Modelling · Topic 5 of 9
Errors And Tolerance
Theory
In mathematical modelling, measurements are rarely perfect. We must account for the fact that data often has a built-in level of uncertainty, which we call an error or tolerance.
1. Absolute and Relative Errors
- Intended Value: The exact target value you are aiming for.
- Absolute Error: The physical amount by which a measurement is allowed to vary above or below the intended value. For example, if a bottle contains 500ml 10ml, the absolute error is 10ml.
- Limits / Tolerance: Using the absolute error gives us the acceptable range. The Upper Limit is the Intended Value + Absolute Error, and the Lower Limit is the Intended Value - Absolute Error.
- Relative Error: Because a 10ml error is huge for a small medicine bottle but tiny for a swimming pool, we use relative error to make fair comparisons. It is usually expressed as a percentage: Relative Error = (Absolute Error Intended Value) 100.
2. Errors in Compound Measures (Min/Max Values)
When calculating a compound measure (like Speed = Distance Time), the errors of the input variables dictate the limits of the output variable.
- To find the Maximum possible value, you must use the largest possible numerator and the smallest possible denominator. (e.g., Max Speed = Max Distance Min Time).
- To find the Minimum possible value, you use the smallest possible numerator and the largest possible denominator.
3. Estimating Errors by Adding
When a mathematical model involves multiplying or dividing two or more independent variables, you can estimate the total relative error of the final answer by simply adding the relative errors of the individual variables together.
4. Accuracy vs. Precision
You must be able to clearly distinguish between these two terms:
- Accuracy: How close a measured or estimated value is to the true, real-life value.
- Precision: How exact or detailed a value is (usually indicated by the number of decimal places or significant figures). A highly precise number can still be completely inaccurate.
Worked examples
Example 1
Example 1: Absolute and Relative Errors
A local bakery sells artisan loaves of bread. The intended weight of a loaf is 800g, but the baker allows a tolerance of to account for moisture loss during baking.
(a) State the absolute error of the bread's weight.
(b) Calculate the relative error as a percentage.
(c) State the lower and upper limits of the weight.
Solutions:
- (a) The absolute error is 24g.
- (b) Relative Error = .
- (c) Lower Limit = . Upper Limit = .
Example 2
Example 2: Compound Measures (Max / Min)
A rescue drone is programmed to fly a distance of 12km 0.5km. The flight time is measured as 40 minutes 2 minutes. Calculate the maximum possible average speed of the drone in km/minute.
Solution:
- Find the Maximum Distance: .
- Find the Minimum Time: .
- Calculate Max Speed: Max Distance Min Time = .
Example 3
Example 3: Estimating Errors in a Formula
A landscaper is ordering topsoil for a rectangular garden. The length of the garden is measured with a relative error of 4% and the width is measured with a relative error of 3%. The area is calculated by multiplying the length and the width. Estimate the relative error of the calculated area.
Solution: Because the two variables are being multiplied together to find the area, we estimate the total relative error by adding their individual relative errors: .
Example 4
Example 4: Accuracy vs. Precision
The exact population of a Scottish town is 18,452.
- A local newspaper reports the population as "approximately 18,000".
- A rival newspaper reports the population as "exactly 21,452.5".
Determine which report is more accurate, and which is more precise.
Solution:
- The first report (18,000) is more accurate because the number is much closer to the true, actual population of 18,452.
- The second report (21,452.5) is more precise because it is stated to a much more exact level of detail (one decimal place), even though the number itself is completely inaccurate.