# Probability distribution of measurements and Parameters of measurements

I am new to statistics and recently learned about ISO guidelines for Accuracy & Precision and Uncertainty & Error. I have tried to plot a graph for what I have learnt including all the parameters of a measurement.

I just want to know that my graph is correct or not according to defination of ISO? I am very sure about the error part (systematic error, random error and total error) that I illustrated it properly but want to confirm the relations that I showed between

1. Trueness & Systematic Error (difference between true value and theoretical mean of measurements taken infinite times)
2. Accuracy & Total Error ( difference between true value and measured value that is average of some finite measurement sample )
3. Precision & Standard deviation ( dispersion in mean of finite sample)

are right or wrong?

• You seem to be asking us to reproduce the figures you have posted!
– whuber
Commented Sep 5, 2023 at 17:53
• @whuber Yes, because they lead to arise some confusions as I described in question below first image. I want a single figure in which all the parameters are described according to ISO guidelines. Commented Sep 5, 2023 at 18:01
• It remains unclear what a satisfactory answer would look like. How would it differ from these diagrams?
– whuber
Commented Sep 5, 2023 at 18:03
• @whuber How can we add some more details in first graph about Trueness , accuracy and precision i.e. If we look at first graph there are some terms mentioned Systematic Error, Random error, Total error, Mean, Measured-Value and True-value, which of them and how(formula) will be related to Trueness, accuracy and precision. For example ( may be I am wrong) to describe accuracy we will insert a horizontal line between true value And measured value and relate accuracy to total error. Commented Sep 5, 2023 at 18:46
• It is such a busy graph in trying to illustrate so many things (samples, sample distribution, data generation process, and at least five properties of one or more of those) that it's difficult to decipher. That reveals the basic problem with this thread: it's trying to ask too many questions at once. Consider editing it to focus on one specific concept you wish to have explained or illustrated.
– whuber
Commented Sep 7, 2023 at 12:06

Regarding the first question and first image:

In this graph(above) if systematic error is zero then average value will be the true value! How's that possible?

You are right that this isn't normally possible. In fact the observed average of measurements will not necessarily and not normally be equal to the true value, however the expected value of the average of measurements will be the true value, i.e., if you repeat taking observations and computing their average potentially infinitely often, the average of these averages will be the true value. The latter is called "measurement population mean" here to make you aware that this is not the mean of your observations but a theoretical mean of the underlying "population of measurements" assuming you could take infinitely many measurements.

Furthermore this holds under certain statistical model assumptions but not necessarily always in reality, however chances are that for understanding these images these assumptions should be taken to hold.

And why random error here is described with respect to measured value not to mean of measured value?

The "random error" would be the difference between an actual observation/measurement and the theoretical mean of the "measurement population", i.e., between what you actually observe and the idealised mean of infinitely many observations. In the image this is indeed shown as the difference between these two.

• As you described random error is related to an actual measurement so In the second Image why accuracy and precision are related to mean of observations not to a single measurement that is actually observed. Commented Sep 5, 2023 at 18:34
• If we look at first graph, how trueness, accuracy and precision would be defined ? i. e. There are some terms in first graph systematic error, Random error, total error, mean , measured value and true value, which of them and how(formula) will related to Trueness, accuracy and precision. Commented Sep 5, 2023 at 18:38
• Sorry, I only meant to address the first question. You'll need some others to address the rest (lack of time). The second image would need far more explanation in text than what can be seen there, but I don't have time to read what the original source says. "Trueness" is a strange term and hardly ever used elsewhere in statistics. Commented Sep 5, 2023 at 22:20