Percentage Error

 Percentage error is a measure of how inaccurate a measurement is, compared to the true or accepted value. It is expressed as a percentage and is calculated using the formula:

$\text{Percentage Error} = \left( \frac{|\text{Measured Value} - \text{Actual Value}|}{\text{Actual Value}} \right) \times 100\%$

This formula helps quantify the accuracy of a measurement or estimation. A smaller percentage error indicates a more accurate measurement, while a larger percentage error signifies a less accurate one. citeturn0search0

Worked Examples:

1. Example 1:

   • Problem: A student measures the length of a rod as 12.5 cm, while the actual length is 12.0 cm. Calculate the percentage error.
   • Solution:
     • Measured Value = 12.5 cm
     • Actual Value = 12.0 cm
     • $\text{Percentage Error} = \left( \frac{|12.5 - 12.0|}{12.0} \right) \times 100\% = \left( \frac{0.5}{12.0} \right) \times 100\% \approx 4.17\%$

2. Example 2:

   • Problem: An experiment estimates the boiling point of a liquid to be 102°C, but the known boiling point is 100°C. Determine the percentage error.
   • Solution:
     • Measured Value = 102°C
     • Actual Value = 100°C
     • $\text{Percentage Error} = \left( \frac{|102 - 100|}{100} \right) \times 100\% = \left( \frac{2}{100} \right) \times 100\% = 2\%$

3. Example 3:

   • Problem: A scale shows a person's weight as 70.5 kg, but the actual weight is 68 kg. Find the percentage error.
   • Solution:
     • Measured Value = 70.5 kg
     • Actual Value = 68 kg
     • $\text{Percentage Error} = \left( \frac{|70.5 - 68|}{68} \right) \times 100\% = \left( \frac{2.5}{68} \right) \times 100\% \approx 3.68\%$

4. Example 4:

   • Problem: A thermometer reads the temperature as 25.2°C, while the actual temperature is 24.8°C. Calculate the percentage error.
   • Solution:
     • Measured Value = 25.2°C
     • Actual Value = 24.8°C
     • $\text{Percentage Error} = \left( \frac{|25.2 - 24.8|}{24.8} \right) \times 100\% = \left( \frac{0.4}{24.8} \right) \times 100\% \approx 1.61\%$

5. Example 5:

   • Problem: A laboratory measures the concentration of a solution as 0.85 M, but the true concentration is 0.80 M. Determine the percentage error.
   • Solution:
     • Measured Value = 0.85 M
     • Actual Value = 0.80 M
     • $\text{Percentage Error} = \left( \frac{|0.85 - 0.80|}{0.80} \right) \times 100\% = \left( \frac{0.05}{0.80} \right) \times 100\% = 6.25\%$

Practice Questions:

1. A student measures the mass of a sample as 5.5 g, but the actual mass is 5.0 g. Calculate the percentage error.

2. A car's speedometer shows a speed of 60 km/h, while the actual speed is 58 km/h. What is the percentage error?

3. A rectangular plate is measured to have a length of 15.2 cm, but the true length is 15.0 cm. Find the percentage error.

4. A voltmeter reads 9.8 V, whereas the actual voltage is 10.0 V. Determine the percentage error.

5. A cylinder's diameter is measured as 7.1 cm, but the actual diameter is 7.0 cm. Calculate the percentage error.

6. A thermometer shows a temperature of 37.5°C, while the actual temperature is 37.0°C. What is the percentage error?

7. A chemical reaction yields 20.5 g of product, but the theoretical yield is 20.0 g. Find the percentage error.

8. A scale measures a weight as 250 g, but the standard weight is 245 g. Determine the percentage error.

9. A clock is found to be 2 minutes fast over a 24-hour period. Calculate the percentage error.

10. A survey estimates that 150 people will attend an event, but the actual attendance is 140. What is the percentage error?

Understanding and calculating percentage error is crucial in various fields, including science, engineering, and everyday measurements, as it provides insight into the accuracy and reliability of data.

For a visual explanation and more examples on percentage error, you might find the following video helpful: