# Errors

## Summary

• Random errors: an error that affects only some observed values and can be reduced by taking average of large number of readings.
• Systematic Error: an error which is built in the measurement device, it cannot be reduced, however it can be removed if we know the exact error and then we subtract it from the measured value in the end.
• Absolute Error: it is how far off a number is from a true value, and it is denoted by a where a is the absolute error in a measurement.
• Relative Error: it is calculated to tell how big the error is as compared to the measured value.    $Relative\quad Error\quad =\quad \frac { Absolute\quad Error }{ Measured\quad Value/Known\quad Value }$
• Percentage Error:  Percentage Error  =  Relative Error  x  100

First, let’s recall from GCSE’s, that accuracy is how close a measurement is to the correct value for that measurement. It is often reported quantitatively using relative error, which we will discuss below.

Whereas precision of a measurement is referred to how close the values are between repeated measurements. It is often reported quantitatively by fractional or percentage error. In our calculations, our results may slightly deviate from the actual value, this is known as an uncertainty or error in the value.

To improve accuracy of our results we must know how to work out these uncertainties.

There types of errors that we will go through in this article are:

#### 1. Random errors

These are basically fluctuations in the measured values due to the precision limitations of the measurement device. Only some readings are affected by random errors hence random errors can be reduced by recording values over a large number of measurements and then finding the average of these values.

#### 2. Systematic Errors

The are errors that are built in the device/ apparatus/instrument you are measuring from, and all observations are affected by this type of errors. This could result in either all recorded values being too high or all of them being too low. Therefore, systematic errors cannot be reduced by taking large number of observations. To remove this type of error, we find the actual error of the instrument and subtract it from all the measured values. Absolute errors are a type of systematic errors. This is discussed below.

#### 3. Absolute Errors

Absolute error is basically how far off a number is from a true value or it can also be called an indication of the uncertainty in a number.

E.g if we measure a length of a piece of cloth to be 0.562 m ± 0.002 m then 0.002m is said to be the absolute error associated with the length.

Similarly when we round of a number to certain significant figure, e.g we are provided with a length of an object 40cm to the nearest 10. Now we are uncertain about the actual length, since it could be anything between 35cm and 45cm. Therefore, the length could be written as (40 ± 5)cm, this shows it has an absolute error of ±5cm.

#### 4. Relative Error

To calculate a relative error we need to know the absolute error first as a relative error basically tells us how large the error is as compared to the measured value. It can be expressed as a fraction or further it can also be multiplied by 100 and expressed as a percent. Formula to find relative error is defined as:

$Relative\quad Error\quad =\quad \frac { Absolute\quad Error }{ Measured\quad Value/Known\quad Value }$

E.g For the above measurement of a piece of cloth, we know the length measured was 0.562m and the absolute error in the measurement was recorded as ±0.002m. Therefore the relative error will be calculated in the following way:

$Relative\quad Error\quad =\quad \frac { 0.002 }{ 0.562 } \quad =\quad 0.0035$

#### 5. Percentage Error

As discussed above, percentage error can simply be calculated once you know the relative error by using the formula:

$Percentage\quad Error\quad =\quad Relative\quad Error\quad \times \quad 100$

E.g we find the percentage error from the above calculated relative error:

$Percentage\quad Error\quad =\quad Relative\quad Error\quad \times \quad 100$

$Percentage\quad Error\quad =\quad 0.0035\quad \times \quad 100\quad =\quad 0.35%$