Accuracy and Error

Accuracy and Error Math Concept Explanation:

In mathematics, accuracy refers to how close a measured or calculated value is to the actual value. Error is the difference between the true value and the approximation. There are two main types of errors:

Absolute error: The absolute difference between the measured value and the actual value.

$\text{Absolute error} = | \text{Measured value} - \text{Actual value} |$

Relative error: The absolute error divided by the actual value, often expressed as a percentage.

$\text{Relative error} = \frac{|\text{Measured value} - \text{Actual value}|}{\text{Actual value}} \times 100\%$

Diagram:

A diagram illustrating the difference between actual and measured values on a number line:

Hard Questions:

Q1: A measurement of a length is recorded as 9.5 cm, but the actual length is 10 cm. Find the absolute and relative error.

Step-by-Step Solution:

Absolute error:

$|\text{Measured value} - \text{Actual value}| = |9.5 - 10| = 0.5 \, \text{cm}$

Relative error:

$\frac{0.5}{10} \times 100 = 5\%$

Answer:

The absolute error is 0.5 cm, and the relative error is 5%.

Q2: A scale gives a reading of 55.2 kg for a person whose actual weight is 56 kg. What is the absolute and relative error?

Step-by-Step Solution:

Absolute error:

$|55.2 - 56| = 0.8 \, \text{kg}$

Relative error:

$\frac{0.8}{56} \times 100 \approx 1.43\%$

Answer:

The absolute error is 0.8 kg, and the relative error is approximately 1.43%.

Q3: A runner’s time is measured as 12.8 seconds, but the actual time is 13 seconds. Calculate the absolute and relative errors.

Step-by-Step Solution:

Absolute error:

$|12.8 - 13| = 0.2 \, \text{seconds}$

Relative error:

$\frac{0.2}{13} \times 100 \approx 1.54\%$

Answer:

The absolute error is 0.2 seconds, and the relative error is approximately 1.54%.

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