Measurements

Measurements

• Accuracy:

The accuracy of a measurement is how close a result comes to the true value.
For example, let’s say you know your true height is exactly 5’9″.You measure yourself with a yardstick and get 5’0″. Your measurement is not accurate.
You measure yourself again with a laser yardstick and get 5’9″. Your measurement is accurate.

Systematic error or Inaccuracy is quantified by the average difference (bias) between a set of measurements obtained with the test method with a reference value or values obtained with a reference method.

• Precision:

Precision is how close two or more measurements are to each other or in other words it refers to how well measurements agree with each other in multiple tests.
If you consistently measure your height as 5’0″ with a yardstick, your measurements are precise.

Random error or Imprecision is usually quantified by calculating the coefficient of variation from the results of a set of duplicate measurements

 

• Significant figures & Rounding:

Significant figures: The significant figures (also known as the significant digits and decimal places) of a number are digits that carry meaning contributing to its measurement resolution. The reliable digits and the first unreliable digit of a measurement are known as significant figure.

Rules for significant figures:

1. All non-zero numbers are significant. The number 33.2 has three significant figures because all of the digits present are non-zero.
2. All zeros between two non-zero digits are significant. 2051 has four significant figures. The zero is between a 2 and a 5.
3. Leading zeros are not significant. They're nothing more than "place holders." the number 0.54 has only two significant figures. 0.0032 also has two significant figures. All of the zeros are leading.
4. In a number with or without a decimal point, Trailing zeros to the right of the decimal are significant. There are four significant figures in 92.00.
5. Trailing zeros in a whole number with the decimal shown are significant. Placing a decimal at the end of a number is usually not done. By convention, however, this decimal indicates a significant zero. For example, "540." indicates that the trailing zero is significant; there are three significant figures in this value.
6. Trailing zeros in a whole number with no decimal shown are not significant. Writing just "540" indicates that the zero is not significant, and there are only two significant figures in this value.
7. Any zero to the right of non-zero digit is significant. All zeros between decimal point and first non-zero digits are not significant. 0.0074 has only two significant digits & 0.06020 has four significant digits.
8. For a number in scientific notation: N x 10^x, all digits comprising N ARE significant by the first 6 rules; "10" and "x" are NOT significant. 5.02 x 10⁴, has THREE significant figures: "5.02." "10 and "4" are not significant.
9. Exact numbers have an INFINITE number of significant figures. This rule applies to numbers that are definitions. For example, 1 meter = 1.00 meters = 1.0000 meters, etc.

Rounding: The basic concept of significant figures is often used in connection with rounding. Rounding to significant figures is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way

Process of Rounding off:

1. Decide which is the last digit to keep.
2. In rounding off numbers, the last figure kept should be unchanged if the first figure dropped is less than 5. Example: 6.422 become 6.4.
3. In rounding off numbers, the last figure kept should be increased by 1 if the first figure dropped is greater than 5. Example: 6.997 become 7.00.
4. In rounding off numbers, if the first figure dropped is 5, and there are any figures following the five that are not zero, then the last figure kept should be increased by 1. Example: 6.6501 become 6.7.
5. In rounding off numbers, if the first figure dropped is 5, and all the figures following the five are zero or if there are no figures after the 5, then the last figure kept should be unchanged if that last figure is even. Example: 6.6500 ≈ 6.6.
6. In rounding off numbers, if the first figure dropped is 5, and all the figures following the five are zero or if there are no figures after the 5, then the last figure kept should be increased by 1 if that last figure is odd. Example: 6.755000 ≈ 6.76.

• Errors:

An error may be defined as the difference between the true or actual value and the measured value. Errors in measurements may happen from the various sources which are generally categorized into the following type. 

• Gross Errors:

The gross error occurs because of the human mistakes. Gross errors can be defined as physical errors in analysis apparatus or calculating and recording measurement outcomes. In general, these types of errors will happen throughout the experiments, wherever the researcher might study or record a worth different from the real one, possibly due to a reduced view. With human concern, types of errors will predictable, although they can be estimated and corrected.

Such type of error is very common in the measurement. The complete elimination of such type of error is not possible. Some of the gross errors easily detected by the experimenter but some of them are difficult to find. Two methods can remove the gross error.

These types of errors can be prohibited by the following couple of actions:

• Careful reading as well as a recording of information.
•  Taking numerous readings of the instrument by different operators. Secure contracts between different understandings guarantee the elimination of every gross error.

• Random Errors:

This type of error is constantly there in a measurement, which is occurred by essentially random oscillations in the apparatus measurement analysis, sudden change in the atmospheric condition or in the experimenter’s understanding of the apparatus reading. These types of errors show up as dissimilar outcomes for apparently the similar frequent measurement, which can be expected by contrasting numerous measurements, with condensed by averaging numerous measurements. These types of error remain even after the removal of the systematic error. Hence such type of error is also called residual error.

• Systematic Errors:

The systematic errors are mainly classified into three categories.

1. Observational Errors
2. Environmental Errors
3. Instrumental Errors

• Observational Errors: The observational errors may occur due to the fault study of the instrument reading, and the sources of these errors are many. For instance, the indicator of a voltmeter retunes a little over the surface of the scale. As a result, a fault happens except the line of the image of the witness is accurately above the indicator. To reduce the parallax error extremely precise meters are offered with reflected scales.
• Environmental Errors: Environmental errors will happen due to the outside situation of the measuring instruments. These types of errors mostly happen due to the temperature result, force, moisture, dirt, vibration otherwise because of the electrostatic field or magnetic. The remedial measures used to remove these unwanted effects include the following.

1. The preparation should be finished to remain the situations as stable as achievable.
2. By the instrument which is at no cost from these results.
3. With these methods which remove the result of these troubles.
4. By applying the computed modifications.

• Instrumental Errors: Instrumental errors will happen due to some of the following reasons.

• An inherent limitation of Devices:-

These errors are integral in devices due to their features namely mechanical arrangement. These may happen due to the instrument operation as well as the operation or computation of the instrument. These types of errors will make the mistake to study very low otherwise very high.
For instance – If the apparatus uses the delicate spring then it offers the high-value of determining measure. These will happen in the apparatus due to the loss of hysteresis or friction. 

• Abuse of Apparatus:-The error in the instrument happens due to the machinist’s fault. A superior device used in an unintelligent method may provide a vast result. For instance – the abuse of the apparatus may cause the breakdown to change the zero of tools, poor early modification, with lead to very high resistance. Improper observes of these may not reason for lasting harm to the device, except all the similar, they cause faults.

• Effect of Loading:- The most frequent type of this error will occur due to the measurement work in the device. For instance, as the voltmeter is associated to the high-resistance circuit which will give a false reading, as well as after it is allied to the low-resistance circuit, this circuit will give the reliable reading, and then the voltmeter will have the effect of loading on the circuit.

 Calculation of Error 

• Error:

Error in i-th observation,

  

Where x_i is the measured value in i-th observation & 

• Absolute Error:

• Relative Error:

• Percentage Error: Percentage error = Relative error × 100% = 
  

                                                              

• Rules of error calculation in different mathematical operation:

·

• Uncertainty analysis: Uncertainty analysis aims at quantifying the variability of the output that is due to the variability of the input. The quantification is most often performed by estimating statistical quantities of interest such as mean, median, and population quarantines. The estimation relies on uncertainty propagation techniques. Because of the limited sample size, these estimated quantities must be provided with the associated confidence intervals.

The main steps of the uncertainty propagation are summarized below:

1. Identify the model input parameters subject to uncertainty
2. Describe the knowledge about the variables by means of probability density functions and, if relevant, account for correlations between the variables by using multivariate Probability Density Functions (PDFs) or by providing a correlation matrix
3. Generate a sample from the original distribution
4. Execute the computer code for this set of sampled values
5. Apply statistical methods to compute the values of the quantities of interest

Statistical Analysis of Data:

• Arithmetic Mean:

• Mean Deviation: The mean deviation (also called the mean absolute deviation or Average Deviation) is the mean of the absolute deviations of a set of data about the data's mean. For a sample size, the mean deviation is defined by

• Standard Deviation: In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that Cumulative probability of a normal distribution with expected value 0 and standard deviation 1 the values are spread out over a wider range. So The Standard Deviation is a measure of how spreads out numbers are.

So the Standard deviation,

Note: Variance = Square of the standard deviation

Guassian Distribution:

In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is,