Ordered Stem And Leaf Plot
Stem and leafage plots
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- Elements of a good stem and leaf plot
- Tips on how to describe a stalk and leaf plot
- Example i – Making a stalk and leaf plot
- The main advantage of a stalk and leaf plot
- Example 2 – Making a stem and leaf plot
- Example iii – Making an ordered stem and leaf plot
- Splitting the stems
- Example iv – Splitting the stems
- Example five – Splitting stems using decimal values
- Outliers
- Features of distributions
- Using stem and foliage plots as graphs
- Case 6 – Using stalk and leaf plots as graph
A stem and leaf plot, or stem plot, is a technique used to classify either discrete or continuous variables. A stem and leafage plot is used to organize data equally they are collected.
A stalk and leafage plot looks something similar a bar graph. Each number in the data is cleaved down into a stalk and a leafage, thus the proper name. The stem of the number includes all but the last digit. The leaf of the number will always be a single digit.
Elements of a good stem and leaf plot
A good stem and leafage plot
- shows the first digits of the number (thousands, hundreds or tens) as the stalk and shows the final digit (ones) every bit the leaf.
- usually uses whole numbers. Anything that has a decimal signal is rounded to the nearest whole number. For instance, exam 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, lowest number, most common number and outliers (a number that lies exterior the main group of numbers).
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Tips on how to draw a stem and foliage plot
Once y'all have decided that a stem and leaf plot is the all-time way to show your data, depict it as follows:
- On the left hand side of the page, write down the thousands, hundreds or tens (all digits only the terminal one). These will be your stems.
- Draw a line to the right of these stems.
- On the other side of the line, write down the ones (the concluding digit of a number). These volition be your leaves.
For instance, if the observed value is 25, then the stem is two and the leafage is the 5. If the observed value is 369, then the stem is 36 and the leaf is 9. Where observations are accurate to 1 or more decimal places, such as 23.7, the stem is 23 and the leafage is 7. If the range of values is besides great, 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 considering the bodily data are used.
Not quite getting it? Try some exercises.
Example i – Making a stem and leaf plot
Each morning, a teacher quizzed his class with 20 geography questions. The class marked them together and everyone kept a tape of their personal scores. As the twelvemonth passed, each student tried to better 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:
| Stem | Leaf |
|---|---|
| 0 | 3 6 5 |
| 1 | 0 i 4 3 5 6 5 vi eight 9 7 9 |
| 2 | 0 0 0 0 |
Analyse Elliot's stem and leaf plot. What is his most mutual score on the geography quizzes? What is his highest score? His everyman score? Rotate the stem and leaf plot onto its side and then that it looks like a bar graph. Are about of Elliot'south scores in the 10s, 20s or under 10? It is hard to know from the plot whether Elliot has improved or not considering we do not know the club of those scores.
Endeavour 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 flavour.
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The main reward of a stalk and foliage plot
The chief advantage of a stalk and leaf plot is that the data are grouped and all the original data are shown, likewise. 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. Nevertheless, the table does non tell y'all what those bodily observations are. A stem and leaf plot would show that information. Without a stalk and leaf plot, the two values (363 and 369) can only be found by searching through all the original data—a tedious job when you take lots of data!
When looking at a information prepare, each observation may exist considered as consisting of two parts—a stem and a leaf. To brand a stem and leaf plot, each observed value must first exist separated into its ii parts:
- The stalk is the first digit or digits;
- The leaf is the final digit of a value;
- Each stem tin consist of any number of digits; just
- Each foliage can have only a unmarried digit.
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Example 2 – Making a stem and leaf plot
A teacher asked 10 of her students how many books they had read in the concluding 12 months. Their answers were equally follows:
12, 23, 19, half dozen, 10, 7, 15, 25, 21, 12
Gear up a stem and leaf plot for these information.
Tip: The number 6 can be written every bit 06, which means that it has a stem of 0 and a leaf of 6.
The stalk and leafage plot should look like this:
| Stem | Leafage |
|---|---|
| 0 | 6 7 |
| 1 | 2 9 0 5 2 |
| ii | 3 5 ane |
In Table two:
- stem 0 represents the course interval 0 to ix;
- stem 1 represents the form interval x to 19; and
- stem 2 represents the class interval 20 to 29.
Commonly, a stem and leaf plot is ordered, which simply means that the leaves are bundled in ascending society from left to right. Besides, there is no need to split the leaves (digits) with punctuation marks (commas or periods) since each leaf is ever a unmarried digit.
Using the data from Tabular array ii, nosotros fabricated the ordered stem and foliage plot shown below:
| Stem | Leaf |
|---|---|
| 0 | 6 7 |
| 1 | 0 2 2 5 ix |
| ii | one 3 5 |
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Instance three – Making an ordered stem and leafage plot
15 people were asked how oft they drove to piece of work over x working days. The number of times each person drove was as follows:
v, vii, nine, 9, three, 5, 1, 0, 0, iv, three, 7, two, 9, 8
Brand an ordered stem and leafage plot for this table.
It should be fatigued equally follows:
| Stem | Leafage |
|---|---|
| 0 | 0 0 one 2 3 3 4 5 5 7 vii 8 ix 9 ix |
Splitting the stems
The organization of this stem and leaf plot does not requite much information nigh the data. With simply one stem, the leaves are overcrowded. If the leaves become as well crowded, then it might be useful to split each stem into two or more components. Thus, an interval 0–nine can be dissever into ii intervals of 0–4 and 5–nine. Similarly, a 0–9 stalk could be split into five intervals: 0–1, 2–3, iv–v, vi–vii and 8–9.
The stem and foliage plot should and then look like this:
| Stem | Leafage |
|---|---|
| 0(0) | 0 0 1 ii 3 iii four |
| 0(5) | 5 5 7 vii 8 9 nine 9 |
Note: The stem 0(0) means all the data within the interval 0–four. The stem 0(5) ways all the information inside the interval five–9.
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Example 4 – Splitting the stems
Britney is a swimmer training for a contest. 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
- Set an ordered stem and leaf plot. Make a brief comment on what it shows.
- Redraw the stem and foliage plot by splitting the stems into v-unit intervals. Make a brief comment on what the new plot shows.
Answers
- The observations range in value from 10 to 39, so the stem and leafage plot should have stems of one, 2 and 3. The ordered stem and leaf plot is shown below:
The stem and leaf plot shows that Britney commonly swims betwixt twenty and 29 laps in training each day.Tabular array 6. Laps swum by Britney in thirty days Stem Leaf 1 0 8 9 2 0 1 2 2 4 iv four 5 5 6 6 6 seven 7 7 7 viii 8 8 8 8 9 9 3 ane ane 2 9 - Splitting the stems into five-unit intervals gives the following stalk and leaf plot:
Table 7. Laps swum past Britney in 30 days Stem Leaf 1(0) 0 1(5) 8 9 2(0) 0 1 2 2 four 4 4 2(5) 5 5 6 six half-dozen 7 seven 7 7 eight 8 8 viii 8 nine 9 3(0) i one two 3(5) 9 Note: The stem ane(0) means all data between 10 and 14, 1(five) ways all data between fifteen and 19, and so on.
The revised stem and leaf plot shows that Britney unremarkably swims between 25 and 29 laps in grooming each mean solar day. The values 1(0) 0 = 10 and iii(five) 9 = 39 could be considered outliers—a concept that volition be described in the next department.
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Example five – 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.v, 62.3, 61.4, 60.9, 59.viii, threescore.5, 59.0, 61.1, 60.seven, 61.six, 56.three, 61.9, 65.7, 60.four, 58.9, 59.0, 61.2, 62.1, 61.4, 58.4, 60.viii, 60.two, 62.seven, 60.0, 59.3, 61.ix, 61.7, 58.four, 62.ii
Fix an ordered stem and leaf 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 will be the decimal values. The data range from 56.3 to 65.7, and then the stems should start at 56 and stop at 65.
| Stem | Leaf |
|---|---|
| 56 | 3 |
| 57 | |
| 58 | four four 9 |
| 59 | 0 0 2 3 8 |
| 60 | 0 ii 4 v 7 eight nine |
| 61 | one 2 four 4 5 half-dozen 7 nine 9 |
| 62 | 1 2 three 7 |
| 63 | |
| 64 | |
| 65 | 7 |
In this instance, it was non necessary to split up stems because the leaves are not crowded on too few stems; nor was it necessary to round the values, since the range of values is not big. This stem and leaf 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 information. It is an observation value that is significantly dissimilar from the rest of the data. In that location may exist more than one outlier in a set of information.
Sometimes, outliers are significant pieces of information and should not exist ignored. Other times, they occur because of an fault or misinformation and should be ignored.
In the previous example, 56.3 and 65.seven could be considered outliers, since these two values are quite dissimilar from the other values.
By ignoring these two outliers, the previous instance'southward stalk and leaf plot could be redrawn as beneath:
| Stem | Leaf |
|---|---|
| 58 | 4 four ix |
| 59 | 0 0 2 three 8 |
| 60 | 0 2 4 5 7 viii 9 |
| 61 | 1 2 4 4 5 half dozen vii nine 9 |
| 62 | 1 2 3 7 |
When using a stem and leaf plot, spotting an outlier is often a matter of judgment. This is considering, 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 set to authorize as an outlier.
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Features of distributions
When y'all assess the overall pattern of any distribution (which is the design formed past all values of a detail variable), wait 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. (See the department on line graphs for details on this blazon of graph.)
The showtime characteristic that can be readily seen from a line graph is the number of high points or peaks the distribution has.
While almost distributions that occur in statistical information have but one main peak (unimodal), other distributions may have two peaks (bimodal) or more than than ii peaks (multimodal).
Examples of unimodal, bimodal and multimodal line graphs are shown below:
Full general shape
The 2d main characteristic of a distribution is the extent to which it is symmetric.
A perfectly symmetric curve is i 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:
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 be skewed to the correct, or positively skewed, when virtually of the data are full-bodied on the left of the distribution. Distributions with positive skews are more common than distributions with negative skews.
Income provides one example of a positively skewed distribution. Well-nigh people make under $twoscore,000 a year, but some make quite a bit 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 mode, whereas the negative (left) skew tail stops at nada. The correct tail clearly extends farther from the distribution's center than the left tail, every bit shown below:
A distribution is said to exist 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's centre than the correct tail, as shown below:
Centre and spread
Locating the heart (median) of a distribution tin can be washed by counting one-half the observations upwards from the smallest. Patently, this method is impracticable for very large sets of data. A stem and foliage plot makes this piece of cake, withal, because the data are arranged in ascending social club. The mean is another measure of central tendency. (Meet the chapter on central tendency for more detail.)
The corporeality of distribution spread and any big deviations from the general design (outliers) can exist rapidly spotted on a graph.
Using stem and leaf plots as graphs
A stalk and foliage plot is a simple kind of graph that is made out of the numbers themselves. It is a means of displaying the principal features of a distribution. If a stem and leaf plot is turned on its side, it will resemble a bar graph or histogram and provide similar visual information.
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Example 6 – Using stem and leaf plots as graph
The results of 41 students' math tests (with a best possible score of 70) are recorded beneath:
31, 49, 19, 62, 50, 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, 4, 54, 57, 39, 52, 45, 35, 51, 63, 42
- Is the variable discrete or continuous? Explicate.
- Set an ordered stem and leaf plot for the information and briefly describe what it shows.
- Are there whatsoever outliers? If so, which scores?
- Look at the stem and leafage plot from the side. Depict the distribution'due south chief features such as:
- number of peaks
- symmetry
- value at the middle of the distribution
Answers
- A test score is a discrete variable. For example, it is not possible to accept a examination score of 35.74542341....
- The lowest value is 4 and the highest is 67. Therefore, the stem and foliage plot that covers this range of values looks like this:
Tabular array 10. Math scores of 41 students Stem Leaf 0 4 1 8 9 two 3 iv 6 iii 1 2 4 5 five 7 nine 4 0 1 2 3 4 5 v 8 9 5 0 0 0 1 1 2 iii 4 4 5 5 6 7 7 6 0 2 3 v seven Note: The notation 2|four represents stem 2 and foliage 4.
The stem and leaf plot reveals that near students scored in the interval between 50 and 59. The large number of students who obtained high results could hateful that the test was likewise like shooting fish in a barrel, that virtually students knew the material well, or a combination of both.
- The result of 4 could be an outlier, since there is a large gap between this and the side by side result, 18.
- If the stalk and leaf plot is turned on its side, it will look like the following:
The distribution has a single peak within the 50–59 interval.
Although at that place are just 41 observations, the distribution shows that nearly data are clustered at the right. The left tail extends further from the data middle than the right tail. Therefore, the distribution is skewed to the left or negatively skewed.
Since there are 41 observations, the distribution heart (the median value) will occur at the 21st ascertainment. 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 downward from the highest observation.)
Ordered Stem And Leaf Plot,
Source: https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch8/5214816-eng.htm#:~:text=Usually%2C%20a%20stem%20and%20leaf,is%20always%20a%20single%20digit.
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