Graphing and analysing income data
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Introduction
This set of resources provides opportunities for students to use and extend their data display and analysis skills. They will use technology (spreadsheets) to assist in their calculations and graphing of data and investigate the relationship between data and the mean and median in the context of taxable income.
Achievement standard
By the end of Year 7, students represent natural numbers in expanded form and as products of prime factors, using exponent notation. They solve problems involving squares of numbers and square roots of perfect square numbers. Students solve problems involving addition and subtraction of integers. They use all 4 operations in calculations involving positive fractions and decimals, choosing efficient calculation strategies. Students choose between equivalent representations of rational numbers and percentages to assist in calculations. They use mathematical modelling to solve practical problems involving rational numbers, percentages and ratios in financial and other applied contexts, justifying choices of representation. Students use algebraic expressions to represent situations, describe the relationships between variables from authentic data and substitute values into formulas to determine unknown values. They solve linear equations with natural number solutions. Students create tables of values related to algebraic expressions and formulas and describe the effect of variation.
They apply knowledge of angle relationships and the sum of angles in a triangle to solve problems, giving reasons. Students use formulas for the areas of triangles and parallelograms and the volumes of rectangular and triangular prisms to solve problems. They describe the relationships between the radius, diameter and circumference of a circle. Students classify polygons according to their features and create an algorithm designed to sort and classify shapes. They represent objects two-dimensionally in different ways, describing the usefulness of these representations. Students use coordinates to describe transformations of points in the plane.
They plan and conduct statistical investigations involving discrete and continuous numerical data, using appropriate displays. Students interpret data in terms of the shape of distribution and summary statistics, identifying possible outliers. They decide which measure of central tendency is most suitable and explain their reasoning. Students list sample spaces for single step experiments, assign probabilities to outcomes and predict relative frequencies for related events. They conduct repeated single-step chance experiments and run simulations using digital tools, giving reasons for differences between predicted and observed results.
Content descriptions
Acquire data sets for discrete and continuous numerical variables and calculate the range, median, mean and mode; make and justify decisions about which measures of central tendency provide useful insights into the nature of the distribution of data (AC9M7ST01).
Create different types of numerical data displays including stem-and-leaf plots using software where appropriate; describe and compare the distribution of data, commenting on the shape centre and spread including outliers and determining the range, median, mean and mode (AC9M7ST02).
Student learning resources
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Income distribution
Income distribution
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Suggested activity sequence
This sequence is intended as a framework to be modified and adapted by teachers to suit the needs of a class group. If you assign this activity to a class, your students will be assigned all student resources on their 'My learning' page. You can also hand-pick the resources students are assigned by selecting individual resources when you add a work item to a class in 'My classes'.
Part A: Modelling
Introduce the concepts of central tendency, mean and median and provide a definition (write them on the board).
Central tendency: The tendency for the values of a random variable to cluster round its mean, mode, or median.
Mean: The average, found by adding the numbers and dividing the sum by the number of numbers in the list.
Median: Is the middle value in a list ordered from smallest to largest.
Do a simple example as a whole class to model the task. If you’re all working together on this, everyone will need access to a computer with spreadsheet software.
This is the weekly wage of 5 people in a netball team. Two people in the team didn’t want to share their information.
\$1,845 \$1,529 \$2,135 \$1,150 \$986
Set up a spreadsheet in Excel and arrange the Weekly Pay values numerically from lowest to highest
A B 1 Weekly pay 2 986 3 1150 4 1529 5 1845 6 2135 7 Average *mean) 8 Median - Use formulas to calculate average [=AVERAGE(B2:B6)] and median [=MEDIAN(B2B6)].
- Discuss with students about the average and the median being the same.
- The rest of the team have decided they will share what they are paid. One gets $950, and the other gets $3428.
- Add rows to your spreadsheet to include these values, correct the formulas and examine the new mean and median.
- Discuss what’s happened to these measures of central tendency.
Set up this table on your spreadsheet.
Weekly pay ($) Number of people 0-999 1000-1999 2000 + Total = sum formula - Add the formula to calculate total people – discuss with students what they expect the figure to be and what they should do if it’s not.
Display this data using:
i. A line graph.
ii. A column graph.
- Add a chart. Try both a line graph and a column graph, change the colours and add titles.
Note: You could model how to construct a histogram. You might find slides 3-12 of the visualiser: Graphing distributions and correlations useful.
Part B: Investigating
- Walk the students through the instructions of the income distribution investigation. reminding them of the similarities to the problem you modelled.
- Students complete the investigation. To do so, they will need copies of the income distribution data sheet. Students may benefit from working individually and/or in small discussion groups.
Part C: Discussing results
When students have completed their investigations, discuss reasons why the average taxable income is so much higher than the median. (Because the income of some exceptionally well-paid people drags the average up). If necessary, provide an example:
For example, imagine there are 10 people in a room: nine of them earn $10,000 a year, and one of them earns $500,000 a year.
What's the average?
Altogether, they earn $590,000 in a year. So, the average income in that room would be $59,000 ($590,000 divided by 10 people).
Notice how, because of the very high income of a single person, the average income for the group is much higher than the typical income of 90 per cent of people in the group? The person who earned $500,000 is an outlier. Outliers in data cause the averages to be misleading!
Discuss frequency and distribution and what students have learned.