Lesson Video: Appropriate Diagrams for Displaying Data Mathematics

In this video, we will learn how to select appropriate diagrams to display data.

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Video Transcript

In this video, we will learn how to select appropriate diagrams to display data. We will begin by recalling 10 different ways we can display data using graphs and charts. Firstly, we have a bar graph or bar chart. This is used when comparing various items in terms of frequency. The frequency is the number of times the item occurs.

A pictograph or pictogram can also be used to show frequency or compare items. This is especially useful when we have a large number of items. And we can use a key so that each symbol represents a certain frequency or number of people. A line graph or line plot allows us to show change, usually over a time period. A conversion graph allows us to compare two units, usually in a linear manner, for example, miles and kilometers. We can use a conversion graph to convert larger values than are on the scale of the graph.

Pie charts or circle graphs can be used to show fractions or percentages of a whole. A scatter graph or scatter plot can be used to determine whether a correlation or link exists between two data sets and how strong it is. Our correlation can be positive, negative, or there can be no or zero correlation. A stem-and-leaf diagram can be used to show frequency. Here, data is grouped according to place value. In the example drawn, the key shows us that three slash four is equal to 34. This means that the number in the stem is the tens value and the number in the leaf is the ones or units value.

Venn diagrams allow us to compare two or more sets of data. The overlap of the circles indicates the items that occur in both sets. A box plot or box-and-whisker plot is used to show the range of values, as well as the median and quartiles of a data set. The two ends of the whiskers are the minimum and maximum values. The two ends of the box represent the lower and upper quartiles. Finally, the line inside the box represents the median.

A cumulative frequency graph is similar to a box-and-whisker plot as it allows us to calculate the median, quartiles, and interquartile range of a data set. The cumulative frequency is the running total, which means that a cumulative frequency graph can never slope downwards.

We will now look at some questions where we need to choose the most appropriate way of displaying data.

Which of the following would be most appropriate to display data if we want to show how frequently each number occurs? Is it option (A) line plot, (B) pie chart, (C) scatter plot, (D) Venn diagram, or (E) box plot?

A line plot shows us how frequently something occurs usually over a period of time. This suggests that option (A) could be the right answer. A pie chart is used to show parts or percentages of a whole. It does not give us the frequency or number of each item. A scatter plot is used to compare two sets of data and identify whether there is a correlation between them. A Venn diagram is also used to compare two or more data sets. Therefore, this is not the correct answer either. A box plot or box-and-whisker plot is used to show the range as well as the median and quartiles of a data set. It does not tell us the frequency of each item.

We can therefore conclude that the most appropriate way to display data if we want to show how frequently each number occurs is option (A) a line plot. We could also use bar graphs or pictographs to display this. However, they were not options in this question.

In our next question, we need to find the most appropriate time to use a particular diagram.

A pie chart is best used to display data when you want to do which of the following? Is it (A) show how data changes over time, (B) compare the frequency of data grouped into equal intervals, (C) see the correlation in data, or (D) compare parts of the data to the whole.

We recall that a pie chart is a circle split into different sections as shown. If we consider the entire circle to be the whole, the data has been split into fractions or percentages. This means that a pie chart enables us to compare parts of the data to the whole. Option (D) is the correct answer.

If we consider the wrong answers, option (A) wanted us to show how data changes over time. In order to do this, we would use a line graph or line plot. In option (B), we wanted to compare the frequency of data grouped into equal intervals. This will be done using a bar graph or a pictograph. In order to see the correlation between two data sets, we would need to draw a scatter diagram or scatter graph. This means that option (A), option (B), and option (C) are incorrect. A pie chart is best used to display data when comparing parts of the data to the whole.

Our next question involves statistical analysis in context.

Students in Madison’s class can speak English, French, Spanish, or a collection of all three. She asks each person to list which of the three languages they speak. She wants to analyze how many people in her class speak one language and how many speak more. Which of the following would be the best choice to display the data she collects? Is it (A) line plot, (B) bar graph, or (C) Venn diagram?

Both the line plot and bar graph would be able to show which of the three languages each child speaks. They would tell us the total number of students who speak English, the total number of students who speak French, and the total number of students who speak Spanish. What they do not tell us, however, is which students speak a collection of all three.

Madison wanted to compare how many people speak one language and how many speak more. This means that line plot and bar graph are not the correct answer. A Venn diagram, on the other hand, will show which students speak more than one language. The area shaded would show the number of students that speak Spanish and English. We could also identify those who speak Spanish and French and French and English.

The middle section of our Venn diagram, where all three circles overlap, would show the students that speak all three languages. Those students that are found in the areas where the circles do not overlap would only be able to speak one language, either English, Spanish, or French. The correct answer is therefore option (C) a Venn diagram. This would be the best way for Madison to analyze how many people in her class speak one language and how many speak more than one language.

In our next question, we will look at two different ways of displaying data.

The following graphs display data about the pets owned by the students in a class. Which of these displays should we use if we want to analyze the most common type of pet? Which of these displays should we used if we want to compare the number of students who own exactly one pet?

In this question, we are given two types of graph. Firstly, we have a bar graph or bar chart. Our second graph is a Venn diagram. The bar graph gives us the frequency or number of students who own each type of pet. We can see that 11 students own a cat, 14 students own a dog, and 10 students own a rabbit. Whilst we could also calculate this from the Venn diagram, we would need to add all the numbers inside each of the three circles. For example, with cat, we would need to add six, two, two, and one to give us a total of 11. We could repeat this process for dogs and rabbits.

This means that the best display to use if we want to analyze the most common type of pet is the bar graph. This is because we don’t need to carry out any calculations. We can read the values straight from the graph. The highest bar, in this case dogs, indicates that this is the most common type of pet, with 14 students owning a dog.

The second part of this question asked us which display we should use to compare the number of students who own exactly one pet. The bar graph does not tell us if any of the students own more than one pet. They might own a cat and a dog, a dog and a rabbit, a cat and a rabbit, or even all three pets. As a result, they would’ve been included in both bars. The Venn diagram, on the other hand, shows us that six people own just a cat; seven people, just a dog; and four people, just a rabbit. This means there were a total of 17 people who own exactly one pet.

The overlaps of the circles in the Venn diagram show us those students who own two or more pets, either a cat and a dog, a dog and a rabbit, a cat and a rabbit, or the two in the center, which shows that two students own all three pets. We can therefore conclude that the Venn diagram is the best way of comparing the number of students who own exactly one pet.

Our final example is a three-part question that compares the merits of different types of diagrams.

A statistician collected data on the masses of a population of birds. She then produced a box plot and a histogram of the masses. From which graph can the exact value of the median be identified? From which graph could you calculate the total population of birds? From which graph could you calculate the exact value of the interquartile range?

We recall that a box plot or box-and-whisker plot can be drawn as shown. It is drawn to show the range of values as well as the median, quartiles, and interquartile range. The minimum and maximum values are the ends of the whiskers. In this case, this will be the lowest and highest mass of the population of birds. The ends of the box identify the lower quartile and upper quartile, and the line inside the box represents the median.

The interquartile range is equal to the upper quartile minus the lower quartile. This means that we can also calculate this easily from a box plot. This means that the correct answer for the first and third parts of this question is the box plot, as the median and interquartile range can be identified and calculated from the box plot.

A histogram is used to show frequency and compare items. Each bar represents an interval of values. In some locations, the 𝑦-axis is represented by the frequency density instead of the frequency. Either way, we can calculate the total population by adding all the frequencies. The correct answer for the second part of this question is the histogram.

We will now summarize the key points from this video. After collecting and organizing data, it is important that we display it in the most appropriate and relevant manner. We can choose how to display our data from a variety of different diagrams, for example, a bar graph, line plot, Venn diagram, scatter plot, and so on. The appropriate diagram will allow us to analyze our data, answer specific questions, and make broader conclusions.

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