Whenever presented with a collection of data, a first step in understanding that data is to visualize it in some manner. An easy way to visualize the distribution of the data is to construct some form of frequency distribution diagram. The term frequent means happening often The term frequency means the rate at which something is happening. The frequency of an AM radio station at 900 KHz is 900,000 cycles per second. In statistics, the term frequency is used to refer to the number of times that a data value falls into a specified range.

A stem and leaf diagram is an easy way to manually construct a special type of frequency diagram.

A frequency diagram for a set of data values shows how often a data value falls into a given interval.

A stem-and-leaf diagram provides a quick manual way to create a visual distribution of a list of numbers. It appears much like a horizontal histogram.

The method for constructing a stem and leaf diagram is now described by way of an example.

Here is the raw data values that represent the score of each student taking an exam that was worth 70 points.

The dashes indicate a student that did not take the exam.

It is not easy to see a pattern to the data when it is presented as an unordered collection of data values.

Here is the raw data as an ordered list of data values.

How can we make some sense of this data?

One way is to create a frequency distribution diagram.

First, a stem and leaf diagram will be created. Then a spreadsheet will be used to create a traditional frequency distribution diagram.

The typical steps to manually create a stem and leaf diagram, or to use a spreadsheet to create a traditional frequency distribution diagram from a set of data values, is as follows.

• Decide how many intervals to use.
• Determine the range of the data.
• Split up the range into the desired number of intervals.
• Count and record the number of data values in each interval.
• Display the results in some form such as a tabular listing, chart, etc.

In general, you want enough intervals, but not too many. Usually, 5 to 15 intervals are used, unless you have a good reason to use fewer or more intervals.

In this example, the data ranges from 0 to 70 points. The scores are counts (i.e., nonnegative integer values).

The maximum value in the data is 65 while the maximum possible score an the exam is 70. That is, no student scored 100.0% on the exam.

The minimum value in the data is 9 points. That is, no student scored 0.0% on the exam.

Once the number of intervals and the range is determined, the buckets into which each data value will fall can be determined. Suppose that each interval is to be 10.0% of 70 points, or 7 points. Then the buckets into which scores are to be placed could be as follows.

These buckets can be written in a more visually informative and concise manner as follows.

The data values are then placed into the buckets. Each data value counts as one value in the bucket. The more data values in a bucket, the more that bucket contributes to the frequency (count of data values) in that data range.

Next, each score is added to the appropriate bucket.

In standard form, the data values in each row of a stem-and-leaf diagram are in ascending order, from left to right. When constructing the diagram manually, however, one just adds the scores as they are encountered. Note that the form used here is that generated by the author's software, so that the "0 to 6" range is not shown as no one scored in that range, and the "Also:" line indicates that two students (indicated by dashes) did not take the exam. Given a list of scores, the maximum value for any score (70 for the above example), and a point interval (10 points for the above example), you should be able to quickly construct a stem-and-leaf diagram in the form shown above and answer questions about such a diagram.