For the students of science and economics it is absolutely necessary to learn how to calculate errors. So, here is an easy guide to learning different types of errors with a special focus on the various ways to calculate percent error.

Errors are introduced in calculations during the measurement of a quantity. These are mainly of three types:

- Systematic errors
- Random errors
- Human errors

Systematic errors arise because of a fault or imperfection of the measuring instrument. For example, if the instrument is handled in an incorrect way it might give incorrect values or there is a fault in the construction of the instrument.

Random errors are unpredictable changes. These can arise due to change in the measuring device or the environment. For example, electronic noise, irregular heat loss due to wind, etc.

Human errors are the most common source of error. These are also called blunders. For example, there might be an error because of recording an incorrect value, misreading a scale, forgetting to note down a digit, etc.

Sometimes it happens that when the same object is measured using two different devices or when it is measured twice with the same device, the measurement values are not exactly similar. This is called variation in measurement.

You can also note that the variation in measurement is also called an error. However, it is better termed as the uncertainty in measurement

The uncertainty or variation in measurement is actually an indication that the actual value is different from the measured or calculated one. This points towards the fact that regardless of the care with which the readings were taken, there might be one or more errors in the measurement. These errors are unavoidable.

To nullify the effects of these errors, we need to know about the percent error formula.

The variation in measurement is calculated in this way:

- Variation in measurement = True Value – Measured Value

Generally, the uncertainty in measurement occurs because of the aforementioned errors or due to the precision of the measuring instrument.

The precision of the instrument is the smallest possible value that can be measured by an instrument.

The accuracy of a measuring instrument is a way to indicate how close will the measured value be to the actual or true value.

The absolute error quantifies the physical error in measurement. It is calculated by the formula:

- Absolute error = |Measured Value - Actual Value|

The relative error describes the absolute error in relation with the actual value. It is calculated by the formula:

- Relative Error = |Measured Value-Actual Value|True Value

The relative error gives a measure of accuracy.

Percent error is also a comparison formula. The percent error formula compares the original, true, or actual value with the measured one. Then, the difference is converted into a percentage of the true value.

The percent error is a type of approximation error. The approximation error reveals the difference between the actual value and the measured one. The discrepancy occurs because of:

- Lack of precision of a measuring instrument
- Approximation of data (using 3.14 instead of the actual value of

The percent error formula is obtained by multiplying the relative error by 100. Mathematically,

Percent error = |Measured Value-Actual Value|True Valuex 100%

To obtain the percent error formula:

- Subtract the two values (real and measured). In this step, the sign doesn’t matter. So, ignore the minus sign that might come after you take the difference after whenever you’re using the percent error formula.
- Next, divide the difference by the actual or ideal value to obtain a decimal number.
- Now, multiply the answer by 100 to obtain the percent error.

It’s easy to calculate error when you use the percent error formula.

For example, you have to measure the length of a cylinder. You measure the length through the Vernier calipers. The length comes out to be 2.68 mm. However, the actual value was 2.70 mm. Here’s how you can use the percent error formula.

Now,

Percent Error = 2.70-2.682.70*100%

Percent Error = 0.022.70*100%

Percent Error = 0.74%

Therefore, the result obtained using the percent error formula is 0.74%.

Other formulas associated with the percent error formula are given below:

actual value=measured1+percent error100%

actual value=measured1-percent error100%

NOTE: As we used the absolute value of the difference in the percent error formula, there will be two different values of the actual value.

The measured value is calculated by the following formulas. Notice that it too has two different possible values because of the absolute in the percent error formula.

measured value=actual value+percent error x actual100%

measured value=actual value-percent error x actual100%

Another percent error formula can thus be obtained. It is given as follows:

percent error=(measure1-measured 2)measured1+measured22x 100%

Example 1: A student, while performing an experiment, measured the boiling temperature of a liquid as 54.9 degree C. The actual temperature was 54.0 degree C. Calculate the absolute error, relative error, and the percentage error by using the percent error formula.

Solution:

Absolute error=54.9-54.0=0.9

Relative error=54.9-54.054.0=0.02

Percent error=54.9-54.054.0x 100%=2%

Example 2: A student, while performing an experiment, calculates the mass of bromine as 36.3. The actual value is known to be 36.2. Calculate the absolute error, relative error, and the percentage error by using the percent error formula.

Solution:

Absolute error=36.3-36.2=0.1

Relative error=0.136.2=0.0276

Percent error=0.0276x 100%=2.76%

Example 3: If the actual length of wire is known to be 3.53 m and it is also known that the percentage error in the measurement is 5%. Calculate the measured value of the length of the wire using percent error formula.

Solution:

measured value=actual value+percent error x actual100%

measured value=3.53+5% x 3.53100%=3.71 m

OR

measured value=actual value-percent error x actual100%

measured value=3.53-5% x 3.53100%=3.35 m

Example 4: During an experiment, the inner diameter of a cylinder was found to be 8.03 mm. If the percent error was 4%, calculate the actual value. Hint: Use percent error formula.

Solution:

actual value=measured1+percent error100%

actual value=8.031+40%=7.72

OR

actual value=measured1-percent error100%

actual value=8.031-40%=7.71

Example 5: A survey conducted by the organizers estimated a total number of 70 people would attend a concert. However, 80 people bought the tickets and attended the event. Calculate the percent error in the estimation of audience by using the percent error formula.

Solution:

Absolute error=80-70=10

Relative error=1080=0.125

Percent error=0.125x 100%=12.5%

If the sign obtained after the subtraction is taken into account, we can estimate whether the measurement was higher or lower with respect to the actual value.

Example 6: A student observes the time it takes a ball to fall down a distance of 2 m. The stopwatch records 0.62 seconds. However, using the equations of motion, the theoretical value of time comes out to be 0.64 seconds. Calculate the percent error in the measurement by using the percent error formula.

Solution:

Percent error=Measured Value-Actual ValueTrue Value x 100%

Percent error=0.62-0.640.64 x 100%=-3.12%

So, the reading observed was 3.12% lower than the actual value.

Example 7: The forecast suggested that a certain area would receive 20 mm of rain. The rain reached 25 mm. Calculate the percent error in the forecast by using the percent error formula.

Solution:

Percent error=Measured Value-Actual ValueTrue Value x 100%

Percent error=25-2020 x 100%=2.5%

The actual amount of rain was 2.5% higher than the forecast.

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