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Level Three > Statistics

Data Distortion

Student Activity: 

    

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Achievement Objectives:

Achievement Objective: S3-2: Evaluate the effectiveness of different displays in representing the findings of a statistical investigation or probability activity undertaken by others.
AO elaboration and other teaching resources

Purpose: 

This is a level 3 statistics activity from the Figure It Out series.

Specific Learning Outcomes: 

critiquing and improving graphs

Description of mathematics: 

This diagram shows the areas of Statistics involved in this activity.

stats diagram.

The bottom half of the diagram represents the 5 stages of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) statistics investigation cycle.

Statistical Ideas

Data Distortion involves evaluating data displays.

Required Resource Materials: 
FIO, Levels 3+ -4, Statistics in the Media, Data Distortion, pages 17-19
A computer spreadsheet/graphing program (optional)
A classmate
Activity: 

Activity One

This activity epitomises statistical literacy at level 3. Unfortunately, not all graphs from various media sources are effective or even accurate, and some may even be distorted (possibly deliberately).
A useful starting point is to revise what a good graph should include (see page 19 of the students’ book) or visit the Statistics New Zealand website (www.stats.govt.nz) and go to the schools corner. Other key features of quality graphs are:
• scales divided into equal intervals
• starting at zero (where appropriate)
• bars of the same style and width
• accurate labels and titles
• data displayed simply
• no distortions.

Activity Two

It is important that the students collect examples of good graphs. Encourage them to include nontraditional data displays (invented graphs) that are just as clear as regular graphs. There are often opportunities for students to invent their own graphs, and seeing examples of these will give them plenty of ideas.
Many graphs use colour to show a variable (for example, in population density). For some interesting data displays, see the National Geographic, The New York Times, scientific journals, or websites devoted to statistical graphics. Encourage your students to look out for maps that convey statistical information.

Answers to Activities

Activity One
1. Discussion and evaluation will vary. In some  cases, the intention may be to distort the data to  support the headline, or it may just be a poor  choice of graph style or wording. Points may  include:
For i.:
a. Flaws:
• Although it is acceptable for the  vertical axis to not start at 0, this needs to be indicated (for example, by a jagged or wavy line). This axis only goes up in ones, which hardly suggests that crime rates are “skyrocketing”.
• The graph itself doesn’t have a title (a separate element from the headline)
or a vertical axis label. (Does it mean the number of criminals caught or
the number of crimes reported?)
b. The author of the graph may have been inexperienced in using a graphing
program or may have wished to exaggerate the increase in crime rates.
c. Possible changes:
• Provide a title for the graph. (This will indicate context. For example, if the graph is about the number of crimes reported, the title could be: Reported Crimes 2006–08.)
• Start the vertical axis at 0 or use a jagged line to indicate that it doesn’t start at 0. Decide on an appropriate vertical axis interval (increment).
• A better headline might be: Reported crime rates edge upwards.
For ii.:
a. Flaws:
• The graph does not have a title.
• The bars (drawn as potatoes) expand in width as well as height (thus visually exaggerating the increase). It is not clear what represents the value – is it the height, the area, or the volume?
• The potatoes are not sitting on the same starting point (the horizontal axis).
• The intervals on the horizontal axis are not equal.
• The label on the vertical axis is not informative (greater than 3 per week or per day?), and in any case, the information on that label should be part of the missing title, with the vertical label just saying Percentage.
• The graph doesn’t say how many children were surveyed, how the sample was chosen, or how reliable or accurate the data source is.
• It’s a very bad graph!
b. The author of the graph might not have been aware that objects should start at the
same point on the vertical axis or that increasing both width and height is misleading. However, it could have been deliberate if the author wanted to emphasise the 2008 figures.
c. Possible changes:
• Change the graph to a bar graph with consistent widths for each year.
• Decide on a title that is based on the current vertical axis label.
• Change the vertical axis label to Percentage.
• A possible headline could be: Big increase in children’s TV watching.
For iii.:
a. Flaws:
• Bars (buildings) are not of equal width.
• The use of colour and shading detracts from meaning (some colours stand out
more than others).
• It doesn’t say what the other buildings are.
b. The author may have been using colour and a drawing of each building to give
interest to the graph.
c. Possible changes:
• Change the graph to a bar graph or to something else (for example, a vertical line with heights and names on it).
• Label the buildings with their names and location.
• Make the graph title more accurate. Research would show that many skyscrapers around the world are over 600 metres high, so the Sky Tower is not “up there” in world terms, even though it is the tallest structure in New Zealand.
• A possible headline is: Heights of Tall Buildings.
For iv.:
a. Flaws:
• The 3D graph does not accurately reflect proportions.
• The front segment of the pie looks bigger because it is closer and lighter in colour.
• The categories are too vague. The first label in the key doesn’t say if
those students actually do drink water at school (for example, out of a
drinking fountain); the fact that other students bring either another type of
drink or a bottle of water to school doesn’t mean they actually drink it.
• There is no title for the graph to indicate what the graph is showing,
for example: What group of students? How many?
b. The author may have hoped that people
would read the headline and relate it to the large top segment without reading the
key or thinking about other ways in which those students have access to drinking
water.
c. Possible changes:
• Draw the graph as a two-dimensional pie graph or as a bar or strip graph.
• Give the graph a title, for example, Percentage of Students Who Bring
Drinks to School.
• A possible headline is: Most students rely on school drinking water.
For v.:
a. Flaws:
• The pictures of the different items are different sizes, skewing the results (visually, it looks like more burgers are sold than other food).
• There is no key: does 1 symbol = 1 person or 1 purchase? 10 purchases?
• The headline seems to be based on the length of the item lines, not the number of them (bananas and apples make up more than half the number of items sold).
b. If the author wanted to highlight the non-healthy food choices, using different
scales is one way to achieve this.
c. Possible changes:
• Use the same symbol for all pictures, or at least make the existing symbols
the same size. The pictogram could also be arranged as a bar graph (still using symbols).
• A possible headline (based on the limited information provided) is:
More healthy snacks being bought at interval (because you don’t know if a
particular student bought more than one kind of snack).
For vi.:
a. Flaws:
• The vertical axis doesn’t show the gap between 0 and 1 042 as a break
or in any other way.
• The interval on the vertical axis is 1, which is pretty meaningless when you are talking about over 1 042 sales.
• The use of a line graph in combination with the flawed vertical axis makes the increases or decreases look to be much more than they are.
• The intervals on the horizontal axis are uneven (some months are very close together).
• The title is uninformative (where is the data from: New Zealand? Auckland? 1 shop?)
• The headline describes the information shown, not the actual trend.
b. The author may have been trying to show the values (a graph with a scale starting at zero would be practically flat between 1 040 and 1 050); on the other hand, the
author may have wanted to exaggerate the variation in the sales.
c. Possible changes:
• Use a wavy line on the vertical axis to indicate the gap between 0 and 1 042.
• Space the months evenly on the horizontal axis.
• Use a bar graph so that monthly totals can be clearly read.
• Given the closeness of the figures, the headline bears little resemblance to the graph because there is only a difference of 5 consoles between the highest and lowest sales figures. A more accurate headline would be: Sales of games consoles flat.
2. Answers and graphs will vary. An example for each of the first two students’ book scenarios is given below.
The first graph is about a particular crime, in this example, burglaries. The graph has been redrawn with the vertical axis starting at zero. This shows a slight increase in numbers over the 2 years, but it is not exaggerated.

graph.

The graph below on the amount of time children spend watching TV shows an increase over time, but more clearly than the potato graph.

graph.

Activity Two

Answers will vary. 


Key Competencies

Data Distortion can be used to develop these key competencies:
• thinking: interpreting, thinking critically, analysing, making decisions, engaging in making sense
• using language, symbols, and texts: exploring different representations, interpreting visual diagrams
• relating to others: working in groups, communicating thinking
• participating and contributing: sharing equipment and/or resources.

 

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