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The errors, which occur during measurement are known as **measurement errors**. In this chapter, let us discuss about the types of measurement errors.

We can classify the measurement errors into the following three types.

- Gross Errors
- Random Errors
- Systematic Errors

Now, let us discuss about these three types of measurement errors one by one.

The errors, which occur due to the lack of experience of the observer while taking the measurement values are known as **gross errors**. The values of gross errors will vary from observer to observer. Sometimes, the gross errors may also occur due to improper selection of the instrument. We can minimize the gross errors by following these two steps.

- Choose the best suitable instrument, based on the range of values to be measured.
- Note down the readings carefully

If the instrument produces an error, which is of a constant uniform deviation during its operation is known as **systematic error**. The systematic errors occur due to the characteristics of the materials used in the instrument.

**Types of Systematic Errors**

The systematic errors can be classified into the following **three types**.

**Instrumental Errors**− This type of errors occur due to shortcomings of instruments and loading effects.**Environmental Errors**− This type of errors occur due to the changes in environment such as change in temperature, pressure & etc.**observational Errors**− This type of errors occur due to observer while taking the meter readings.**Parallax errors**belong to this type of errors.

The errors, which occur due to unknown sources during measurement time are known as **random errors**. Hence, it is not possible to eliminate or minimize these errors. But, if we want to get the more accurate measurement values without any random error, then it is possible by following these two steps.

**Step1**− Take more number of readings by different observers.**Step2**− Do statistical analysis on the readings obtained in Step1.

Following are the parameters that are used in statistical analysis.

- Mean
- Median
- Variance
- Deviation
- Standard Deviation

Now, let us discuss about these **statistical parameters**.

Let $x_{1},x_{2},x_{3},....,x_{N}$ are the $N$ readings of a particular measurement. The mean or **average value** of these readings can be calculated by using the following formula.

$$m = \frac{x_{1}+x_{2}+x_{3}+....+x_{N}}{N}$$

Where, $m$ is the mean or average value.

If the number of readings of a particular measurement are more, then the mean or average value will be approximately equal to **true value**

If the number of readings of a particular measurement are more, then it is difficult to calculate the mean or average value. Here, calculate the **median value** and it will be approximately equal to mean value.

For calculating median value, first we have to arrange the readings of a particular measurement in an **ascending order**. We can calculate the median value by using the following formula, when the number of readings is an **odd number**.

$$M=x_{\left ( \frac{N+1}{2} \right )}$$

We can calculate the median value by using the following formula, when the number of readings is an **even number**.

$$M=\frac{x_{\left ( N/2 \right )}+x_\left ( \left [ N/2 \right ]+1 \right )}{2}$$

The difference between the reading of a particular measurement and the mean value is known as *deviation from mean*. In short, it is called *deviation*. Mathematically, it can be represented as

$$d_{i}=x_{i}-m$$

Where,

$d_{i}$ is the deviation of $i^{th}$ reading from mean.

$x_{i}$ is the value of $i^{th}$ reading.

$m$ is the mean or average value.

The root mean square of deviation is called **standard deviation**. Mathematically, it can be represented as

$$\sigma =\sqrt{\frac{{d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+....+{d_{N}}^{2}}{N}}$$

The above formula is valid if the number of readings, N is greater than or equal to 20. We can use the following formula for standard deviation, when the number of readings, N is less than 20.

$$\sigma =\sqrt{\frac{{d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+....+{d_{N}}^{2}}{N-1}}$$

Where,

$\sigma$ is the standard deviation

$d_{1}, d_{2}, d_{3}, …, d_{N}$ are the deviations of first, second, third, …, $N^{th}$readings from mean respectively.

**Note** − If the value of standard deviation is small, then there will be more accuracy in the reading values of measurement.

The square of standard deviation is called **variance**. Mathematically, it can be represented as

$$V=\sigma^{2}$$

Where,

$V$ is the variance

$\sigma$ is the standard deviation

The mean square of deviation is also called **variance**. Mathematically, it can be represented as

$$V=\frac{{d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+....+{d_{N}}^{2}}{N}$$

The above formula is valid if the number of readings, N is greater than or equal to 20. We can use the following formula for variance when the number of readings, N is less than 20.

$$V=\frac{{d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+....+{d_{N}}^{2}}{N-1}$$

Where,

$V$ is the variance

$d_{1}, d_{2}, d_{3}, …, d_{N}$ are the deviations of first, second, third, …, $N^{th}$ readings from mean respectively.

So, with the help of statistical parameters, we can analyze the readings of a particular measurement. In this way, we will get more accurate measurement values.

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