Question

distinguish between the absolute error and the relative error in a measurement. give an example in which the absolute error is large but the relative error is small and another example in which the absolute error is small but the relative error is large.
distinguish between the absolute error and the relative error in a measurement.
a. the relative error describes how far a measured (or claimed) value lies from the true value. the absolute error compares the size of the relative error to the true value and is never expressed as a percentage.
b. the absolute error describes how far a measured (or claimed) value lies from the relative error. the relative error compares the size of the absolute error to the relative error and is often expressed as a percentage.
c. the absolute error describes how far a measured (or claimed) value lies from the true value. the relative error compares the size of the absolute error to the true value and is often expressed as a percentage.
d. the relative error describes how far a measured (or claimed) value lies from the absolute error. the absolute error compares the size of the relative error to the true value and is often expressed as a percentage.
give an example in which the absolute error is large but the relative error is small.
a. a chemist has 2.9 mg of substance, but a scale measures 2.1 mg
b. a census says that the population of a town is 72,453, but the true population is 96,000.
c. a runners true weight is 125 pounds, but a scale says he weighs 130 pounds.
d. a company projects sales of $7.30 billion and true sales turn out to be $7.32 billion.

Explanation:

1. Definition - Absolute and Relative Error:

• Absolute error is the difference between the measured (or claimed) value and the true value. For example, if the true value is $x$ and the measured value is $y$, the absolute error $E_a=\vert x - y\vert$.
• Relative error is the ratio of the absolute error to the true - value, often expressed as a percentage. So, the relative error $E_r=\frac{\vert x - y\vert}{x}\times100\%$ (when $x

eq0$).

1. Example - Large Absolute Error, Small Relative Error:

• For option B:
• The true population $x = 96000$ and the measured population $y = 72453$.
• The absolute error $E_a=\vert96000 - 72453\vert=23547$.
• The relative error $E_r=\frac{23547}{96000}\times100\%\approx24.53\%$.
• For option A:
• True value $x = 2.9$ mg and measured value $y = 2.1$ mg. Absolute error $E_a=\vert2.9 - 2.1\vert = 0.8$ mg, relative error $E_r=\frac{0.8}{2.9}\times100\%\approx27.59\%$.
• For option C:
• True weight $x = 125$ pounds and measured weight $y = 130$ pounds. Absolute error $E_a=\vert130 - 125\vert = 5$ pounds, relative error $E_r=\frac{5}{125}\times100\% = 4\%$.
• For option D:
• True sales $x = 7.32$ billion and projected sales $y = 7.30$ billion. Absolute error $E_a=\vert7.32 - 7.30\vert=0.02$ billion, relative error $E_r=\frac{0.02}{7.32}\times100\%\approx0.27\%$. Here, the absolute error of $0.02$ billion is small compared to the true - value of $7.32$ billion, and the relative error is also small. In option B, the absolute error of 23547 in population count is large, and the relative error is relatively small compared to some other cases where the absolute error is much smaller in magnitude.

Answer:

1. C. The absolute error describes how far a measured (or claimed) value lies from the true value. The relative error compares the size of the absolute error to the true value and is often expressed as a percentage.
2. B. A census says that the population of a town is 72,453, but the true population is 96,000.