Pareto charts
Pareto charts are useful in business for analysing the frequency of particular problems and their potential causes. The chart combines a column graph and a cumulative frequency polygon. The column graph has its vertical axis on the left and the cumulative frequency polygon has its vertical axis on the right.
A column is drawn for each category so that the height of a column represents the number of observations that fell into that category. In a Pareto chart, the columns are arranged so that the tallest column is on the left of the chart and the heights go in decreasing order.
This way of displaying the frequencies in each category makes it easy to see which categories are the most significant contributors to the overall result.
Overlaid with the columns in a Pareto chart is a line graph showing the cumulative percentages of observations in the categories, looking from left to right. An example should make this clear.
Worked example
Example 1
The excuses of $50$50 students for non-submission of homework assignments were grouped into $5$5 categories.
- I didn't have time (18)
- I did it but the dog ate it (7)
- I did it but my mother tidied my room and lost it (6)
- It was too hard (15)
- Other (4)
For convenience, we have numbered the categories 1 to 5 but these numbers are nothing more than labels to identify the categories. In the chart, the labels are in another order, in this case, 1-4-2-3-5, depending on the frequency in each category. The frequencies are the numbers in brackets for each category.
The vertical axis on the left gives the frequencies and it relates to the heights of the columns. The vertical axis on the right shows the percentages and it relates to the levels on the line graph.
Note that the line graph in a Pareto chart will always have an upward trend from left to right because it is a display of the cumulative relative frequencies.
One can see from this Pareto chart that the most important factors leading to non-submission of homework assignments for this group of students were predominantly lack of time and the difficulty of the work. The other factors were much less significant.
Practice questions
Question 1
A marine scientist spent the day fishing to collect data on the range of species in an enclosed dam. She recorded the number of each fish species that were caught, and constructed a Pareto chart. For the following questions, you can assume that the bars are in line with the gridlines, or exactly halfway between gridlines.
The marine scientist wanted to compare the population levels of each species.
What is the difference between the amounts of the most and least caught fish species?
It is expected that the two species with the lowest population will have a cumulative percentage of $50%$50%. How different is this to the cumulative percentage the marine scientist observed?
Perch are often targeted in commercial fishing. The marine scientist is concerned about whether their numbers have decreased relative to other species. Determine the percentage of fish caught that were Perch.
Question 2
At Pareto's Burritos, the owners regularly ask their customers if and why they are not happy with their burritos.
They created a chart for last month's feedback.
How many customers in total expressed dissatisfaction last month? You can assume that the bars are in line with the labels on the left-hand $y$y axis, or exactly halfway between two labels.
Using the bar section of the Pareto chart, find the percentage of customer complaints made up by the three most frequent complaints.
Round your answer to the nearest percentage.
Pareto wants to significantly improve customer satisfaction in the next month. What single change would improve customer satisfaction the most?
Increasing the speed of service.
ALowering the price of burritos.
BAdding more guacamole.
CUsing fresher ingredients.
DWhat percentage of customer complaints would be resolved by reviewing how their chefs make their burritos? You can assume that the bars are in line with the labels on the left-hand $y$y axis, or exactly halfway between labels.
Round your answer to the nearest percentage.
Question 3
Bill caught the train and noted what activity each person in his carriage (excluding himself) was doing between the next two stops. The Pareto chart shows the results.
How many other people were in the carriage? You can assume that each bar is either in line with a tick on the left-hand $y$y-axis, or exactly halfway between ticks.
Using the bar section of the Pareto chart, find the percentage of people on the carriage (excluding Bill) that make up the three most common activities. You can assume that each bar is either in line with a tick on the left-hand $y$y-axis, or exactly halfway between ticks.
Round your answer to the nearest percentage.