Function Table Math Definition Continuous Data Math Definition
Your teacher asks you to determine the number of people in your class who are taller than 170 cm. To complete this task, you'll have to do two things: measure the heights of all your classmates and then, from those heights, count how many people are taller than 170 cm. In doing this, you will have collected three different types of data: discrete, continuous and grouped. This article will dive deeper into what exactly discrete, continuous and grouped data is, the graphs associated with them, as well as cover examples on how to identify these different types of data.
Definition of continuous and discrete data
In the scenario above, the number of people taller than 170 cm was an example of discrete data. You will have had to count all your classmates taller than 170 cm to determine the exact number. From this, we arrive at the following definition for discrete data:
Discrete data is data that can be counted.
Using the same scenario, the heights of your classmates was an example of continuous data. You determine their heights by measuring each classmate. Each person's height will fall into a range of possible heights for humans. From this, we can define continuous data:
Continuous data is measured data that can be of any value within a range.
Difference between continuous and discrete data
Discrete data is countable whereas continuous data can only be measured, with the most common examples of continuous data being height and weight.
If a 500 g bag of sweets contains 7 sweets, then the discrete data in this scenario would be the number of sweets as it is a countable value, whereas the weight of the bag would be continuous data as it is a value that you measure.
Continuous data falls within a range of reasonable values, whereas discrete data can only be one finite value (e.g. there can only be a certain number of sweets in a bag).
The table below summarizes the differences between continuous and discrete data:
| Discrete data | Continuous data |
| Data that is counted | Data that is measured |
| Can only be one finite value | Can be any value within a reasonable range. |
An easy way to remember the difference between continuous and discrete data is to think of discrete data as data that you can count on your fingers.
Discrete and continuous data examples
The following examples will demonstrate how to identify whether data is discrete or continuous.
A box of 10 books is placed on a scale. The scale reads 8 kg. Identify the discrete and continuous data in this situation.
Solution
The number of books in the box is the discrete data. The exact number of books would have been determined by counting them, hence why it's discrete data.
The weight of the box is continuous data, as its value of 8 kg was measured using a scale.
Let's look at another example.
A sprinter takes 17.2 s to run 100 m at a speed of 21 km/h. Identify the data in this situation.
Solution
The time of 17.2 seconds is continuous data.
The speed at which the sprinter runs is also continuous data.
Grouped data
Another very common data type is grouped data.
Grouped data is data that is given in intervals.
It is most often used with continuous data types. With grouped data, values are no longer represented individually, but are instead grouped into intervals.
The following example demonstrates how one can go about grouping data:
You are asked to record your classmate's heights and present a summary of the results to your principal. One way of doing this would be to list each student's height next to their name, but an easier way of showing this could be to represent it by choosing ranges of heights and listing the number of students that fall within each range.
The ranges you decide on are as follows:
\[ \begin{align} &150 - 159\text{ cm} \\ &160 - 169\text{ cm} \\ &170 - 179\text{ cm} \\ &180\text{ cm}+ \end{align}\]
After measuring each of your 15 classmates and making a mark next to each range they fall within, you tally the results to get:
| Heights | Number of students |
| \(150 - 159\text{ cm}\) | 3 |
| \(160 - 169\text{ cm}\) | 6 |
| \(170 - 179\text{ cm}\) | 5 |
| \(180\text{ cm}+ \) | 1 |
This is grouped data as you have grouped all the students who fall within a specific interval together instead of representing each of their heights individually.
Please note that the number of students that fall within each interval is discrete data, but their heights are still considered to be continuous data.
It is important to ensure that your intervals do not overlap as that could result in something falling within two intervals and this could cause the data to be misrepresented.
Graphs of discrete and continuous data
There are various graphs that can be used to represent the different types of data.
Discrete data graphs
Scatter graphs are often used to represent discrete data. Each point on the graph represents one data value.
Discrete data can also be represented by Bar graphs.
A class of 20 students was asked to raise their hands when their favourite subject was called. The teacher counted five hands for Mathematics, seven hands for biology, two hands for geography and six hands for chemistry.
The teacher decided that the best way to visualize the data was to make use of a bar graph, so she made the following:
Bar graph showing students favourite subjects.
As you can see, there are multiple values that the teacher counted. Each one of these is of the discrete data type. Each bar on the graph represents a subject, and the top of each bar coincides with the number of students, as shown on the y-axis.
Continuous data graphs
Continuous data is most often represented using line graphs, but can also be represented using scatter plots and bar graphs .
Your teacher asks you to collect a set of continuous data and represent it in a graph. You decide to record the temperature at 9am every day for a week, using the thermometer in your geography classroom.
You record the following values:
| Day | Temperature (\(^\circ\)C) |
| Monday | 22 |
| Tuesday | 25 |
| Wednesday | 19 |
| Thursday | 23 |
| Friday | 26 |
| Saturday | 20 |
| Sunday | 25 |
To represent this data, you could choose a line graph:
Line graph showing the temperatures recorded on each day of the week.
The shape of the graphs helps show how the temperature varies throughout the week.
Alternatively, you could choose to represent the data with a scatter plot and a line of best fit:
Scatter plot showing the temperature recorded on each day of a week.
As shown in the diagram, a line of best fit can be used to see if there is any linear correlation between the data.
Continuous data can also be represented by bar graphs, as shown in the following example:
Using the same values as from the previous example, you can represent the temperatures using a bar graph:
Bar graph showing the temperatures recorded on each day of the week.
Data, such as your classmates heights, could be represented using a scatter graph, but is better suited to grouped data graphs that are covered in the next section.
Grouped data graphs
Grouped data is best represented using cumulative frequency graphs and histograms, but it can also be represented using other graphs such as bar graphs.
Following on from the heights example in the grouped data section above, you can plot a graph of the results.
Remember that these were the heights you recorded:
| Heights | Number of students |
| \(150 - 159\text{ cm}\) | 3 |
| \(160 - 169\text{ cm}\) | 6 |
| \(170 - 179\text{ cm}\) | 5 |
| \(180\text{ cm}+\) | 1 |
The easiest way to show this spread of data would be to use a histogram.
Check out the article on Histograms for an in-depth explanation of how to plot a histogram.
Using the \(x\)-axis to represent the range of heights and the \(y\)-axis to represent the number of students, you will end up with a histogram that looks like the following:
Histogram showing the heights of the students in a class.
Alternatively, you could represent the data using a cumulative frequency graph:
Cumulative frequency graph showing the heights of the students in a class.
You and your family want to go to a trampoline park, but you all need special socks to be allowed to jump. The socks only come in the following sizes:
- Size A - fits shoe sizes 1-3
- Size B - fits shoe sizes 4-6
- Size C - fits shoes sizes 7-9
- Size D - fits shoe sizes 10-12
Your mother tasks you with collecting the shoes sizes of each family member and tallying the total number of pairs of socks needed per size. Anybody who wears half sizes must take the next size up. She is also a school teacher and asks you to represent this data in a graph as well for extra practice for your upcoming exams.
First, you show the data in a table:
| Shoe sizes | Number of family members |
| 1-3 | 2 |
| 4-6 | 5 |
| 7-9 | 3 |
| 10-12 | 1 |
Next, you draw a histogram to represent the data:
Histogram showing the shoe sizes and pairs of socks needed for each member of a family.
Discrete, continuous and grouped data - Key takeaways
- Discrete data is data that is counted and can only be one value.
- Continuous data is data that is measured, and it can be any value within a range.
- Grouped data is data that is given within ranges.
- Bar graphs are frequently used to represent discrete data.
- Line graphs are most often used to represent continuous data.
- Histograms and cumulative frequency graphs are often used to represent grouped data.
Source: https://www.studysmarter.co.uk/explanations/math/probability-and-statistics/continuous-and-discrete-data/
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