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Stem and Leaf Plots, Range of a Data Set

Range, median and distribution from the plot.

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Displaying Univariate Data

Dot plots and stem-and-leaf plots are the most common ways to display univariate data (data with one variable).

Dot Plots

dot plot is one of the simplest ways to represent numerical data. After choosing an appropriate scale on the axes, each data point is plotted as a single dot. Multiple points at the same value are stacked on top of each other using equal spacing to help convey the shape and center.

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Stem-and-Leaf Plots

A stem-and-leaf plot is a visual diagram where you organize numbers according to place value. The data is organized in either ascending or descending order.

To build a stem-and-leaf plot, we use place value as our method of organizing data.

If we had a 15 as our number, the stem would be a ten since that is the tens place value. The leaf would be the 5.

To write it as a stem-and-leaf plot, here is what it would look like.

\begin{align*}1\ \bigg |\ 5\end{align*} This means 15.

A stem-and-leaf plot is most useful when looking at a series of data. When we have a series of data, we can organize them according to place value.

22, 15, 11, 22, 24, 33, 45

Let’s say that we want to organize this data in a stem-and-leaf plot. First, we organize them by the tens place since all of our numbers have tens places as the highest place value.

11, 15, 22, 22, 24, 33, 45

Next, we put each stem on the left side of our vertical line.

Notice that the largest of each place is on the left of the lines. Now we can put the ones or the leaves on the right of the vertical line.

Each number in the data has been organized. The tens place is on the left for each number and the ones places that go with each ten are on the right side of the vertical bar.

This is our completed stem-and-leaf plot.

Note 1: Depending on the size of your numbers, your stem column may represent more than just the single greatest place value, and your leaf column may represent more than just the single smallest place value. If you are listing multiple place values in either column, make a key such as the one below easily visible.

Key: 53,243

53 | 243

Note 2: If you have two different related sets of data to compare (or bivariate data), you can use a back-to-back stem plot (also known as a two-sided stem plot). The back-to-back version places the stem in the center of three columns, and the different leaves on each side, as below. Note that ordering the data on the plot requires that you arrange the values so that they increase from the center out , rather than just from left to right.

Interpreting Stem-and-Leaf Plots
The range is the difference between the least and greatest values in the set. It can be important to note when a set appears to have a very large range, especially when there are very limited values in each stem, because it may indicate questionable data.

The shape of a stem plot carries the same general characteristics as a similar shape would if using a histogram:

Bell-shaped: An obvious single and central area of the stem plot that has notably more members than the extremes do is referred to as a bell-shaped plot. This shape indicates that most of the values cluster around the median, and quickly become less dense as we move away from the median and toward either of the extremes.

Uniform: A consistent width of each leaf suggests data that is not dramatically changed by the input. A very uniform stem plot is not particularly useful for identifying trends in the data, and may suggest a need to increase the number of place values in the stem in order to show more detail in each leaf. For example, a stem plot that appears nearly uniform with a stem only representing the hundreds place may show significantly more detail if the stem were set to represent the hundreds and tens places both, since this would create many more leaves.

Skewed: As with the horizontal skewing of a histogram, stem plots with a obvious skew toward one end or the other tend to indicate an increased number of outliers either lesser than the mode (skewed down – correlating to a left-skew in a histogram) or greater than the mode (skewed up – correlating to a right-skewed histogram). Plots with an upward skew will have a mode that is smaller than either the mean or the median, and a mean that is greater than either the median or the mode. Downward-skewed plots will have a mean lesser than median or mode and a mode greater than either mean or median.

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