**t â€“ Distribution**

The t-distribution method is the most appropriate method to be used when there is a sample size of less than thirty, the population standard deviation is not known and the population is approximately normal. The t-distribution is symmetrical in shape, however, flatter than the normal distribution. Also, as the sample size increases, the t-distribution approaches normality.

For every possible sample size or degrees of freedom, there is a different t-distribution. Look at the figure to observe. Unlike the normal distribution, a t-distribution is lower at the mean and higher at the tails. The t-distribution is used for hypothesis testing.

**Â F â€“ Distribution**

Its Characteristics The F-distribution is a ratio if two chi square distributions. A specific F-distribution is denoted by the degrees of freedom for the numerator chi square and the degrees of freedom for the denominator chi square. The F-test is performed to calculate and observe if the standard deviations or variances of the two processes are significantly different. The project teams are concerned about reducing the process variance.

As per the equation, if the F calculated is equal to one, it implies no difference in the variance. S one and S two are the standard deviations of the two samples. If S one is greater than s2, the numerator must be greater than the denominator. In other words, d f one is equal to n one minus one and d f two is equal to n two minus one. From the F-distribution table, you can be easily find out the critical F-distribution at alpha and the degrees of freedom of the samples of two different processes d f one and d f two.

**Objective of Measurement System**

An important objective is to obtain information about the type of measurement variation associated with the measurement system and establish criterion to accept and release new measuring equipment. It also compares measuring one method against another. It helps to form a basis for evaluating a method which is suspected of being deficient. The measurement system variations are also resolved in order to arrive at the correct baseline.

According to the measurement analysis, an evaluation of the measurement system must be undertaken to ensure effective analysis of any subsequent data generated for a given process or product characteristic. The observed value is equal to the sum of the true value and the measurement error. The measurement error can be a negative or a positive value. The measurement error is a statistical term which means the net effect of all sources of measurement variability that cause and observed value to deviate from the true value.

The true variability is sum of the process variability and the measurement variability. Both the process as well as the measurement variability must be evaluated and improved together. If we begin to work on the process variability while the measurement variability was large, it can never be concluded that there was a significant or correct improvement.

There are two type of measurement errors, the measurement system bias calibration study and the measurement system variation G R study. In the measurement system bias calibration study, the total mean is a sum of the process mean and the measurement mean. On the other hand, in the measurement system variation G R R study, the total variance is a sum of the process variance and measurement variance.

**Sources of Variation**

There are different sources of variation. The observed process variation is divided into two main parts, the actual process variation and the measurement variation. Both these variations have a common factor which is the variation within a sample.

Also, the actual process variation can be divided into long term process variation and short term process variation. On the other hand, the measurement variation can be further divided into variation due to operators and variation due to gage. The variation due to operators is due to reproducibility. On the other hand, variation due to gage has factors like accuracy, repeatability, stability, and linearity attached to it.

**Gage Repeatability and Reproducibility**

The gage repeatability and reproducibility is a measure of the capability of a gauge or gage to obtain the same measurement reading every time the measurement process is undertaken for the same characteristic or parameter.

On one hand, the gage repeatability is the variation in its measurements obtained when one operator uses the same gage for measuring the identical characteristics of the same part.Look at the figure closely and observe. The gage reproducibility is the variation in the average of measurements made by different operators using the same gage when measuring identical characteristics of the same part. A closer look at the image helps to understand this.

**Component of GRR Study**

The figure shows the repeatability and reproducibility on six different parts represented by the numbers from one to six for two different trial readings by three different operators.If there is a difference in reading for part one represented by the green box, by three different operators, it is known as reproducibility error.On the other hand, if there is a difference in reading of part four represented by the red box by the same operator in two different trials, it is known as the repeatability error.

**Key Concepts**

An important thing to note is that the gage repeatability and reproducibility studies are referred to as the GRR studies.

The GRR studies should be performed over the range of expected observations. One should be careful to note that the actual equipments should be used for the GRR studies and already written procedures should be allowed.In other words, it should be business as usual. The measurement variability should be represented as is, not the way it was designed to be.

After GRR, the measurement variability is separated into casual components, sorted as per priority and then targeted for action.

What do we understand by measurement resolution?

The measurement resolution is the capability of the measurement system to detect the smallest tolerable changes.

Also, it is important to have a number of increments in the measurement system at full range. For example, the usage of a truck weighing scale for measuring the weight of a tea pack is to be avoided. Most importantly, to make sure that as a pre-requisite for the GRR, one must ascertain that the gage has acceptable resolution.

**Data Collection**

There are usually three operators with around ten units to measure. The general sampling techniques must be used to represent the population and each unit must be measured two to three times by each operator.

It is important that the gage should be properly calibrated and resolution should be ensured. Additionally, the first operator should measure all the units in random order and the same order must be maintained for all other operators. All the trials must be repeated.

There are various methods to perform the GRR studies. However, the ANOVA method is the best method.

**Interpretation of Measurement System Analysis**

Let us interpret the measurement system analysis now. Let us assume two cases now. In one case, the reproducibility error is larger compared to the repeatability error.

The possible causes could be so that the operators are not properly trained in using and reading the gage. In addition to this, the calibrations on the gage dial may not be clear.In another case, the repeatability error is larger compared to the reproducibility error.

Causes for such a situation are that the gage or instrument needs maintenance, the gage needs to be more rigid, and the location for gaging needs improvement, and the statement of purpose for measurement are not clear.

Let us now focus on measurement correlation. Measurement correlation is the comparison of the measurement values from one measurement system with the corresponding values reported by one or more measurement systems.

Additionally, it also means the comparison of values obtained using different measurement methods to measure different properties. For example, the correlation of hardness and strength of the metal, temperature, linear expansion of a given item being heated and weight and piece count of small points. The measure system analysis is an experiment which seeks to identify the components of variation in the measurement. It is important to note that the measure system analysis or the M S A classifies the measurement system error into five different categories. The categories are bias, stability, linearity, percent agreement, and precision or tolerance.