the data. There may be more than one outlier in a set of information.

Sometimes, outliers are significant pieces of data and should non be ignored. Other times, they occur considering of an error or misinformation and should be ignored.

In the previous example, 56.iii and 65.7 could be considered outliers, since these ii values are quite unlike from the other values.

Past ignoring these two outliers, the previous example'due south stem and leaf plot could be redrawn as below:

Tabular array nine. Weights of 30 students except for outliers

Stem	Leaf
58	4 four 9
59	0 0 2 three viii
lx	0 2 four 5 7 viii nine
61	1 two four 4 5 6 7 9 9
62	i 2 3 seven

When using a stem and foliage plot, spotting an outlier is often a matter of judgment. This is because, except when using box plots (explained in the section on box and whisker plots), there is no strict dominion on how far removed a value must be from the rest of a data fix to qualify equally an outlier.

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Features of distributions

When you appraise the overall pattern of whatsoever distribution (which is the pattern formed by all values of a item variable), expect for these features:

• number of peaks
• general shape (skewed or symmetric)
• center
• spread

Number of peaks

Line graphs are useful considering they readily reveal some feature of the data. (See the section on line graphs for details on this type of graph.)

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

While well-nigh distributions that occur in statistical data take only one main top (unimodal), other distributions may have two peaks (bimodal) or more than 2 peaks (multimodal).

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

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 below:

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

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

A distribution is said to be skewed to the right, or positively skewed, when almost 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 i example of a positively skewed distribution. Almost people make under $40,000 a twelvemonth, but some brand quite a scrap more, with a smaller number making many millions of dollars a year. 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 right tail clearly extends farther from the distribution'due south centre than the left tail, as shown beneath:

A distribution is said to be skewed to the left, or negatively skewed, if most of the data are concentrated on the correct of the distribution. The left tail clearly extends farther from the distribution'south centre than the correct tail, every bit shown beneath:

Centre and spread

Locating the centre (median) of a distribution can be done by counting half the observations upwards from the smallest. Plain, this method is impracticable for very big sets of information. A stem and leaf plot makes this easy, however, because the data are arranged in ascending order. The mean is another measure of fundamental tendency. (Run into the chapter on cardinal tendency for more detail.)

The amount of distribution spread and whatsoever large deviations from the general pattern (outliers) tin can be chop-chop spotted on a graph.

Using stem and leafage plots as graphs

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

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Case 6 – Using stalk and leafage plots equally graph

The results of 41 students' math tests (with a all-time possible score of 70) 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, 50, 34, 67, 56, 44, iv, 54, 57, 39, 52, 45, 35, 51, 63, 42

1. Is the variable detached or continuous? Explain.
2. Prepare an ordered stem and leaf plot for the data and briefly draw what information technology shows.
3. Are there any outliers? If so, which scores?
4. Expect at the stem and leaf plot from the side. Describe the distribution's principal features such as:
   1. number of peaks
   2. symmetry
   3. value at the heart of the distribution

Answers

1. A test score is a detached variable. For case, information technology is not possible to accept a test score of 35.74542341....
2. The everyman value is 4 and the highest is 67. Therefore, the stem and leafage plot that covers this range of values looks like this:
   

   Table 10. Math scores of 41 students

   Stalk	Foliage
   0	4
   one	8 nine
   2	3 4 half-dozen
   3	1 2 4 5 5 7 9
   4	0 ane ii iii four v 5 8 9
   5	0 0 0 i one 2 three four 4 5 5 half dozen 7 7
   6	0 2 3 five vii

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

   The stem and foliage plot reveals that near students scored in the interval between fifty and 59. The large number of students who obtained high results could mean that the examination was also easy, that most students knew the material well, or a combination of both.

3. The result of 4 could be an outlier, since at that place is a large gap betwixt this and the adjacent result, xviii.
4. If the stem and leaf plot is turned on its side, it volition expect similar the following:

   

   The distribution has a single peak inside the 50–59 interval.

   Although there are but 41 observations, the distribution shows that most data are clustered at the correct. The left tail extends farther from the data centre than the right 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 same value would have been obtained if 21 observations were counted down from the highest observation.)

How to Draw Stem and Leaf Diagram

Source: https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch8/5214816-eng.htm

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