(taken from 14 Exams in Preparation and Practice Exam p. 243)
]]>(taken from 14 Exams in Preparation and Practice Test, p. 241)
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(taken from 14 Exams in Preparation and Practice Exam for TOEFL p. 238-239)
]]>(taken from 14 Exams in Preparation & Practice Exam TOEFL)
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(TOEFL Practice)
]]>Bar graphs generally have categories on the x-axis, and numbers on the y-axis. This means that you can compare numbers between different categories. The categories need to be independent, that is changes in one of them do not affect the others.
Here is a summary of ‘some data’ in a data table:
Some Data | |
Category 1 | 4.1 |
Category 2 | 2.5 |
Category 3 | 3.5 |
Category 4 | 4.7 |
Histogram – Worked Example
You have been given a list of ages in years, and you need to show them in a graph.
The ages are:
5, 12, 23, 22, 28, 17, 11, 21, 25, 23, 7, 16, 13, 39, 35, 42, 24, 31, 35, 36, 35, 34, 37, 44, 51, 53, 46, 45, and 57.
You can choose to group them into ten-year age categories, 0–10, 11–20, 21–30 and so on:
Age | Number of people |
0-10 | 2 |
11-20 | 5 |
21-30 | 7 |
31-40 | 8 |
41-50 | 4 |
51-60 | 3 |
To show this data in a histogram, your x-axis would be numbered in 10s from 0 to your highest age, your y-axis from 0 to 8 (the highest number of people in any group), and there would be no gaps between the bars, because there are no gaps between the age ranges.
A pictogram is a special type of bar graph. Instead of using an axis with numbers, it uses pictures to represent a particular number of items. For example, you could use a pictogram for the data above about ages, with an image of a person to show the number of people in each category:
A pie chart looks like a circle (or a pie) cut up into segments. Pie charts are used to show how the whole breaks down into parts.
For example, this data shows the sales figures for a year, broken down by quarters:
Quarterly Sales Figures | 1^{st} Qtr | 2^{nd} Qtr | 3^{rd} Qtr | 4^{th} Qtr |
8.2 | 3.2 | 1.4 | 1.2 |
From the pie chart you can see immediately that sales in Quarter 1 were much bigger than all the others: more than 50% of total annual sales.
Quarter 2 was next, with around one quarter of sales.
Without knowing anything more about this business, you might be concerned about the way that sales appeared to have dropped over the year.
Pie charts, unlike bar graphs, show dependent data.
The total sales in the year have to have occurred in one quarter or another. If you’ve got the figures wrong, and Q1 should be smaller, one of the other quarters will have sales added to compensate, assuming that you haven’t made a mistake with the total.
Pie charts show percentages of a whole – your total is therefore 100% and the segments of the pie chart are proportionally sized to represent the percentage of the total. For more on percentages see our page: Introduction to Percentages.
Usually it is not appropriate to use pie charts for more than 5 or 6 different categories. Lots of segments are difficult to visualise and such data may be better displayed on a different type of chart or graph.
Line graphs are usually used to show dependent data, and particularly trends over time.
Line graphs depict a point value for each category, which are joined in a line. We can use the data from the pie chart as a line graph too.
You can see even more obviously that sales have fallen rapidly over the year, although the slow-down is levelling out at the end of the year. Line graphs are particularly useful for identifying the point in time at which a certain level of sales, revenue (or whatever the y value represents) was reached.
In the example above, suppose we want to know during which quarter sales first fell below 5. We can draw a line across from 5 on the y-axis (red line on the example), and see that it was during quarter 2.
Cartesian graphs are what mathematicians really mean when they talk about graphs. They compare two sets of numbers, one of which is plotted on the x-axis and one on the y-axis. The numbers can be written as Cartesian coordinates, which look like (x,y), where x is the number read from the x-axis, and y the number from the y-axis.
John is two years older than Mary, and their ages added together equal 12. What age are they both now?
We can solve this by drawing two lines, one of John’s age compared with Mary’s, and one of the ages that add together to give 12.
Line 1: John’s age when Mary is different ages between 1 and 9
Mary’s Age | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
John’s Age (=Mary + 2) |
3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
Line 2: John’s age when Mary is different ages between 1 and 9 if their ages add up to 12
Mary’s Age | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
John’s Age (=12 – Mary’s Age) |
11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 |
Plotting the two lines on graph, with Mary’s age as the x-axis, you can see that there is a point at which the lines cross. This is the only point at which a) John is two years older than Mary and b) their ages add up to 12. This must be their current ages, which are therefore 5 for Mary and 7 for John.
For more about the uses of Cartesian graphs to solve problems in maths, take a look at our pages on Simple Equations and More Complicated Equations.
You can use various computer software packages, including Word and Excel, to draw graphs.
However, be aware that these applications are somewhat limited in the type of charts that they can draw, and you may not find the results entirely what you expected! You really need a basic understanding of graphs and charts so that you can compare what the computer has created to what you want to show.
Computer applications also make it easy to produce overly complicated graphs. A 3D exploding pie chart may look ‘cool’ but does it help you or others to visualise the data? It is often best to keep graphs and charts simple with neat, clear formatting.
Whatever way you choose to draw your graphs, once you have the knack of reading them, you will almost certainly find that the old saying is right: a picture really can tell a thousand words.
Whether a graph is worth a thousand numbers is a moot point, but it is certainly an effective way of showing several numbers together, and demonstrating the relationships or differences between them.
https://www.skillsyouneed.com/num/graphs-charts.html
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http://www.wikihow.com/Write-a-Letter-of-Application-for-a-Job
]]>(taken from 14 Exams in Preparation and Practice Exam TOEFL, by Mr. Kent Ang-zie)
]]>(TOEFL Exercise)
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