18.04 Stem-and-leaf and dot plots

The dot plot is a useful way to express discrete data in a visually simple manner. The main advantages of the dot plot are that we can find the mode and range very easily, as well as quickly see how the data is distributed. The main disadvantages are that we need to count each dot when finding the median and it is often easier to convert to a table to find the mean. The dot plot is particularly suited to discrete data where the frequency of results are often greater than one.

In a dot plot, each dot represents one data point belonging to the result that it is placed above. The modes of a dot plot will be the results with the most dots. Since a dot plot stacks vertically, the greatest columns will belong to the modes.

Examples

Example 1

A group of adults is asked: "How old were you when you passed your driving test?". The responses were: 22,\,17,\,17,\,17,\,19,\,21,\,17,\,22,\,21,\,18,\,18,\,17,\,18,\,22,\,18

The following dot plot represents the responses.

a

What is the range of this data set?

Worked Solution

Create a strategy

To find the range, use the formula: \text{Range}=\text{Highest score}-\text{Lowest score}

Apply the idea

\displaystyle \text{Range}	\displaystyle =	\displaystyle 22-17	Find the difference
	\displaystyle =	\displaystyle 5	Evaluate the subtraction

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b

What is the mode of this data set?

Worked Solution

Create a strategy

Choose the score which has the most dots above it.

Apply the idea

\text{Mode}=17

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c

What is the median of this data set?

Worked Solution

Create a strategy

Order the scores and find the middle score.

Apply the idea

Arrange the scores in order:17,\,17,\,17,\,17,\,17,\,18,\,18,\,18,\,18,\,19,\,21,\,21,\,22,\,22,\,22

The middle score is: 18

\text{Median}=18

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d

How many people passed their driving test on or after their 19\text{th} birthday?

Worked Solution

Create a strategy

Count the number of dots that are greater or equal to 19.

Apply the idea

\displaystyle \text{No. People}

\displaystyle =

\displaystyle 6

Count the scores

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The stem-and-leaf plot is an example of a way to express data in a more complicated way so that we can express more information visually. In particular, the stem-and-leaf plot is used when we have lots of numerical data points.

A stem-and-leaf plot is made up of two components, the stem and the leaf. The stem is usually used to represent the tens part of a score while the leaf is used to represent the ones part of the score.

What is useful about the stem-and-leaf plot is that we can record as many scores as we like by writing the leaves in the appropriate rows.

Notice that the leaves have been arranged in ascending order from left to right. We need to do this so that we can find the median without jumping back and forth across our rows.

It is also worth noting that if there is more than one of the same score, in this case 52 appears twice, each score should have its own leaf.

While stem-and-leaf plots are used primarily to store data of two digit numbers, there are some cases where the stem and leaf might mean something different. It is for this reason that we should always check the key before translating the leaves into scores.

There are also cases of the stem representing the number of tens as usual, except it uses two digit numbers in the stem to express three digit numbers.

In both cases, we need the key to tell us how to interpret the stem-and-leaf plot since the data is different from our usual two digit scores.

Examples