Representing Numerical Data

There are many ways to represent numerical data. For instance bar graphs, pie charts, and line graphs. Depending on the data and the context in which it is presented, one representation might be a better choice than the other.

Concept

Numerical Data

Numerical data is data that is measurable, such as time, speed and distance. It is described with numbers that can be either discrete or continuous. When the data is continuous, in theory, there are infinitely many possibilities.

Concept

Dot Plot

A dot plot is a way to represent numerical data in which each data point is represented with a dot above a horizontal number line. The dots representing the same measurements are stacked above each other. Consider the following data set: $\begin{gathered} \{\ 4,\ 4,\ 3,\ 1,\ 4,\ 4,\ 1,\ 4\ \} \end{gathered}$

There are two $1$ s in this data set, so on the corresponding dot plot two dots are stacked above the number $1$ of the number line.
There is one $3$ in this data set, so on the corresponding dot plot a single dot is drawn above the number $3$ of the number line.
There are five $4$ s in this data set, so on the corresponding dot plot five dots are stacked above the number $4$ of the number line.

A dot plot illustrating the data set given in the text.

Dot plots are normally used for discrete data. For data sets containing more than

20

data points, dot plots are often inconvenient and other representations are preferred.

Draw the dot plot

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Exercise

Sarah likes the candy Frutty. They are sold in packs of thirty, with the different flavors: apple, orange, and banana. Sarah wanted to know how many banana-flavored candies there are in each pack. She bought ten packs and counted the number of banana bars in each. Her results are as follows. $10, \; 8, \;10, \;9, \;12, \;9, \;10, \;10, \;12, \;10$ Draw a dot plot to represent the data.

Show Solution

Solution

First, we should notice the minimum and maximum values of the data set. It can be seen that they are $8$ and $12,$ respetively. We can draw a horizontal number line for the dot plot from $7$ to $13.$

Next, we can mark a dot for each data point one by one until all points are marked.

From the dot plot, we can see that several packs had $10$ banana-flavored candies and none had $11.$

Concept

Histogram

A histogram is a graphical illustration of a data set.

The data is grouped into specific ranges of values — or intervals — and this grouping is marked on a horizontal line. All intervals must have the same size.
The histogram is the collection of rectangles drawn above the intervals. The height of these rectangles are proportional to the frequency of the data in the corresponding interval.

Say a certain fruit store wants to examine the weights of the apples they sell. To see the distribution, it is not necessary to show each apple's weight individually; instead, the apples can be grouped by their weights in intervals: $70 \text{-} 79\text{g},$ $80 \text{-} 89\text{g},$ and so on.

A histogram showing the distribution of the weight of apples

From the histogram, it can be seen that

65

apples weigh between

100

and

109

grams.

Method

Drawing a Histogram

When drawing a histogram, it is necessary to first decide the intervals. Each interval must have the same length and all data points must lie in an interval. Consider the following data set. $13, \; 11, \; 4, \; 11, \; 21, \; 25, \; 37, \; 17, \; 8, \; 19, \; 26, \; 15$ One method to find a suitable number of intervals is to take the square root of the number of data points. Here that number is $\sqrt{12}=3.46410\ldots$ Thus, the histogram can have either three or four intervals. Here, it will have four. Next, it is necessary to determine the size of the intervals. Since the lowest data value in the set is $4$ and the highest is $37,$ using four intervals with a range of $10$ will encompass all data points. The intervals are then $1 - 10, \quad 11 - 20, \quad 21 - 30, \quad \text{and} \quad 31 - 40.$ Next it is recommended to make a frequency table showing how many data points lie in each interval.

Interval	Frequency
$1-10$	$2$
$11-20$	$6$
$21-30$	$3$
$31-40$	$1$

From the frequency table, the histogram can be constructed by drawing a bar over each interval a height corresponding to the frequency.

Example

Draw the dot plot