How to calculate a random measurement error?

In summary, there are two formulas for calculating the measure of dispersion, the standard deviation and the mean absolute deviation. The first formula, which uses the squared differences from the mean, is the standard deviation and is commonly used. The second formula, which uses the absolute differences from the mean, is the mean absolute deviation and is rarely used. It may be used in some machine learning applications for its computational efficiency, but in experimental physics, the standard deviation is preferred.
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How to calculate a random measurement error? If we did a measurement and want to calculate a random error, what formula are we to use?
I have seen this formula

$$\sigma=\sqrt{\frac {\sum_{i=1}^{N}{(X_i- \bar{X})^2}}{N(N-1)}}$$

but also this formula $$\sigma =\frac{\sum_{i=1}^{N}{|X_i- \bar {X}|}}{N}.$$ Which of them is correct?
 
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The first one :smile:

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The first one is the standard deviation. That is the usual one. The second one is the mean absolute deviation and it is rarely used at all.
 
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Dale said:
The second one is the mean absolute deviation and it is rarely used at all.
Some machine learning applications seem to use it - I think because it's cheaper to calculate, so it gives you a time saving if you can live with the less mathematically nice behaviour. But in experimental physics I agree it's a no, you want the standard deviation.
 
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1. What is a random measurement error?

A random measurement error is an unpredictable and uncontrollable variation in the measured value of a quantity. It can occur due to various factors such as human error, equipment malfunction, or environmental influences.

2. How is a random measurement error different from a systematic error?

A systematic error is a consistent deviation from the true value of a quantity, while a random measurement error is an unpredictable fluctuation around the true value. Systematic errors can be corrected, but random measurement errors cannot be eliminated entirely.

3. How do you calculate a random measurement error?

To calculate a random measurement error, you need to take multiple measurements of the same quantity and then calculate the average of these measurements. The difference between each individual measurement and the average is the random measurement error.

4. What is the purpose of calculating a random measurement error?

The purpose of calculating a random measurement error is to assess the reliability and accuracy of the measurements. It helps to identify any sources of variation and determine the level of uncertainty in the measured value.

5. Can a random measurement error be reduced?

No, a random measurement error cannot be completely eliminated. However, it can be minimized by using precise and accurate measurement techniques, taking multiple measurements, and controlling external factors that may affect the measurement.

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