Stem And Leaf Plot Range
Stem and leaf plots
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- Elements of a good stem and leaf plot
- Tips on how to describe a stem and leaf plot
- Example 1 – Making a stem and leafage plot
- The main advantage of a stem and leaf plot
- Example 2 – Making a stem and leaf plot
- Case 3 – Making an ordered stem and leafage plot
- Splitting the stems
- Case 4 – Splitting the stems
- Example 5 – Splitting stems using decimal values
- Outliers
- Features of distributions
- Using stalk and leaf plots as graphs
- Example 6 – Using stem and leafage plots equally graph
A stalk and leaf plot, or stem plot, is a technique used to allocate either discrete or continuous variables. A stalk and leaf plot is used to organize data every bit they are nerveless.
A stalk and leaf plot looks something like a bar graph. Each number in the data is broken down into a stalk and a leaf, thus the name. The stalk of the number includes all but the terminal digit. The leaf of the number will e'er be a unmarried digit.
Elements of a skillful stalk and leaf plot
A good stem and leaf plot
- shows the first digits of the number (thousands, hundreds or tens) every bit the stem and shows the last digit (ones) as the foliage.
- ordinarily 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 information 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 stalk and leafage plot
Once you take decided that a stem and foliage plot is the best way to show your data, describe it as follows:
- On the left hand side of the page, write downwards the thousands, hundreds or tens (all digits just the last one). These will be your stems.
- Depict a line to the right of these stems.
- On the other side of the line, write downwards the ones (the terminal digit of a number). These will be your leaves.
For example, if the observed value is 25, so the stem is 2 and the leafage is the 5. If the observed value is 369, so the stem is 36 and the leaf is 9. Where observations are accurate to i or more decimal places, such as 23.7, the stalk is 23 and the leaf is 7. If the range of values is too nifty, the number 23.7 can be rounded up to 24 to limit the number of stems.
In stem and leaf plots, tally marks are non required because the actual data are used.
Not quite getting it? Try some exercises.
Case 1 – Making a stem and leafage plot
Each morning time, a teacher quizzed his class with xx geography questions. The class marked them together and everyone kept a record of their personal scores. As the yr passed, each student tried to improve 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:
| Stem | Leafage |
|---|---|
| 0 | 3 6 five |
| ane | 0 one 4 3 5 6 five half dozen viii 9 seven 9 |
| two | 0 0 0 0 |
Analyse Elliot's stem and foliage 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 so that information technology looks like a bar graph. Are near of Elliot'south scores in the 10s, 20s or under ten? It is difficult to know from the plot whether Elliot has improved or not because nosotros practice not know the guild of those scores.
Try making your own stalk and leaf plot. Use the marks from something like all of your examination results last year or the points your sports squad accumulated this season.
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The main advantage of a stem and leaf plot
The main advantage of a stalk and leaf plot is that the data are grouped and all the original data are shown, besides. In Example three on battery life in the Frequency distribution tables section, the table shows that 2 observations occurred in the interval from 360 to 369 minutes. All the same, the table does not tell you what those bodily observations are. A stem and leaf plot would bear witness that information. Without a stem and leaf plot, the ii values (363 and 369) can but be found past searching through all the original information—a tedious job when you have lots of information!
When looking at a data set, each observation may be considered equally consisting of two parts—a stem and a leaf. To make a stem and foliage plot, each observed value must first exist separated into its ii parts:
- The stem is the outset digit or digits;
- The leaf is the final digit of a value;
- Each stem can consist of whatsoever number of digits; merely
- Each leaf can have but 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 final 12 months. Their answers were as follows:
12, 23, xix, vi, 10, seven, 15, 25, 21, 12
Prepare a stem and leafage plot for these data.
Tip: The number vi tin be written every bit 06, which means that it has a stem of 0 and a leaf of half-dozen.
The stalk and leaf plot should look similar this:
| Stem | Foliage |
|---|---|
| 0 | 6 seven |
| 1 | two 9 0 5 2 |
| 2 | 3 v one |
In Table 2:
- stem 0 represents the form interval 0 to 9;
- stem 1 represents the course interval 10 to 19; and
- stalk 2 represents the form interval xx to 29.
Commonly, a stem and foliage plot is ordered, which simply means that the leaves are arranged in ascending order from left to correct. Also, there is no need to separate the leaves (digits) with punctuation marks (commas or periods) since each leaf is always a single digit.
Using the data from Table 2, nosotros made the ordered stem and leaf plot shown below:
| Stem | Leafage |
|---|---|
| 0 | 6 7 |
| i | 0 2 two 5 9 |
| two | i 3 5 |
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Example 3 – Making an ordered stalk and leaf plot
Fifteen people were asked how often they drove to piece of work over 10 working days. The number of times each person collection was every bit follows:
5, 7, 9, ix, iii, five, 1, 0, 0, four, 3, vii, 2, 9, viii
Make an ordered stem and leaf plot for this table.
It should exist drawn every bit follows:
| Stem | Leaf |
|---|---|
| 0 | 0 0 1 2 3 3 4 5 5 7 7 8 nine 9 9 |
Splitting the stems
The system of this stalk and leaf plot does non give much data near the data. With only i 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 0–9 can be split into two intervals of 0–4 and five–9. Similarly, a 0–9 stem could exist split into five intervals: 0–one, 2–3, 4–5, half-dozen–7 and 8–9.
The stem and leaf plot should so look like this:
| Stem | Leaf |
|---|---|
| 0(0) | 0 0 ane 2 three 3 4 |
| 0(v) | v 5 seven seven 8 9 ix 9 |
Note: The stem 0(0) means all the data inside the interval 0–four. The stalk 0(v) means all the data within the interval 5–ix.
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Example four – Splitting the stems
Britney is a swimmer training for a competition. The number of l-metre laps she swam each 24-hour interval for 30 days are as follows:
22, 21, 24, xix, 27, 28, 24, 25, 29, 28, 26, 31, 28, 27, 22, 39, 20, 10, 26, 24, 27, 28, 26, 28, 18, 32, 29, 25, 31, 27
- Fix an ordered stem and leaf plot. Brand a brief comment on what it shows.
- Redraw the stem and leaf plot past splitting the stems into five-unit intervals. Make a brief comment on what the new plot shows.
Answers
- The observations range in value from ten to 39, then the stalk and leaf plot should have stems of 1, 2 and 3. The ordered stalk and leaf plot is shown below:
The stem and leaf plot shows that Britney usually swims between xx and 29 laps in training each day.Tabular array 6. Laps swum past Britney in 30 days Stem Leaf 1 0 eight nine ii 0 i 2 2 4 4 iv 5 5 vi six vi 7 7 7 7 eight viii 8 eight viii 9 9 3 1 ane 2 9 - Splitting the stems into five-unit intervals gives the following stalk and leaf plot:
Table seven. Laps swum by Britney in 30 days Stem Leafage 1(0) 0 1(five) 8 9 two(0) 0 ane 2 ii 4 4 4 2(5) 5 5 six vi half dozen 7 7 7 7 8 8 8 viii 8 nine 9 3(0) 1 1 ii three(5) 9 Note: The stem 1(0) means all data betwixt 10 and 14, i(5) means all information between 15 and 19, and so on.
The revised stem and leaf plot shows that Britney commonly swims betwixt 25 and 29 laps in training each day. The values one(0) 0 = 10 and 3(v) ix = 39 could be considered outliers—a concept that volition be described in the next section.
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Example five – Splitting stems using decimal values
The weights (to the nearest tenth of a kilogram) of 30 students were measured and recorded as follows:
59.two, 61.5, 62.3, 61.4, sixty.ix, 59.eight, lx.5, 59.0, 61.ane, 60.7, 61.6, 56.3, 61.9, 65.7, 60.4, 58.9, 59.0, 61.two, 62.1, 61.4, 58.4, 60.eight, 60.2, 62.7, lx.0, 59.3, 61.9, 61.7, 58.iv, 62.ii
Gear up an ordered stem and foliage plot for the information. Briefly comment on what the analysis shows.
Answer
In this case, the stems will 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 finish at 65.
| Stalk | Leaf |
|---|---|
| 56 | 3 |
| 57 | |
| 58 | 4 iv 9 |
| 59 | 0 0 2 3 8 |
| sixty | 0 2 4 5 seven 8 nine |
| 61 | one 2 four 4 five 6 7 9 9 |
| 62 | 1 two 3 vii |
| 63 | |
| 64 | |
| 65 | 7 |
In this example, information technology was non necessary to dissever 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 stalk and leafage plot reveals that the group 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 ascertainment value that is significantly different from the rest of the data. There may be more than one outlier in a prepare of data.
Sometimes, outliers are significant pieces of information and should not be ignored. Other times, they occur because of an mistake or misinformation and should exist ignored.
In the previous example, 56.three and 65.7 could be considered outliers, since these ii values are quite unlike from the other values.
By ignoring these 2 outliers, the previous example's stem and leaf plot could be redrawn as beneath:
| Stem | Leaf |
|---|---|
| 58 | 4 4 ix |
| 59 | 0 0 2 3 8 |
| sixty | 0 2 4 five vii eight nine |
| 61 | 1 2 four 4 5 half dozen vii 9 9 |
| 62 | one two 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 rule on how far removed a value must exist from the rest of a data set to qualify as an outlier.
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Features of distributions
When you 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 considering they readily reveal some characteristic of the data. (Run into the section on line graphs for details on this blazon of graph.)
The first characteristic that tin 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 i principal peak (unimodal), other distributions may have 2 peaks (bimodal) or more than two 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 information technology 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 key point. An case is shown beneath:
A symmetric, unimodal, bell-shaped distribution—a relatively mutual occurrence—is called 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 correct, or positively skewed, when most of the data are concentrated on the left of the distribution. Distributions with positive skews are more mutual than distributions with negative skews.
Income provides one instance of a positively skewed distribution. Most people make under $twoscore,000 a year, just some make 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 manner, 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:
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 tin be done by counting half the observations up from the smallest. Plainly, this method is impracticable for very large sets of data. A stalk and leaf plot makes this easy, still, because the data are arranged in ascending order. The mean is another measure of central tendency. (Come across the chapter on central tendency for more detail.)
The amount of distribution spread and any large deviations from the general pattern (outliers) tin be quickly spotted on a graph.
Using stem and leaf plots as graphs
A stalk 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 main 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 like visual information.
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Example half-dozen – Using stalk and leaf plots as graph
The results of 41 students' math tests (with a all-time possible score of 70) are recorded below:
31, 49, xix, 62, 50, 24, 45, 23, 51, 32, 48, 55, 60, 40, 35, 54, 26, 57, 37, 43, 65, fifty, 55, eighteen, 53, 41, 50, 34, 67, 56, 44, 4, 54, 57, 39, 52, 45, 35, 51, 63, 42
- Is the variable discrete or continuous? Explain.
- Set an ordered stem and foliage plot for the data and briefly describe what it shows.
- Are there any outliers? If so, which scores?
- Look at the stem and foliage plot from the side. Describe the distribution's chief features such equally:
- number of peaks
- symmetry
- value at the center of the distribution
Answers
- A test score is a discrete variable. For case, it is not possible to have a test score of 35.74542341....
- The everyman value is 4 and the highest is 67. Therefore, the stalk 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 nine 2 iii 4 half-dozen 3 1 two 4 5 v vii nine 4 0 1 two iii 4 v 5 viii ix 5 0 0 0 1 1 2 3 iv 4 5 five half dozen 7 7 6 0 ii 3 v 7 Note: The note 2|4 represents stem two and leaf 4.
The stem and leaf plot reveals that well-nigh students scored in the interval between 50 and 59. The large number of students who obtained high results could mean that the examination was as well easy, that most students knew the material well, or a combination of both.
- The result of iv could be an outlier, since there is a large gap between this and the next result, xviii.
- If the stem and leaf plot is turned on its side, it will look like the post-obit:
The distribution has a single peak inside the fifty–59 interval.
Although there are just 41 observations, the distribution shows that most data are clustered at the correct. The left tail extends farther from the information middle than the right 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 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 ascertainment.)
Stem And Leaf Plot Range,
Source: https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch8/5214816-eng.htm
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