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8.1: Statistical Graphs and Tables - Mathematics


8.1: Statistical Graphs and Tables - Mathematics

8.1: Statistical Graphs and Tables - Mathematics

Tables, graphs, and charts are an easy way to clearly show your data.
Be sure to consider how to best show your results with appropriate graph forms.

Tables

is a very useful way of organizing numerical information or data.

2010 South Africa
2006 Germany
2002 South Korea / Japan
1998 France
1994 USA
1990 Italy
1986 Mexico
1982 Spain
1978 Argentina
1974 Germany
1970 Mexico
1966 England
1962 Chile
1958 Sweden
1954 Switzerland
1950 Brazil



A graph is a chart or drawing that shows the relationship between changing things.
It is a diagram displaying the relationship between numbers or amounts.

Most graphs use bars, lines, or parts of a circle to display data.

A line graph is a diagram, usually a line or curve, which shows how two or more sets of numbers

or measurements are related.
The names of the axes on a graph are the vertical axis and the horizontal axis
The vertical axis is sometimes called the y axis , and the horizontal axis is sometimes called the x axis .

Here is an example of a common graph that will use x and y coordinates for different points on the graph
(notice that negatives numbers used)

The different types of lines commonly used in line graphs are:

solid line ________________________

dotted line . . . . . . . . . . . . . . . . . . . .

broken line -- -- -- -- -- -- -- -- -- -- --

Pictographs are similar to a bar chart but use pictures to symbolize a countable unit of items

A Bar graph is a diagram that makes information easier to understand

by showing how two or more sets of data are related.

A bar graph is usually divided into vertical columns

A Histogram is a bar chart such as these, used to show how different sets of information compare.

Histograms use a sequence of numbers as units to compare.

Circle graph or Pie charts

A pie chart is a circle divided into segments (usually represented with percentages)


Difference Between Table and Chart

A table is a means of displaying data or information in rows and columns. Rows are also called a record or vector, columns are also known as parameters, fields, or attributes. The point of intersection between a column and a row is called a cell.

A table is used in research, data analysis, and communication, and it can be seen in various media from signs to notes in print and in computer software and several other locations. It is used to keep track of information in terms of quantities and numbers as well as names and addresses and other details.

Tables can be simple, consisting only of a few columns and rows, or they can be multi-dimensional consisting of ordered hierarchies. An example of a multi-dimensional table is the multiplication table. Tables are used in:

Publishing – example is the Table of Contents
Mathematics – example is the Multiplication Table
Natural Sciences – example is the Periodic Table
Information Technology – example is one that is supported by software applications such as word processing and presentation software.

A chart, on the other hand, is a graphical display of information wherein the information is illustrated in symbols such as bars, lines, or slices. It is used to define the relationship between a large quantity of data and its parts and makes it easier to read and understand.

Texts are seldom used in a chart they are mostly used in titles which appear above the chart describing data that is being referred to in the chart. Data are displayed in a horizontal (x) axis or a vertical (y) axis each consisting of a scale. A chart also consists of either a major or a minor grid of lines. With data that have multiple variables, the chart must have a legend which lists the variables in the chart for easy identification.

Charts have several types:

Common charts: histogram, bar chart, pie chart, line chart, timeline chart, organizational chart, tree chart, flow chart, area chart, cartogram, and pedigree chart.
Less common charts: bubble chart, polar area diagram, radar chart, waterfall chart, and tree map.
Field specific charts: open-high-low-close chart, candlestick chart, Kagi chart, and sparkline.
Well known charts: Nolan chart, Gantt chart, PERT chart, and Smith chart.
Other charts: control chart, natal chart, nomogram, run chart, structure chart, and strip chart.

1.A table is the representation of data or information in rows and columns while a chart is the graphical representation of data in symbols like bars, lines, and slices.
2.A table can be simple or multi-dimensional. While there are several types of charts, the most common are pie charts bar charts, and line charts.
3.Texts are seldom used in charts while they are often used in tables.
4.A chart is used to help understand a large amount of data and its components while a table is used to keep track of information such as quantities, numbers, names, addresses, and other details.


What is known as a histogram?

A histogram is another kind of graph that uses bars in its display. This type of graph is used with quantitative data. Ranges of values, called classes, are listed at the bottom, and the classes with greater frequencies have taller bars.

A histogram often looks similar to a bar graph, but they are different because of the level of measurement of the data. Bar graphs measure the frequency of categorical data. A categorical variable is one that has two or more categories, such as gender or hair color. Histograms, by contrast, are used for data that involve ordinal variables, or things that are not easily quantified, like feelings or opinions.


Tables and graphs

What are tables and graphs?
Tables and graphs are visual representations. They are used to organise information to show patterns and relationships. A graph shows this information by representing it as a shape. Researchers and scientists often use tables and graphs to report findings from their research. In newspapers, magazine articles, and on television they are often used to support an argument or point of view.

Why do we want students to know about tables and graphs?
Tables and graphs can be useful tools for helping people make decisions. However, they only provide part of a story. Inferences often have to be made from the data shown. As well as being able to identify clearly what the graph or table is telling us, it is important to identify what parts of the story are missing. This can help the reader decide what other information they need, or whether the argument should be rejected because the supporting evidence is suspect. Students need to know how to critique the data and the way it is presented. A table or graph can misrepresent information by

  • leaving out important information. Student absences gives an example of a graph with missing features.
  • constructing it in such a way that it misrepresents relationships. This may be because of poor skills, or it may be done deliberately to bolster a particular argument, for example using 2-dimensional shapes to inflate apparent growth. See Newspaper stories for examples of misleading graphs.

It is easy, if students are not skilful at reading graphs and tables, to interpret them incorrectly. They can make wrong decisions because they are basing them on false inferences. When constructing graphs and tables, it is also possible to misrepresent the data. Research suggests that students often regard tables and graphs as an end in themselves. Few refer to them as a source of evidence, or as a way of exploring patterns and relationships in data or information.

The curriculum
Tables and graphs are relevant to almost all areas of the curriculum. The conventions of tables and graphs are consistent across all curricula. It is the context in which they are used that identifies them as science, social sciences, geography, etc. The table below gives examples of English, mathematics, and science ARB resources that include tables or graphs.

Examples from the Assessment Resource Banks

Tables Graphs
English
Construct a graph to show trends in a character's emotions related to a particular event: Cuthbert’s Babies.
Mathematics
Complete a table to show the amount of staff members' Christmas bonus, then use this to calculate how much money is left over: $200 bonus.
Mathematics
Complete and use a table to graph the cost per hour of repairing a car: Car maintenance.
Science
Complete a table about properties of paper towels: The best mopper upper.
Science
Interpret a graph of a car's journey and add to the graph to represent a further description of the journey: A car journey.

Key competencies
Investigating tables and graphs potentially strengthens several key competencies.

  • Using language, symbols, and texts: Knowing about graphs and tables strengthens students' ability to access and critique others' ideas. It also helps them to effectively communicate their own. The statement that students' "confidently use ICT" (p. 12) reinforces the role assistive technology has for tables and graphs. This should include organising, analysing, and making sense of information as well as being able to "access and provide information and to communicate with others" (Ministry of Education, 2007, p. 12).
  • Participating and contributing: Interpreting and critiquing sometimes conflicting data is a necessary skill for making decisions. Tables and graphs are a useful tool for organising available data for decision making. They are also a useful way of providing evidence to convince others towards a particular argument.
  • Thinking: Analysing and synthesising data from various sources is an important part of developing arguments and decision making.

What are the problem areas for students?
The National Education Monitoring Project (NEMP) identified two relevant sets of skills

NEMP (2003) reported that many New Zealand students

  • did not give their graphs and tables an appropriate title
  • did not label the axes appropriately
  • had difficulty with working with more than one variable at a time, i.e. comparing, calculating, and working with multiple sources.

Trials of ARB resources have identified further areas of difficulty for students.

Language
Some technical vocabulary can cause problems for some students.

Some students are only familiar with the everyday meaning of table. These students draw a kitchen table when asked to "draw a table". Go to Language barriers.

  • In mathematics and science the term range is often used to refer to a single number whereas in everyday situations we tend to use the word "range" to refer to the set of numbers between a lower and upper limit. For example in the question "What is the range of temperature fluctuations shown in this place?" the correct answer would be 14 °C if range is interpreted in a mathematical sense and 35°C to 49°C if range is interpreted in an every day context. Mathematics also refers to the y-axis of a graph as the range (the x-axis is called the domain).

Tables
In our ARB trials we have found that most students can complete simple table-reading tasks. Students have had difficulty with:

  • constructing more complex tables, e.g. two-way tables
  • transforming data from texts
  • interpolating and extrapolating information
  • answering questions that involve calculations.
  • comparing (for example, identifying differences)
  • answering questions about the least (as opposed to the most)
  • considering a number of features to make a decision
  • using information in a table if they haven't also some contextual knowledge
  • using the information in the table to justify decisions.

However, many students also do not complete these sorts of tasks well in contexts other than tables. Organising the information into a table is a helpful strategy for assisting students to develop these skills.

Graphs
In our ARB trials we have found students may have difficulty with

  • selecting an appropriate graph to communicate their findings
  • providing a title for the graph
  • naming the axes
  • reading the scale of the axes, and relating them to the shape of or trends in the graph
  • deciding on the appropriate scale to use when constructing graphs
  • marking sub-units on the axes at regular intervals (although occasionally marking at irregular intervals may be acceptable)
  • including the units of measure (plus any multipliers) on each axis of a graph
  • answering questions that involve calculations
  • plotting information from an article/ written text
  • identifying trends, explaining or synthesising relationships between two graphs, or two or more variables
  • reading the overall shape or trend of a graph
  • interpreting time/distance graphs. They read or construct them as a picture of what happened, for example
    interpreting when the line goes up as going uphill
    going back to the starting point to reach "home".

At Year 4 most students can read the information on a simple graph. Pie graphs may be more difficult than bar or line graphs. At Year 8 many students can extrapolate information from a simple line graph. At Year 10 most students are reasonably successful at converting a straight-forward table to a graph.

When making decisions about students' interpretation of graphs, it is important to also consider their familiarity with the context.
Lack of knowledge about the context may affect their ability to interpret the graph.

Variables – what are they?
In graphs and tables the components that are being compared or measured are called variables. For example, if the question is: How does shadow length vary during a day? The length of the shadow is one variable, and the time of day is the other. It is often useful to describe variables as either dependent or independent. The dependent variables are what can be seen to be changing in relation to the particular levels of the independent variables. In the above example

  • the independent variable is the time of day
  • the length of the shadow is the dependent variable as it depends upon the time of day.

In many instances, however, there is no obvious connection of this type between the variables. In other situations we are interested in how the many variables interact with each other. There are 4 main types of variables:

  • categoric variable – described by a word label, not a number, e.g., different brands of paper towel
  • ordered variable – categoric variables that can be put in order, e.g., cool, warm, hot
  • discrete variable – described by whole numbers only, e.g., 1, 2, 3 teaspoons
  • continuous variable – described by any number or part number, e.g., 35.5°.

An investigation can have any combination of variables. This is defined by the question. Variables which are subject to some sort of random, statistical errors are known as random variables. Most variables in real investigations are of this type (and are usually just referred to as variables).

Constructing tables
Tables are

  • an organiser for an investigation
  • a way of presenting data in a report
  • an organiser to assist comprehension and thinking.

For investigations with no numerical data it is usually better to use a table to present the data. A table with numerous variables can be broken down into smaller tables that look at each variable separately. The interaction between the various variables can then be explored.

  • The independent variables (if they have been identified) go in the left hand columns, the dependent variables on the right.
  • Any column heading should have all the information needed to define the table's meaning. A categoric variable should include a description of the class. A discrete or continuous variable should identify units and any multipliers (e.g., hundreds of people, millions of dollars, kilometres).
  • A title summarises what the table is showing.
  • When investigating, the order of the entries is arbitrary. When reporting results, they should be sorted into an order.
  • Sometimes it is better to put data into bands, e.g., < 10 years, 10-15 years, 16-20 years… this makes it more manageable, and easier to see trends and patterns.
  • A table helps organise information so it is easier to see patterns and relationships.
  • If a variable is continuous the table reveals a lot more information. It may show the range, interval, and number of readings.
  • Tables with multiple variables can provide a lot of information. They can be read by selecting and controlling factors to search for patterns in the data.
  • It can be difficult to see numerical relationships and patterns. A graph may make these clearer.
  • When clumping information into bands, there is no indication of how many are in each category.

Constructing graphs
Purposes
Graphs are

  • a way of exploring the relationships in data
  • a way of displaying and reporting data, making it easier to report patterns and relationships, shapes of distributions, and trends.

Structure
Any graph used to report findings should show

  • the significant features and findings of the investigation in a fair and easily read way
  • the underlying structure of an investigation in terms of the relationships between and within the variables
  • the units of measurement
  • the number of readings (though sometimes these will be in the accompanying text)
  • the range and interval of readings, where appropriate.

It is good practice (but only a convention) to put the dependent variable on the horizontal (x) axis and the independent on the vertical (y) axis.

Bar graphs
Bar graphs should be used for categoric, ordered, and discrete variables. If the number of units in a discrete variable is large it may be displayed as a continuous variable.

Line graphs
Line graphs should be used for continuous variables.

Pie graphs
Pie graphs (sometimes called pie or circle charts) are used to show the parts that make up a whole. They can be useful for comparing the size of relative parts. Because it is difficult to compare different circle graphs, and often hard to compare the angles of different sectors of the pie, it is sometimes better to choose other sorts of graphs.

Histograms
Use histograms when y-axis gives the frequency of, or occurrences for continuous data that has been sorted into groups, for example, 20-24 metres. All bars are usually of equal width. They can be turned into line graphs by connecting the middle of the top section of each vertical bar. Histograms are not joined up bar graphs and should not be used for categoric data (unless the number of units in each group is large).

What a graph can tell you
On a graph you get an overall shape of a variable or the relationships between variables. A line graph represents a numerical or mathematical relationship and so has more information "buried" in it than other graphs. Line graphs can sometimes be used to make predictions for values that were not measured, by interpolating or extrapolating the trend, or by looking at the shape.

  • Graphs can tell you a lot about the design of an investigation, but they don't tell you everything. For example, they don't usually tell you which variables were controlled, the sample size, or the method of measurement. So there are lots of questions to ask to find out about validity and reliability, and also about the actual context of the investigation.
  • The scales on the axes can be stretched or shrunk to emphasise one side of a relationship or to make a point that may not be justified by the data.
  • A graph implies a relationship but not necessarily a cause. For example, a graph may show that houses cost less in March than they did in February, but it does not show why this happened. We may infer it is because the interest rates have gone up.

Interpreting tables and graphs
Gott and Duggan identified layers of complexity to reading, interpreting, and analysing data shown in tables and graphs. These include

  • reading off particular data from points on the table or graph (easy)
  • selecting sections of relevant data from complex data sets (more complex)
  • identifying and interpreting patterns within different types of data (most complex).

A study of 12 and 14 year olds found that individual students noticed the patterns of line graphs in different ways. Their responses were grouped into five categories.

How students interpreted line graphs:

Category Description Example of student response
No pattern - -
Numerical patterns Identified numerical patterns generated from one or both axes. These were irrelevant to the graphs' "messages". Some students were distracted from generalising about the relationships by obvious numerical patterns. The numbers if put in order go even, odd, odd, even and so on.
Graphical patterns Described the shape or direction of the line. These students did not relate this shape to what the axes represented. It goes down and then back up.
Unrelated trends in variable Described a general trend in the separate variables but did not relate these to one another, or described a trend in one variable but not the other. The length of the shadow decreased.
Generalised relationships between variables Were able to generalise a relationship between dependant and independent variables. Prior experience of the context appeared to be a factor in being able to make generalisations. The higher the ball was dropped from the higher it bounced back.

Key questions to ask
The questions below

  • assist students to critique their own and others' tables and graphs
  • provide useful teaching points for teachers as they plan for next learning

Teachers need to consider the age of their students and reword at an appropriate level.

  • Is the information presented appropriately for the design of the investigation?
  • What does the table or graph not tell us about the design of the investigation?
  • What does the information in the table or graph tell us? (Are there any patterns in the data?)
  • What does the data shown not tell us that might invalidate our interpretation?
  • Do the patterns suggest an association, a difference, or a change between the variables?
  • Can we use the pattern in the data to predict and generalise? (This includes being aware of the limitations of the presentation of the data.)
  • Are there alternative interpretations for the pattern of the data? Might other factors be causing the pattern? Have the limitations of the data been clearly identified?

Implications for teaching
Construction

  • Teach not only the agreed conventions, but also the reasons for these.
  • Students are more likely to include titles and name the axes if they understand their purpose.
  • Support students to think about the most appropriate type of table or graph to present their data.
  • Assist students to select an appropriate scale for their data.
  • Younger students may benefit from physically representing data with props, including themselves.
  • Interactive whiteboards, Excel spreadsheets, graphic calculators and other forms of ICT have been used successfully by some teachers for developing skills in constructing graphs. Some computer programmes can be used to generate graphs. Discussion is an important part of using these props and tools.
  • Develop the vocabulary to describe parts of tables and graphs.

Interpretation

  • Assist students to investigate the story the table or graph tells.
  • Give students practice in investigating the relationships presented in tables and graphs, and making inferences from them.
  • Teach students to break a graph into sections, read separately, and then reconstruct to tell the story.
  • Encourage students to decide what other information they need to know before they can make decisions based on the data represented.
  • Develop vocabulary that describes and compares.

Possible progressions in teaching about graphs
Progressions in constructing and interpreting graphs should be used with caution. Students' skill levels are likely to be influenced by


Also Read

4. Stem and Leaf Plot

A stem and leaf plot is one of the best statistics graphs to represent the quantitative data. This graph breaks each value of a quantitative data set into two pieces.

These pieces are often known as the stem and the leaf. Furthermore, the higher places values are known as the stem, and the other places values are known as the leaf.

We can list all the data values in a compact form with the help of this graph. It is a device that is used to represent the data set. It evolved in the early 1900s from Arthur Bowley’s work. Most statisticians use it for data analysis work.

AdvantageDisadvantage
You can represent all the data on a single graph. Moreover, it provides a visual interpretation of distributed data.It is not beneficial to use this graph if you have many stems. It provides less information for scattered data.

5. Dot Plot

It is not that much of a famous statistics graph. Most of the experts say that it is a hybrid of the histogram and a stem and leaf plot.

In this type of graph, each value is represented as the dot, and this dot is placed above the appropriate class. We use this graph to represent quantitative data values. Likewise, we use rectangles and bars in histograms. In the same way, we use the dots, which are joined with the help of simple lines. We use these graphs to compare the data of many individuals.

AdvantageDisadvantage
It is suitable for small and moderate data sets as it highlights clusters and outliers of the data.It is difficult to compare the number of data sets. Moreover, you can not read exact data as the data is being categorized into groups.

6. Scatterplots

Scatterplot graphs are one of the famous statistics graphs that use in the most powerful statistics software. It is used to display data based on the horizontal axis and vertical axis.

I have mentioned earlier that the statistics tools of correlation of regression are used to show trends with the scatterplot. In the scatterplot, the lines or curve is used to show the data.

This chart goes upside down and left to right. Scatter means to place the points at different places to each other. It is the statistics chart to uncover the potential of the dataset.

AdvantageDisadvantage
Because of its visual size, it is possible to make relative comparisons effectively.It isn’t very easy to determine actual values. Sometimes it is difficult to read & understand the data of a scatterplot graph.

7. Time-Series Graphs

The time-series graph is one of the most popular statistics graphs among statisticians. It is used to represent the data points in time. It is the statistics graph that is used for a certain kind of paired data.

We use this graph to measure the trends over a certain period of time. Here in this statistics graph, the timeframe can contain the minutes, hours, days, months, years, decades, or even centuries.

AdvantageDisadvantage
It is good to show how data gets changed over time. Sometimes, the data change is complicated to plot because of its ups and downs result.


8.1: Statistical Graphs and Tables - Mathematics

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        Analyzing and Interpreting Data

        Once collected, data must be presented in a form that can reveal any patterns and relationships and that allows results to be communicated to others. Because raw data as such have little meaning, a major practice of scientists is to organize and interpret data through tabulating, graphing, or statistical analysis. Such analysis can bring out the meaning of data—and their relevance—so that they may be used as evidence.

        Engineers, too, make decisions based on evidence that a given design will work they rarely rely on trial and error. Engineers often analyze a design by creating a model or prototype and collecting extensive data on how it performs, including under extreme conditions. Analysis of this kind of data not only informs design decisions and enables the prediction or assessment of performance but also helps define or clarify problems, determine economic feasibility, evaluate alternatives, and investigate failures. (NRC Framework, 2012, p. 61-62)

        As students mature, they are expected to expand their capabilities to use a range of tools for tabulation, graphical representation, visualization, and statistical analysis. Students are also expected to improve their abilities to interpret data by identifying significant features and patterns, use mathematics to represent relationships between variables, and take into account sources of error. When possible and feasible, students should use digital tools to analyze and interpret data. Whether analyzing data for the purpose of science or engineering, it is important students present data as evidence to support their conclusions.