5.02 Pictographs, bar and misleading graphs

Bar graphs

Practice questions

Question 1

Question 2

Pictographs

A pictograph is very similar to a bar graph in that is displays the relative size of data in different categories. It differs by using using pictures or symbols to represent data rather than a bar. Pictographs have a key, which explains what each symbol represents.

For example the following pictograph displays the number of glasses of water $6$6 students drank in a week. In this case, each glass represents $6$6 glasses of water, thus, we can see that Conrad consumed $5.5\times6=33$5.5×6=33 glasses of water.

Practice questions

Question 3

Question 4

Question 5

Statistics are usually included in media to support facts, reinforce arguments or provide additional information to the viewer. However, we must not forget that they are a powerful tool of persuasion, and must be interpreted with caution, as statistics can be deliberately manipulated or skewed by the author to shape the opinions of viewers. Data may be manipulated in the following common ways:

• Cherry-picking data - discarding results that are unfavourable.
• Wrong measure used - we have seen that different the measures of centre and spread may give different impressions of the data if outliers are present.
• Biased surveys - the phrasing of survey questions and the interviewer can often influence the responses given.
• Sampling technique - a sample should be representative of the population of interest. A poor sampling technique can introduce bias to the results. Practices likely to produce bias include very small samples or samples not selected randomly, such as from volunteer phone polls.
• Over-generalisation - when asserting a conclusion from a particular group to a wider population when the initial group is not representative of the larger population.
• Overstating significance of findings - sometimes it is not clearly stated how large the sample was or how large the margin of error is. If the error margin is large then then result may be significantly off the stated figure.
• Correlation and causation - if a relationship is observed between two variables A and B, it is tempting to jump to the conclusion that A causes B. However, it may be that B causes A, the two may in part cause each other, there may be a third factor that causes both or the relationship may be purely by chance.
• Comparing apples and oranges - when comparing two data sets that cannot be meaningfully compared against each other. (Example)
• Misinterpreting statistics or probability - probabilities, in particular, are commonly misunderstood and misrepresented in media court cases. (see prosecutor's fallacy)

Misrepresentation of statistics often happens unintentionally by a source who is reporting on a subject for which they are not an expert or are not familiar with the statistics quoted. However, misrepresentation may also be intentional to lead viewers to a particular conclusion. Critical evaluation of figures given in articles is very important in our information rich community. This video has some tips for looking deeper than the attention grabbing headlines.

Practice questions

Question 6

Question 7

 

 

Misleading graphs

Another way to commonly misrepresent data is using an inappropriate visual representation. Graphs are an important tool to convey statistics, however, a poorly constructed graph may lead a reader to an incorrect conclusion. A misleading graph may be created to intentionally bias the reader to a particular interpretation or accidentally by someone unfamiliar with creating an effective way to display the given data. Some common features of graphs that may lead to incorrect interpretations are:

1. Showing an inappropriate or irregular scale
2. Scale or labels not clearly given
3. Leaving data out
4. Using pictures or three-dimensional graphics that distort differences
5. Using the wrong graph for a given data type

Let's look at an example of each characteristic above.

 

Showing an inappropriate or irregular scale

The scale of given for the graph should be even and in proportion to the data. The scale should not be compressed or expanded to exaggerate or diminish change. 

Misleading: the range of values shown on the vertical axis is disproportionate to the data. This makes it appears as though the temperature is constant - no variation or overall increasing trend.	Accurate: the range of values shown on the vertical axis is in proportion to the data. We can now see the data has high variability and an overall increasing trend.

The above three graphs show the same data but by compressing or expanding the horizontal axis it can appear as if the trend is gradual or quite abrupt.

This is not to say the vertical axis always has to start at zero, as we can see in the above example. However, caution should be taken to not exaggerate differences. When not starting the vertical axis at zero a clear indication of a broken scale should be given and it is not recommended for graphs such as bar charts where the viewer takes in the comparative areas of the different categories.
 

Scale or labels not clearly given

If a scale is omitted or missing units then the reader can not interpret if the trend seen is significant or not. 

Without a scale and units on the vertical axis we cannot tell if the sales are in single units sold, $100$100 units sold, dollars, or so forth, this means we have no means to tell if the difference and upward trend seen is significant. There may also be a truncated axis.

 

Leaving data out

A common way to mislead the audience is to cherry-pick the data shown, so it only includes statistics that supports a particular conclusion. Whenever a broad range of information exists, appearances can be manipulated by highlighting some facts and ignoring others. 

This graph only shows a few months data and displays an overall negative trend.	Expanding the data to three years, we see a bit more variability and an overall upward trend.

Selecting temperature data from the years 1997 - 2012 we see relatively stable temperature with no obvious upward trend. Compare this to the graph given earlier.

 

Using pictures or three-dimensional graphics that distort differences

Using graphics in perspective can make it appear as though the section of the graph at the front is larger in comparison to sections further back even when they are the same size. Improperly scaling graphics can also distort the difference in categories significantly. 

In this bar chart the front bar looks far larger than the bar at the back due to perspective. However, both columns represent the same value.	Three dimensional bars can also lead to confusion when reading the scale.

 

Here again we see a three dimensional graphic in perspective with one piece brought forward from the chart. The perspective makes the front piece of 7% look far larger than the equal valued slice at the rear.	Here is the same data without the perspective and we can now clearly compare the size of the slices. The two equal valued slices now appear the same size.

The vertical axis indicates that twice as many houses were sold. However, by scaling both dimensions the house on the right appears far larger than twice the house on the left.	This image shows comparatively the house on the right is in fact four times the area of the house on the left.

This graph shows the correct use of a pictograph with the icons of equal size. Easy to compare by eye.	This picture shows the correct scaling if the area of the house represents a given number of sales, then the house on the right has twice the area.

Infographics often use areas of pictures or bubbles to represent values. This is not misleading if correct scaling is used. Above are scaled squares, notice the side lengths of the second and third square are not twice and three times the side length of the original square. Mistakenly scaling the side lengths will give an area that is the scale factor squared larger. So if we doubled the side lengths we would get an area four times larger and if we tripled the side length we would obtain an area nine times larger.

1 icon = 1 animal	1 icon = 1 animal
Using different sized icons makes categories appear larger in comparison. It looks as if there are equal numbers of tigers and pandas for instance.	This graph correctly uses the same sized icon for each animal and they are aligned so we can compare numbers at a glance.

Using the wrong graph for a given data type

The type of graph used to visualise the data depends on the type of data you have and the characteristic of the data you wish to highlight. Choosing an inappropriate graph type may lead to the reader to misinterpret the data. 

Circle graphs are used to show the composition of a whole and not comparison across groups. Here we can see the percentages add to more than $100%$100%, it is likely that respondents to the survey could select more than one option.	Here the data is represented as a bar chart and we can make clear comparison across all the categories.

 

Practice questions

Question 8

Question 9