Using statistics to solve engineering problems
These resources support the use of statistics to solve engineering problems with particular reference to measures of location, measures of spread volume of 3D shapes and common measures. The resources support students to achieve the assessment outcomes of:
- calculate the mean, median and modal averages
- determine cumulative frequency, variance and standard deviation
- describe and explain how statistical data in engineering and quality systems
In some cases the mathematical concepts are those found in the GCSE mathematics syllabus, but the application of these concepts in an engineering context requires skills beyond GCSE level. It is vital that students have the ability to apply the mathematics they know in unfamiliar and more challenging contexts. This will thoroughly test their mathematical understanding in preparation for tackling extension tasks at level 3 in other areas of the mathematics curriculum.
Resistor Production
Modern manufacturing plants make use of engineering quality control to ensure that a product’s quality meets a specified standard and that rejection rates are minimised. When aiming to produce resistors to a specified value the process invariably produces a range of values. Often, products are graded depending on how close to the nominal specification; those that are closer can have a higher sale price.
In this engineering resource, students are asked: "How can production outcome data from a manufacturing process be analysed to optimise the process?"
Students consider the outputs of resistors to see the outcomes form a normal distribution. Students have to decide whether a machine producing the resistors needs calibrating. The interactive file can be used to analyse the significance of changes to the mean and standard deviation when dealing with normally distributed data.
In completing this activity students are required to:
• display data using a bar chart
• extract numerical information from a dataset
• calculate probability
• estimate probability as a relative frequency
Monitoring Vibration Levels in Steam Turbines
In this resource students explore the application of mathematics within the mechanical and electrical engineering power industry. Students use statistical models to set alarms which detect changes in vibration above normal values. The alarms highlight that something is changing and allow action to be taken before a major failure takes place, or the machine has to be taken out of service for investigation, thus avoiding major loss of revenue.
The extension activity requires students to calculate the probability of a false alarm and calculate the expected number of false alarms. In order to complete these tasks students are required to:
· understand the idea of mathematical modelling
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understand the mathematical structure of a range of functions and be familiar with their graphs
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understand how to describe situations using statistics and use probability as a measure of likelihood
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construct rigorous mathematical arguments and proofs in engineering context
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comprehend translations of common realistic engineering contexts into mathematics
The Study of Engineers' Data 1
In this resource students explore the application of mathematics when investigating the volume of registration of engineers. Statistical calculations and graphical representations are used to explore the variations, over time, of membership by area of expertise.
To complete the activity students will be required to:
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find mean and median of a given data set
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plot a graph using appropriate software
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interpret the data in real world language
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understand how to describe situations using statistics and use probability as a measure of likelihood
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comprehend translations of common realistic engineering contexts into mathematics
Winning Medals: Does Engineering Design Make a Difference?
In this resource students create a presentation that provides a justified answer to the question "Does engineering design make a difference to a wheelchair athlete's performance?" It is intended that students arrive at their answer having investigated the practical challenges of making a model wheelchair and, through developing their knowledge of the science and mathematics used by engineers, to design wheelchairs for sport by investigating torque, track and turning moments, students can develop an understanding of some of the principles engineers apply to make wheelchairs easier to accelerate and control.
The students are required to make a presentation which should include: text, photographs, diagrams, charts, graphs, and data.
Power Demand
One important area of civil engineering is electrical power production. In order to plan for future building, which may take many years to prepare, design and construct, demand forecasts are often used to indicate the quantity and size of new power stations required.
Students are asked: "How can you predict future power requirements?" To solve the problem, students are required to complete a table by substituting values into a formula and plot a graph. The interactive file can be used to demonstrate some of the important aspects of growth and decline. The activity offers good opportunities to consolidate work on geometric progression.
The mathematics covered in this activity is:
• be able to write the rule for a sequence in symbolic form
• change the subject of a formula
• be able to plot data
• be able to draw a graph by constructing a table of values
• solve problems using the laws of logarithms
• solve problems involving exponential growth and decay
Gas Compression and Expansion
Many machines compress or expand gas or fluid as part of their working design. Examples include a simple bicycle pump, a refrigerator, and an internal combustion engine. To compress gas energy needs to be expended to reduce its volume. When gas is allowed to expand energy is released.
In this engineering resource students are asked the question: "How can you calculate the energy used, or made available, when the volume of a gas is changed?"
Boyle’s law is used and students need to be able to integrate to complete the activities. Isothermal change and adiabatic change are considered. An interactive file graphs the motion of a piston in a cylinder.
The mathematics students will be required to use in this activity is to:
• be able to interpret data
• distinguish between definite and indefinite integrals and interpret a definite integral as an area
Heat Loss from Buildings
In this activity, students are asked the question "How can the most efficient design be determined, taking both building and running costs into account?"
Students consider thermal conductivity of different materials graphically to help decide which material should be used. There follows an explanation of the concept of kilowatt hours.
A video accompanies the resource explaining thermal conductivity.
The mathematics covered in this activity is:
• solve problems involving area, perimeter and volume
• use scale drawings
• work with formulae for the areas and perimeters of plane shapes
• work with formulae for surface areas and volumes of regular solids
• be able to draw graphs by constructing a table of values
• be able to extract information from a graph