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**T****cal Calibration**

*Calibration/Validation of Data/Transient Loggers, Recorders and Acquisition Systems*

*This is a Tcal calibration article. Follow this link to return to the Tcal homepage.*

**RMS Noise**

*RMS calculations are often done of absolute values (the difference from zero) but in that case it is generally not considered to be noise. In this article and in Tcal in general the noise is calculated from the average. If a measured value is supposed to be zero but has a non-zero average then that error will be picked up elsewhere in the analysis.*

RMS Noise is the root mean square of the differences from the average. RMS stands for "Root Mean Square" meaning that the differences from the mean are all squared, an average taken and then a square root done of the average. The square has to be done because otherwise all the positive and negative noise would cancel out: and the square root is done to undo it.

RMS Noise is equivalent to standard deviation. The standard deviation is the square root of the variance which is the sum of the squares of the differences from an average value. This is the same definition.

If you are looking at the calculations in the Tcal spreadsheets you will see the RMS Noise calculated using an expansion of the brackets. Instead of calculating the sqrt[sum of the {average - data points}^2] it calculates the sqrt[average^2 - 2*average*(sum of the {data points}) + n*(sum of the {data points}^2)]. These two are exactly the same with the longer version just being the shorter version with the brackets multiplied out. It is written like this on the spreadsheet as a clearer way could not be found without taking up a lot more space.

**Implications for Sample Size**

*Averaging can overcome noise. RMS Noise is no different.*

The higher the RMS Noise the more data points that are needed for the average to get close to the mean of the underlying population. If a signal is rapidly changing this can be a problem. However, if a signal is static or has a slow/steady trend then a running average can smooth out the noise in the signal. If fast changes are being measured then the running average can smooth out these features which can be a bad thing.

The RMS Noise can be used to make an error bar when taking measurements from the same instrument. Multiply the RMS Noise by 1.645 and then divide it by the square root of the number of samples: this is then the size of the positive and the size of the negative error bar. So, if (RMS Noise)*1.645/sqrt(n)=0.04 and you measure a voltage as 3V then ignoring other sources of error you could give the result as 3V+/-0.04%.

An explanation of the use of 1.645 can be found on the Margin of Error Wikipedia page or other resources about errors from normal distributions. An explanation of the use of the square root of the number of samples can be found on the Standard Error Wikipedia page or other resources about standard error/normal distribution error.

This issue impacts on both the number of data points to be collected during calibration and during day-to-day use of the instrument. The Tcal software gives feedback on this issue during operation.

RMS Noise as calculated for the Tcal sample data set. A data set with high noise was used to give more to see during practice.

Here is an excerpt from the Tcal sample data calibration certificate. The top row of the table shows the error for the +/-10V range for different numbers of samples. If just one sample is taken then the error is 0.55%. As more samples are averaged the error settles to 0.35% which is the remaining error due to other factors.

Averaging is obviously most important where the RMS Noise error is significantly larger than the other errors and less important where it is significantly smaller than the other errors.

**Additional Noise Issues**

*Instrument noise will be the main source of noise only in situations where the cables are short and/or shielded, electromagnetic noise is low, environmental noise that would register on sensors is low and so on. A ground plug is often chosen for calibration to give an accurate idea of the noise generated by the instrument excluding other issues.*

Noise can also come from wires acting as aerials, a noisy earth or simply background noise naturally present in the environment being measure. This means that for some applications you need to add the noise due to the instrument, the other sources of electrical and hardware noise and any environmental noise.