Question

What is the percentage error of a length measurement of 0.229 \(\mathrm{cm}\) if the correct value is 0.225 \(\mathrm{cm} ?\)

Short answer

The percentage error is approximately 1.78%.

Step-by-step solution

Identify the Measured and True Values

Determine the measured value and the true (correct) value. Here, the measured value is 0.229 cm and the true value is 0.225 cm.

Calculate the Absolute Error

Find the absolute error by subtracting the true value from the measured value. \[ \text{Absolute Error} = |0.229 - 0.225| = 0.004 \text{ cm} \]

Calculate the Percentage Error

Use the formula for percentage error: \[ \text{Percentage Error} = \frac{\text{Absolute Error}}{\text{True Value}} \times 100 \] Substitute the values: \[ \text{Percentage Error} = \frac{0.004}{0.225} \times 100 \] Then, simplify the calculation to get the percentage error. \[ \text{Percentage Error} \approx 1.78 \text{ \text{(rounded to two decimal places)}} \]

Key concepts

Absolute Error

Absolute error is key in understanding accuracy in measurements. It tells us how far off a measured value is from the true value.

To find the absolute error, simply subtract the true value from the measured value and take the absolute value, ensuring the result is always a positive number.

For instance, in our exercise:

• The measured value is 0.229 cm
• The true value is 0.225 cm

Thus, the absolute error is calculated as: \[ \text{Absolute Error} = |0.229 - 0.225| = 0.004 \text{ cm} \]

Measured Value

The measured value represents the data you obtain from a measurement instrument. It’s the value found during an experiment or observation.

In our exercise, the measured value is 0.229 cm. This is the length obtained using a ruler or any other measuring tool.

Measured values will often have small differences from the true values due to limitations in measurement precision, observer error, or imperfections in the measuring instrument.

True Value

The true value is the actual, accurate value of the quantity being measured. It is considered the standard or correct value in an experiment.

In this case, the true value is given as 0.225 cm.
This value serves as the benchmark against which the measured values are compared.

Understanding the true value is crucial for calculating errors and analyzing the accuracy of your measurements.

Percentage Error Formula

The percentage error formula helps in determining the extent and significance of the absolute error in comparison to the true value. This is essential for evaluating the precision of measurements.

The percentage error is calculated using the formula: \[ \text{Percentage Error} = \frac{\text{Absolute Error}}{\text{True Value}} \times 100 \] By substituting the values from our example:
\[ \text{Percentage Error} = \frac{0.004}{0.225} \times 100 \] When you calculate this, it simplifies to approximately 1.78%.

Therefore, on average, our measured value deviates from the true value by 1.78%, indicating the accuracy level of our measurement process.