Stemplot
Encyclopedia
A stemplot in statistics
, is a device for presenting quantitative data in a graphical
format, similar to a histogram
, to assist in visualizing the shape
of a distribution
. They evolved from Arthur Bowley's work in the early 1900s, and are useful tools in exploratory data analysis
. Stemplots became more commonly used in the 1980s after the publication of John Tukey
's publication of Exploratory Data Analysis in 1977. The popularity during those years is attributable to their use of monospaced (typewriter) typestyles that allowed computer technology of the time to easily produce the graphics. Modern computers' superior graphic capabilities have meant these techniques are less often used.
Unlike histograms, stemplots retain the original data to at least two significant digits, and put the data in order, thereby easing the move to order-based inference and non-parametric statistics
.
A basic stemplot contains two columns separated by a vertical line. The left column contains the stems and the right column contains the leaves.
44 46 47 49 63 64 66 68 68 72 72 75 76 81 84 88 106
Next, it must be determined what the stems will represent and what the leaves will represent. Typically, the leaf contains the last digit of the number and the stem contains all of the other digits. In the case of very large numbers, the data values may be rounded to a particular place value (such as the hundreds place) that will be used for the leaves. The remaining digits to the left of the rounded place value are used as the stem.
In this example, the leaf represents the ones place and the stem will represent the rest of the number (tens place and higher).
The stemplot is drawn with two columns separated by a vertical line. The stems are listed to the left of the vertical line. It is important that each stem is listed only once and that no numbers are skipped, even if it means that some stems have no leaves. The leaves are listed in increasing order in a row to the right of each stem.
It is important to note that when there is a repeated number in the data (such as two 44's) then the plot must reflect such (so the plot would look like 4 | 4 4 6 7 9 if it had the numbers 44 44 46 47 49)
4 | 4 6 7 9
5 |
6 | 3 4 6 8 8
7 | 2 2 5 6
8 | 1 4 8
9 |
10 | 6
key: 6|3=63
leaf unit: 1.0
stem unit: 10.0
Rounding may be needed to create a stemplot. Based on the following set of data, the stem plot below would be created:
-23.678758, -12.45, -3.4, 4.43, 5.5, 5.678, 16.87, 24.7, 56.8
For negative numbers, a negative is placed in front of the stem unit, which is still the value X / 10. Non-integers are rounded. This allowed the stem and leaf plot to retain its shape, even for more complicated data sets. As in this example below:
-2 | 4
-1 | 2
-0 | 3
0 | 4 6 6
1 | 6
2 | 4
3 |
4 |
5 | 7
s and finding the mode
. However, stem and leaf plots are only useful for moderately sized data sets (around 15-150 data points). With very small data sets a stem and leaf plot can be of little use, as a reasonable number of data points are required to establish definitive distribution properties. A dot plot
may be better suited for such data. With very large data sets, a stemplot will become very cluttered, since each data point must be represented numerically. A box plot
or histogram
may become more appropriate as the data size increases.
The ease with which histograms can now be generated on computers has meant that stemplots are less used today than in the 1980s, when they first became widely utilized as a quick method of displaying information graphically by hand.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, is a device for presenting quantitative data in a graphical
Information graphics
Information graphics or infographics are graphic visual representations of information, data or knowledge. These graphics present complex information quickly and clearly, such as in signs, maps, journalism, technical writing, and education...
format, similar to a histogram
Histogram
In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson...
, to assist in visualizing the shape
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
of a distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
. They evolved from Arthur Bowley's work in the early 1900s, and are useful tools in exploratory data analysis
Exploratory data analysis
In statistics, exploratory data analysis is an approach to analysing data sets to summarize their main characteristics in easy-to-understand form, often with visual graphs, without using a statistical model or having formulated a hypothesis...
. Stemplots became more commonly used in the 1980s after the publication of John Tukey
John Tukey
John Wilder Tukey ForMemRS was an American statistician.- Biography :Tukey was born in New Bedford, Massachusetts in 1915, and obtained a B.A. in 1936 and M.Sc. in 1937, in chemistry, from Brown University, before moving to Princeton University where he received a Ph.D...
's publication of Exploratory Data Analysis in 1977. The popularity during those years is attributable to their use of monospaced (typewriter) typestyles that allowed computer technology of the time to easily produce the graphics. Modern computers' superior graphic capabilities have meant these techniques are less often used.
Unlike histograms, stemplots retain the original data to at least two significant digits, and put the data in order, thereby easing the move to order-based inference and non-parametric statistics
Non-parametric statistics
In statistics, the term non-parametric statistics has at least two different meanings:The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:...
.
A basic stemplot contains two columns separated by a vertical line. The left column contains the stems and the right column contains the leaves.
Constructing a stem plot
To construct a stem plot, the observations must first be sorted in ascending order: this can be done most easily if working by hand by constructing a draft of the stem and leaf plot with the leaves unsorted, then sorting the leaves to produce the final stem and leaf plot. Here is the sorted set of data values that will be used in the following example:44 46 47 49 63 64 66 68 68 72 72 75 76 81 84 88 106
Next, it must be determined what the stems will represent and what the leaves will represent. Typically, the leaf contains the last digit of the number and the stem contains all of the other digits. In the case of very large numbers, the data values may be rounded to a particular place value (such as the hundreds place) that will be used for the leaves. The remaining digits to the left of the rounded place value are used as the stem.
In this example, the leaf represents the ones place and the stem will represent the rest of the number (tens place and higher).
The stemplot is drawn with two columns separated by a vertical line. The stems are listed to the left of the vertical line. It is important that each stem is listed only once and that no numbers are skipped, even if it means that some stems have no leaves. The leaves are listed in increasing order in a row to the right of each stem.
It is important to note that when there is a repeated number in the data (such as two 44's) then the plot must reflect such (so the plot would look like 4 | 4 4 6 7 9 if it had the numbers 44 44 46 47 49)
4 | 4 6 7 9
5 |
6 | 3 4 6 8 8
7 | 2 2 5 6
8 | 1 4 8
9 |
10 | 6
key: 6|3=63
leaf unit: 1.0
stem unit: 10.0
Rounding may be needed to create a stemplot. Based on the following set of data, the stem plot below would be created:
-23.678758, -12.45, -3.4, 4.43, 5.5, 5.678, 16.87, 24.7, 56.8
For negative numbers, a negative is placed in front of the stem unit, which is still the value X / 10. Non-integers are rounded. This allowed the stem and leaf plot to retain its shape, even for more complicated data sets. As in this example below:
-2 | 4
-1 | 2
-0 | 3
0 | 4 6 6
1 | 6
2 | 4
3 |
4 |
5 | 7
Usage
Stemplots are useful for displaying the relative density and shape of the data, giving the reader a quick overview of distribution. They retain (most of) the raw numerical data, often with perfect integrity. They are also useful for highlighting outlierOutlier
In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....
s and finding the mode
Mode (statistics)
In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
. However, stem and leaf plots are only useful for moderately sized data sets (around 15-150 data points). With very small data sets a stem and leaf plot can be of little use, as a reasonable number of data points are required to establish definitive distribution properties. A dot plot
Dot plot (statistics)
A dot chart or dot plot is a statistical chart consisting of data points plotted on a simple scale, typically using filled in circles. There are two common, yet very different, versions of the dot chart. The first is described by Wilkinson as a graph that has been used in hand-drawn graphs to...
may be better suited for such data. With very large data sets, a stemplot will become very cluttered, since each data point must be represented numerically. A box plot
Box plot
In descriptive statistics, a box plot or boxplot is a convenient way of graphically depicting groups of numerical data through their five-number summaries: the smallest observation , lower quartile , median , upper quartile , and largest observation...
or histogram
Histogram
In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson...
may become more appropriate as the data size increases.
The ease with which histograms can now be generated on computers has meant that stemplots are less used today than in the 1980s, when they first became widely utilized as a quick method of displaying information graphically by hand.