Create a stem display for the data set 13

Stalk 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. Please contact the states to request a format other than those available.

• Elements of a good stem and leafage plot
• Tips on how to draw a stem and leaf plot
  • Example 1 – Making a stem and leaf plot
• The main advantage of a stem and leaf plot
  • Example ii – Making a stem and leafage plot
  • Example 3 – Making an ordered stalk and leaf plot
• Splitting the stems
  • Example 4 – Splitting the stems
  • Instance v – Splitting stems using decimal values
• Outliers
• Features of distributions
• Using stem and leafage plots every bit graphs
  • Instance half-dozen – Using stem and leafage plots as graph

A stem and leafage plot, or stalk plot, is a technique used to classify either discrete or continuous variables. A stem and leaf plot is used to organize data as they are collected.

A stalk and leaf plot looks something similar a bar graph. Each number in the data is broken downward into a stem and a leafage, thus the name. The stalk of the number includes all merely the last digit. The leaf of the number volition always be a single digit.

Elements of a proficient stem and leaf plot

A adept stem and leaf plot

• shows the first digits of the number (thousands, hundreds or tens) as the stem and shows the last digit (ones) as the leafage.
  
• usually uses whole numbers. Anything that has a decimal point is rounded to the nearest whole number. For example, test results, speeds, heights, weights, etc.
  
• looks like a bar graph when it is turned on its side.
  
• shows how the data are spread—that is, highest number, everyman number, most mutual number and outliers (a number that lies outside the principal group of numbers).

Tiptop of Page

Tips on how to depict a stalk and leaf plot

Once you accept decided that a stalk and leaf plot is the best way to show your data, draw information technology equally follows:

• On the left mitt side of the page, write downward the thousands, hundreds or tens (all digits only the concluding one). These will be your stems.
• Draw a line to the right of these stems.
• On the other side of the line, write downwardly the ones (the last digit of a number). These will be your leaves.

For example, if the observed value is 25, so the stem is 2 and the leaf is the 5. If the observed value is 369, then the stem is 36 and the leaf is 9. Where observations are accurate to one or more decimal places, such as 23.vii, the stem is 23 and the leaf is vii. If the range of values is too corking, the number 23.seven tin be rounded upwardly to 24 to limit the number of stems.

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

Non quite getting it? Try some exercises.

Example ane – Making a stem and leaf plot

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

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

Stem	Leaf
0	three half-dozen 5
i	0 i 4 3 5 half dozen v 6 8 9 seven 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 everyman score? Rotate the stem and leaf plot onto its side so that it looks similar 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 not know the order of those scores.

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

Top of Page

The main reward of a stalk and leaf 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 department, the table shows that two observations occurred in the interval from 360 to 369 minutes. Notwithstanding, the table does not tell y'all what those actual observations are. A stem and leaf plot would show that information. Without a stem and foliage plot, the two values (363 and 369) can only be establish past searching through all the original data—a tedious chore when yous have lots of information!

When looking at a data set up, each observation may be considered as consisting of two parts—a stalk and a leaf. To brand a stem and leaf plot, each observed value must first be separated into its 2 parts:

• The stem is the outset digit or digits;
• The leaf is the final digit of a value;
• Each stem can consist of any number of digits; just
• Each leafage tin have only a single digit.

Pinnacle of Folio

Example two – Making a stalk and leaf plot

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

12, 23, nineteen, 6, x, 7, xv, 25, 21, 12

Set up a stem and leafage plot for these data.

Tip: The number half-dozen can be written as 06, which means that information technology has a stem of 0 and a foliage of 6.

The stem and leaf plot should look like this:

Table 2. Books read in a year past x students

Stem	Leafage
0	half dozen vii
ane	2 9 0 5 ii
2	3 five 1

In Table 2:

• stem 0 represents the class interval 0 to 9;
• stem 1 represents the course interval 10 to 19; and
• stem ii represents the class interval 20 to 29.

Usually, a stem and foliage plot is ordered, which simply means that the leaves are arranged in ascending order from left to right. Also, there is no need to separate the leaves (digits) with punctuation marks (commas or periods) since each foliage is ever a single digit.

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

Table 3. Books read in a year past 10 students

Stem	Foliage
0	6 7
1	0 ii ii 5 ix
2	1 3 5

Top of Page

Instance 3 – Making an ordered stem and leaf plot

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

5, 7, 9, 9, iii, five, ane, 0, 0, 4, 3, 7, 2, nine, 8

Make an ordered stem and leafage plot for this tabular array.

It should be drawn every bit follows:

Table 4. Number of drives to work in 10 days

Stem	Foliage
0	0 0 ane 2 iii 3 4 v 5 seven 7 eight nine ix 9

Splitting the stems

The organization of this stem and leaf plot does not give much information about the information. With merely one stem, the leaves are overcrowded. If the leaves become also crowded, then it might be useful to split each stem into two or more than components. Thus, an interval 0–9 can be split into two intervals of 0–iv and five–nine. Similarly, a 0–9 stalk could be split into five intervals: 0–1, 2–3, 4–5, 6–7 and 8–ix.

The stem and leaf plot should and so look like this:

Table 5. Number of drives to work in ten days

Stalk	Leafage
0(0)	0 0 1 2 3 3 four
0(5)	v 5 7 7 eight 9 9 9

Note: The stem 0⁽⁰⁾ means all the data within the interval 0–four. The stem 0⁽⁵⁾ ways all the data within the interval 5–ix.

Top of Folio

Example 4 – Splitting the stems

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

22, 21, 24, nineteen, 27, 28, 24, 25, 29, 28, 26, 31, 28, 27, 22, 39, xx, ten, 26, 24, 27, 28, 26, 28, 18, 32, 29, 25, 31, 27

1. Set up an ordered stem and leaf plot. Brand a brief comment on what it shows.
2. Redraw the stalk and leaf plot by splitting the stems into five-unit intervals. Make a cursory annotate on what the new plot shows.

Answers

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

   Table 6. Laps swum past Britney in 30 days

   Stalk	Leaf
   i	0 eight 9
   2	0 1 two two 4 iv 4 5 5 6 6 6 7 7 7 seven eight 8 8 eight 8 9 9
   3	i one two 9

   The stem and leaf plot shows that Britney commonly swims betwixt 20 and 29 laps in training each day.
2. Splitting the stems into five-unit intervals gives the following stalk and foliage plot:

   Table 7. Laps swum by Britney in xxx days

   Stalk	Leaf
   ane(0)	0
   1(5)	viii 9
   2(0)	0 1 2 2 four iv 4
   two(v)	5 5 6 6 6 7 7 vii 7 8 8 8 eight 8 nine 9
   3(0)	i 1 2
   3(5)	9

   Note: The stem 1⁽⁰⁾ means all data between 10 and 14, one⁽⁵⁾ means all data betwixt 15 and 19, and and so on.

   The revised stem and leaf plot shows that Britney unremarkably swims between 25 and 29 laps in grooming each twenty-four hour period. The values 1⁽⁰⁾ 0 = ten and 3⁽⁵⁾ ix = 39 could be considered outliers—a concept that will be described in the next section.

Top of Page

Case 5 – Splitting stems using decimal values

The weights (to the nearest tenth of a kilogram) of 30 students were measured and recorded as follows:

59.ii, 61.5, 62.3, 61.4, 60.ix, 59.8, 60.5, 59.0, 61.ane, 60.vii, 61.vi, 56.3, 61.9, 65.seven, 60.4, 58.ix, 59.0, 61.2, 62.1, 61.iv, 58.4, sixty.viii, 60.2, 62.7, 60.0, 59.3, 61.9, 61.vii, 58.4, 62.2

Ready an ordered stem and leaf plot for the data. Briefly comment on what the assay shows.

Answer

In this case, the stems will exist the whole number values and the leaves will be the decimal values. The data range from 56.iii to 65.7, so the stems should start at 56 and end at 65.

Tabular array 8. Weights of 30 students

Stem	Leafage
56	iii
57
58	four 4 9
59	0 0 two iii 8
60	0 2 4 5 7 8 9
61	1 ii 4 4 5 half-dozen 7 ix 9
62	i 2 3 7
63
64
65	vii

In this example, information technology was not necessary to divide stems because the leaves are not crowded on too few stems; nor was it necessary to circular the values, since the range of values is not big. This stem and leaf plot reveals that the group with the highest number of observations recorded is the 61.0 to 61.nine grouping.

Acme of Page

Outliers

An outlier is an extreme value of the information. It is an ascertainment value that is significantly different from the rest of the data. In that location may exist more than 1 outlier in a set of data.

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

In the previous example, 56.3 and 65.7 could be considered outliers, since these ii values are quite different from the other values.

Past ignoring these 2 outliers, the previous case's stem and foliage plot could be redrawn as beneath:

Table 9. Weights of 30 students except for outliers

Stem	Foliage
58	4 4 9
59	0 0 2 three 8
lx	0 2 four 5 7 eight 9
61	1 two four 4 five 6 7 9 ix
62	1 2 3 7

When using a stalk and foliage 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), there is no strict rule on how far removed a value must be from the balance of a data set to qualify as an outlier.

Top of Page

Features of distributions

When y'all assess the overall pattern of any distribution (which is the pattern formed by 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 feature of the data. (See the section on line graphs for details on this type of graph.)

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

While most distributions that occur in statistical data have only 1 main pinnacle (unimodal), other distributions may accept two peaks (bimodal) or more than ii peaks (multimodal).

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

General shape

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

A perfectly symmetric curve is 1 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 be skewed.

A distribution is said to exist skewed to the correct, or positively skewed, when almost of the data are concentrated on the left of the distribution. Distributions with positive skews are more than common than distributions with negative skews.

Income provides one case of a positively skewed distribution. About people brand under $twoscore,000 a year, only some make 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 right tail clearly extends farther from the distribution's eye 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 right of the distribution. The left tail clearly extends farther from the distribution'southward centre than the right tail, as shown below:

Centre and spread

Locating the centre (median) of a distribution can be done by counting one-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 easy, however, considering the data are arranged in ascending order. The mean is another measure out of central trend. (See the chapter on central trend for more particular.)

The amount of distribution spread and any large deviations from the general pattern (outliers) tin exist apace spotted on a graph.

Using stem and leafage plots every bit graphs

A stem and leaf plot is a simple kind of graph that is made out of the numbers themselves. It is a means of displaying the main 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.

Tiptop of Page

Example 6 – Using stem and leaf plots every bit graph

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

31, 49, 19, 62, fifty, 24, 45, 23, 51, 32, 48, 55, lx, 40, 35, 54, 26, 57, 37, 43, 65, 50, 55, 18, 53, 41, 50, 34, 67, 56, 44, 4, 54, 57, 39, 52, 45, 35, 51, 63, 42

1. Is the variable discrete or continuous? Explain.
2. Fix an ordered stem and leaf plot for the data and briefly describe what information technology shows.
3. Are in that location any outliers? If so, which scores?
4. Look at the stem and leaf plot from the side. Describe the distribution's main features such equally:
   1. number of peaks
   2. symmetry
   3. value at the centre of the distribution

Answers

1. A test score is a discrete variable. For example, it is not possible to have a test score of 35.74542341....
2. 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:
   

   Tabular array x. Math scores of 41 students

   Stem	Leaf
   0	iv
   1	8 9
   2	3 four 6
   iii	1 2 4 5 5 7 ix
   4	0 i 2 3 4 5 5 8 9
   v	0 0 0 one 1 two 3 iv four 5 5 6 7 7
   6	0 2 3 5 7

   Note: The notation 2|4 represents stalk 2 and leaf iv.

   The stem and leaf plot reveals that about students scored in the interval between fifty and 59. The large number of students who obtained loftier results could mean that the examination was besides like shooting fish in a barrel, that most students knew the textile well, or a combination of both.

3. The effect of 4 could be an outlier, since in that location is a big gap between this and the adjacent outcome, eighteen.
4. If the stem and leaf plot is turned on its side, it will look similar the following:

   

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

   Although there are just 41 observations, the distribution shows that nigh information are amassed at the right. The left tail extends further from the data eye than the correct tail. Therefore, the distribution is skewed to the left or negatively skewed.

   Since there are 41 observations, the distribution center (the median value) volition occur at the 21st observation. Counting 21 observations upwardly from the smallest, the middle is 48. (Note that the same value would take been obtained if 21 observations were counted downwards from the highest observation.)

christiansorinut.blogspot.com

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

Share this post