In the context of measurements, when the value obtained from a measurement deviates from the real value of a physical quantity, it is termed as error in measurement. Mathematically it can be written in the below form.

Error = Measured value – Actual value

Actual value is that value which is given to us by theory or formula while the value which is obtained by a physical experiment is called measured value.

## Explanation

Physics is the science of measurement. No measured value is absolutely perfect. Even if the value is measured more carefully, there is a margin of doubt, which is known as an error. Thus every measurement is approximate due to errors in measurement. Suppose we have a measurement of the actual value of 10cm. After measurement, we get the measured value of 10.5cm. This difference is known as an error.

## Types Of Errors

The error in measurement can be broadly classified into three types

- Personal error
- Random error
- Systematic error

The details of these three types of errors are discussed below

### Personal Error

The error which is produced by an experimenter (person) is called a personal error. It is considered a bidirectional error.

#### Causes of Personal Error

Personal errors arise in measurement due to the following factors.

##### 1. Carelessness

Carelessness is one of the major causes of personal error. The experimenter does not focus on the measured value carefully which leads to causing an error known as personal error.

##### 2. Improper Knowledge

The personal error may also arise in the measurement due to the insufficient knowledge of the experimenter about the instrument. This improper knowledge of instruments leads to cause personal error in measurement.

##### 3. Incorrect Reading

Incorrect reading of the scale also leads to cause personal error. This incorrect reading is either obtained due to the carelessness of the experimenter or the experimenter has insufficient knowledge about the instrument use.

#### Reduction of Personal Error

The personal error can be minimized if the experimenter is careful and have sufficient knowledge about an instrument.

#### Example

Let’s suppose an example that in a measurement the actual value is 15m and we get the measured value of 15.4 m either due to carelessness, improper knowledge, or incorrect reading of the scale, the error is known as personal error. To reduce the error perform the experiment carefully having sufficient knowledge.

### Systematic Error

Systematic error is one of the common types of errors. It can be stated as the error which is produced by the system (apparatus and instrument) of the measurement is called systematic error. There is a property of systematic error that this error is always unidirectional either positive or negative. It is a predictable error.

#### Causes of Systematic Error

Systematic errors arise in measurement due to the following factors.

##### 1. Faulty Apparatus

Faulty apparatus is considered one of the major causes of systematic error. In such a case, the experimenter gets a constant error every time.

##### 2. Poor Calibration

Calibration is an important factor of measurement using instruments. Usually, most of the systematic error arises in the measurement due to poor calibration of the instruments. In this case, an experimenter may get different results.

#### Reduction of Systematic Error

Systematic error can be reduced by improving experimental techniques and selecting better instruments. Improving the experimental techniques leads to the good calibration of the apparatus which will lead to the relatively accurate measurement.

#### Example

Let’s consider we have to measure the diameter of the nut using the screw gauge. The screw gauge may show different results than the original reading due to poor calibration or faulty apparatus. To reduce the error use an advanced apparatus having good calibration.

**Also Explore**: **What are Supplementary units? How many radians account for circle’s circumference**?

### Random Error

The error that occurs randomly due to environmental factors is called random error. It is usually one of the common types of errors that arise in the measurement.

#### Causes of the random error:

Environmental factors such as humidity, pressure, and temperature are the major causes of random error. There are some measurements that are done very carefully using an advanced apparatus but still, the reading of measured value is different than actual measurement. It is due to the environmental factors that cause differences (error) in the measurement, such type of error is called random error.

#### Reduction of Random Error

Random error can be reduced by maintaining strict conditions in the laboratory. Strict conditions in the laboratory mean that there will be constant pressure and temperature, and there will be no change in environmental factors, as a result, the measured value will be comparatively more accurate.

#### Example

For example, a student is performing an experiment in the laboratory. He observe that by opening and closing the door, he obtained different results due to changes in the pressure. Go through the below picture for a better understanding.

## Some Common and Important Questions Regarding Errors

Measurement is a fundamental part of life. We measure different things in our daily life. All these measurements contain errors. Many questions come into an individual mind during the study of errors in which some of the common and important questions are given below.

### Q1 What are the precautions of systematic error and random error?

**Precautions of Systematic Error and Random Error**

We must remember the following points regarding systematic error and personal error.

**For systematic errors**, we should take precautions before the measurement. It is because once they are introduced to readings, they can never be removed.

**For random errors**or those errors which are bidirectional or those whose causes or unknown we cannot take some precautions before measurement but they can be removed by repeat and average.

### Q2 What are the different forms of errors

**Different Forms Of Errors**

There are various forms of errors. The most common and well-known three forms of errors are simple error, relative error, and percentage error.

#### 1. Simple Error

** A simple error, also known as an **absolute error is the positive numerical difference between the measured value and the actual value of a given quantity. Let’s consider an example, a measurement having an actual value of 2.2 m and a measured value is 2.5 m. The difference between them is a positive value. Hence by definition, there is a simple error in the measurement.

Simple error = Measured value – Actual value

Simple error = 2.5 – 2.2

Simple error = 0.3 m

#### 2. Relative Error

** The relative error **may be stated as the ratio of error to the actual value. For example, Suppose a measurement has a 0.3 m error while the actual value is 2.5 m. Hence according to the definition, the ratio between them is known as a relative error.

\text{Relative Error } = \frac{\text{Error}}{\text{Measured Value}}

\text{Relative Error } = \frac{\text{0.3}}{\text{2.5}}

\text{Relative Error} = 0.12

#### 3. Percentage Error

** Percentage error **may be stated as relative error multiplied by 100.

%Error = Relative error X 100

**By putting the value of relative error**

\text{Percentage Error } = \frac{\text{Error}}{\text{Actual Value }} X{100}

Let’s consider an example. A student measured a mass with an error of 0.1 kg while the actual value is 2.6 kg. To determine the percentage error of this measurement put error = 0.1 and actual or exact value = 2.6, in the above equation.

\text{Percentage Error } = \frac{\text{0.1}}{\text{2.6 }} X{100}

\text{Percentage Error} = {3.84\%}

### Q3. What are the types of Personal Error?

**Types of Personal Error**

There are many types of personal errors. Some well-known types include parallax error, gross error, and reaction time error. The details of these three types of personal errors are discussed below.

#### 1. Parallax Error

** Parallax errors **may be defined as when measurements are not taken from the correct line of sight (parallel line of sight). For a measurement to be more accurate, it is necessary to take the measurement from the correct sight line. Usually, the line of sight is taken parallel. The misalignment can lead to inaccurate readings and affect the precision of measurements in various instruments, particularly those with a visual component.

To minimize parallax error in measurements keep the following points in mind.

- Ensure that your eyes are at the same level as the scale or indicator you are reading.
- This may involve positioning yourself directly in front of the measurement device.
- Some instruments have marks or indicators to help align the eye correctly. Always use these reference points when making measurements.

#### 2. Gross Error

In the context of measurement and experimental data analysis, a gross error refers to a significant and obvious mistake or discrepancy that occurs during the data collection or analysis process. It is an error that is typically caused by human error, equipment malfunction, or procedural mistakes. Gross error is commonly related to human blunders. For example, when the measurement is performed which was actually 1.00 but was entered as 100. Such a type of personal error is known as gross error.

#### 3. Reaction time Error

** Reaction time error** is defined as the processing time taken by a human or machine which remains unmeasured. Numerous factors can introduce errors in these measurements, impacting the precision and reliability of the obtained results. Some common factors of reaction time error in measurement include calibration issues, environmental factors, and individual differences with different levels of skills and experiences. This error can be reduced by following the below-given points.

- By averaging time for a large number of experiments.
- Researchers and practitioners conducting reaction time measurements need to consider and control for these sources of error to ensure the validity and reliability of the data.
- Additionally, by performing multiple experiments of the same measurement and employing statistical methods for data analysis, the adverse effects of random errors on reaction time measurements can be minimized, resulting in more accurate estimates.

### Q4 How to calculate percentage error in measuring cylinder?

For the determination of percentage error in measuring cylinder, you must compare the error in your measurement to the true value or actual value you are trying to measure. The formula for percentage error is given below.

\text{Percentage Error } = \frac{\text{Error}}{\text{True Value }} X{100}

**As we know that**

Error = Measured value – Actual value

**Putting the Value of error in the above equation**

\text{Percentage Error } = \frac{\text{Measured Value - Actual Value}}{\text{Actual Value }} X{100}

Here’s a step-by-step guide on how to calculate percentage error using a measuring cylinder:

#### Step 1: Make the Measurement

Using the measuring cylinder, measure the volume of the liquid or substance you are working with. Note down the value you obtained for the volume.

#### Step 2: Obtain the True Value

Find the accepted or known value for the volume of the liquid or substance you are measuring. This value can be obtained from published data or a more accurate measuring device.

#### Step 3: Calculate the Absolute Error (Simple Error)

To determine the absolute error, simply find the positive numerical difference between the measured value and the true value, ignoring any negative signs.

Absolute Error = |Measured Value – True Value|

#### Step 4: Calculate the Percentage Error

To obtain the percentage error, Divide the absolute error (simple error) value by the exact value and then express the value as a percentage by multiplying it by 100.

\text{Percentage Error } = \frac{\text{Measured Value - Exact Value}}{\text{Exact Value }} X{100}

#### Step 5: Express the Result

Express the percentage error as a percentage, including the correct number of significant figures.

Keep in mind that the percentage error provides an indication of the accuracy of your measurement. The magnitude of the percentage error directly corresponds to the accuracy of the measurement; a lower percentage error signifies higher accuracy, while a higher percentage error implies lower accuracy. Always strive for the smallest possible percentage error by using appropriate measurement techniques and instruments.

#### Solved Examples

Following are some important solved examples of how to determine the percentage error in measurement.

##### Example 1:

**The volume of liquid**** recorded on a measuring cylinder is 6.4 cm ^{3} for a value that is actually 6 cm due to marginal errors. Calculate the percentage error in measurement.**

**Given Data** Recorder value (Measured value) = 6.4 cm^{3} Actual value = 6 cm^{3}

**Required Data** Calculate percentage error in measurement.

**Solution**

**By using the Percentage error formula**

\text{Percentage Error } = \frac{\text{Measured Value - Exact Value}}{\text{Exact Value }} X{100}

**Putting the Values**

\text{Percentage Error } = \frac{\text{6.4 - 6}}{\text{6 }} X{100}

\text{Percentage Error} = {6.67\%}

Hence for the given data, the value of the percentage error is 6.67%

##### Example 2

**An experimenter measured the volume of blood using a small measuring cylinder and got the reading with a 2.4 cm ^{3} error. Determine the percentage of the error when the true value is 5 cm^{3}**

**Given Data** Error in measurement = 2.4 cm^{3} Actual value = 5 cm^{3}

**Required Data** Calculate the percentage of error in a smaller measuring cylinder

**Solution**

**Using the formula**

\text{Percentage of Error } = \frac{\text{Error}}{\text{Actual Value }} X{100}

**Putting the Given Values**

\text{Percentage of Error } = \frac{\text{2.4}}{\text{5 }} X{100}

\text{Percentage Error} = {48\%}

The result of the given question ‘calculate the percentage of error in small measuring cylinder’ is 48%

#### Problems

1. A student measures the length of a wooden plank using a ruler with a precision of 1 cm. The measured length is 75 cm. If the real length of the plank is 80 cm, calculate the percentage error in the measurement.

2. A scientist weighs a sample as 18.5 grams on a balance, but the true weight is 20 grams. Determine the percentage of error in the measurement.

3. A pharmacist dispenses 2.1 mL of medicine, but the correct dosage should be 2.5 mL. Determine the percentage of error in the medicine dosage.

### Q5 What is Zero Error?

**Zero Error**

Zero error refers to the systematic deviation of a measuring instrument’s reading from the true or actual value when the measurement is zero. In other words, when the value being measured should be zero, a non-zero reading is observed on the instrument. Zero errors can occur in various types of measuring instruments, such as thermometers, voltmeters, ammeters, weighing scales, and more.

#### Causes of Zero Error

- The instrument might not have been calibrated properly during its manufacturing or may have drifted from its accurate calibration over time.
- Imperfections in the construction of the instrument can cause zero errors.
- Temperature, humidity, and other environmental conditions can influence the instrument’s behavior.
- Rough handling or improper use of the instrument can introduce zero errors.

#### Detection and Correction

Detecting zero errors is crucial to ensure accurate measurements. The following steps can be taken to identify and correct zero errors:

**Calibration**: Regular calibration of measuring instruments by a certified technician can help detect and correct zero errors.**Zero Adjustment:**Many instruments have zero-adjustment knobs or procedures to align the measurement with the true zero.**Check with known values:**Verify the instrument’s accuracy by measuring known values and comparing them with the expected results.**Maintenance:**Regular maintenance and inspection of the instrument can prevent zero errors caused by wear and tear.

In conclusion, zero errors can lead to significant inaccuracies in measurements and should be taken seriously. Proper calibration, regular maintenance, and careful handling of measuring instruments are essential to minimize zero errors and ensure reliable and precise measurements.

### Q6 How to Measure Zero Error in Measurement?

**Measurement of Zero Error**

Measuring zero error involves determining the systematic deviation of a measuring instrument’s reading from the true value when the quantity being measured is zero. The process of measuring zero error depends on the type of instrument, but here are some general steps:

#### Step 1: Understand the Instrument

Familiarize yourself with the measuring instrument and its zero position. For example, with a weighing scale, the zero position is when there is nothing placed on the scale, and the reading should be exactly zero.

#### Step 2: Ensure Proper Setup

Set up the instrument in a stable and controlled environment. Make sure it is clean, free from any debris or substances that could affect its accuracy.

#### Step 3: Zero the Instrument

If the instrument has a zero-adjustment knob or feature, follow the manufacturer’s instructions to zero the instrument. This procedure typically aligns the measurement with the true zero.

#### Step 4: Perform Zero Measurement

Carry out a measurement where the quantity being measured is known to be zero or as close to zero as possible. For example, with a thermometer, immerse the sensor in a container of ice water, which should be very close to 0°C.

#### Step 5: Record the Reading and Determine the Zero Error

Take note of the instrument’s reading. If it shows a value other than zero, this is the zero error. When an instrument consistently indicates values higher than the true zero point, it possesses a positive zero error, and if it consistently indicates values lower than the true zero point, it possesses a negative zero error.

### Q7 How to measure zero error in vernier Calipers?

**Measurement of Zero Error in Vernier Calipers**

Measuring zero error in a vernier caliper involves determining the systematic deviation of the caliper’s reading from the true zero point when its jaws are closed. Zero error in a vernier caliper can be either positive or negative, depending on whether the caliper’s zero mark aligns with the true zero on the scale or not. Here’s how to measure zero error in a vernier caliper:

#### Materials Needed

- Vernier caliper
- A small object to measure (optional, for verification)

#### Steps

Zero error is one of the major types of errors. It leads to a huge difference in the measurement. It is necessary to determine and measure the zero error. Some of the steps to measure the zero error of the vernier caliper are given below.

##### Step 1: Clean and Setup the Vernier Caliper

Ensure the caliper is clean and free from any debris. Close the jaws of the caliper completely.

##### Step 2: Inspect the Zero Mark Alignment

Examine both the main scale and the vernier scale very carefully and attentively. The desired outcome is for the zero mark on the vernier scale to be in complete alignment with the zero mark on the main scale when the jaws are fully closed. Perfect alignment of the zero marks indicates that there is no zero error (i.e., zero error equals 0). If the zero marks do not match, proceed to the next steps.

##### Step 3: Determine the Direction of Zero Error

Observe the orientation of the zero mark on the vernier scale in relation to the zero mark on the main scale to determine whether it is positioned to the left or the right.

- If the zero mark on the vernier scale is to the left of the main scale zero, the zero error is negative.
- If the zero mark on the vernier scale is to the right of the main scale zero, the zero error is positive.

##### Step 4: Measure the Zero Error

o measure the zero error value, count the number of divisions on the vernier scale that are to the left or right of the main scale zero mark. Each division on the vernier scale represents a specific fraction of the main scale division (e.g., 0.1 mm or 0.02 inches, depending on the caliper’s precision).

- For example, if you count 3 divisions to the left of the main scale zero, and each vernier scale division represents 0.02 inches, the zero error would be -0.06 inches (3 divisions * 0.02 inches/division).
- If you count 2 divisions to the right of the main scale zero, and each vernier scale division represents 0.1 mm, the zero error would be +0.2 mm (2 divisions * 0.1 mm/division).

It is essential to be aware of and account for zero error whenever using a vernier caliper to obtain accurate measurements. If you frequently use the caliper for critical measurements, consider having it calibrated professionally.

### Q8 How to Measure Zero Error in Screw Gauge?

**Measurement of Zero error in Screw Gauge**

Measuring the zero error of a screw gauge involves determining the systematic deviation of the gauge’s reading from the true zero point when the spindle and anvil faces are in light contact (no object being measured). The zero error can be either positive or negative, depending on whether the gauge’s zero mark aligns with the true zero on the main scale or not. Here’s how to measure zero error in a screw gauge:

#### Materials

- Screw gauge (micrometer)
- A flat object or a calibration standard (optional, for verification)

#### Steps

The steps that are involved to measure the zero error in the screw gauge are as under.

##### 1. Clean and set up the screw gauge:

Make sure the screw gauge is tidy and clear of any debris or dirt. Close the spindle and anvil faces so that they are in light contact without applying any pressure.

##### 2. Inspect the Zero Mark Alignment

Look closely at the main scale and the thimble scale of the screw gauge. The zero mark on the thimble scale should ideally align perfectly with the zero mark on the main scale when the faces are in light contact. If the zero marks align, there is no zero error (i.e., the zero error is 0). However, if the zero marks do not coincide, proceed to the next steps.

##### 3. Determine the Direction Of Zero Error

Observe whether the zero mark on the thimble scale is to the left or the right of the zero mark on the main scale.

- If the zero mark on the thimble scale is to the left of the main scale zero, the zero error is negative.
- If the zero mark on the thimble scale is to the right of the main scale zero, the zero error is positive.

##### 4. Measure the Zero Error

To measure the zero error value, count the number of divisions on the thimble scale that are to the left or right of the main scale zero mark. Each division on the thimble scale represents a specific fraction of the main scale division (e.g., 0.01 mm or 0.001 inches, depending on the screw gauge’s precision).

- For example, if you count 5 divisions to the left of the main scale zero, and each thimble scale division represents 0.001 inches, the zero error would be -0.005 inches (5 divisions * 0.001 inches/division).
- If you count 3 divisions to the right of the main scale zero, and each thimble scale division represents 0.01 mm, the zero error would be +0.03 mm (3 divisions * 0.01 mm/division).

It is important to be aware of and account for zero error whenever using a screw gauge to obtain accurate measurements.