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, there are an infinite number of possible values that the data can take.

Concept

Dot Plot

A dot plot is a way to represent numerical data in which each data point is represented with a dot on a horizontal number line. The dots representing the measurements are stacked above the line. Consider the data set

44314414 .

To represent this data on a dot plot, it is necessary to count the frequency of each value, or how often it occurs. It can be seen that

1

occurs twice,

3

once, and

4

five times. Thus, the corresponding dot plot can be drawn.

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 representation of a frequency table. The bars represent specific ranges of values — or intervals — and the heights of the bars correspond to the frequency of data points in that interval. All intervals must have the same size.

For example, in a fruit store they want 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 - 79

g,

80 - 89

g, and so on.

From the histogram, it can be seen that $65$ apples weigh between $100$ and $109$ grams. In a histogram all intervals must have the same size.

Concept

Histogram Distributions

Histograms can be described depending on the shape of their distribution. If, by drawing a vertical line through its center, a histogram can be divided in two mirror images, then the histogram is said to be symmetric.

If each interval in the histogram has approximately the same frequency it is said to be uniform.

When a histogram has a peak which is not in the middle, the histogram is skewed.

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 $12 = 3.46410 \dots$ 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 $1 - 10, 11 - 20, 21 - 30, and 31 - 40 .$ 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