# How to understand the variance in this setting?

All -

I have a dataset originated from a practical setting, but it is not clear to me how to interpret it. Let me try to frame the setting and see if it makes sense:

A manufacturer produces a type of product that use noise level (dB) as one of its quality metrics. All products during a production cycle are expected to have similar or consistent noise levels. Further, Let's say this "consistency requirement" is defined here as "the max and min noise level should not differ more than $x$ percentage point from the mean".

My question are two-fold:

• Is it possible to use some distribution metrics, for example, standard devision, to make claim such as: if the $sd < a$, then it will meet the consistency requirement?

• It seems intuitively that the more products produced in the cycle, the harder it is to meet the consistency requirement. Statistically, each product's noise is independent variable, the larger the population, the larger the variation of sum - is this a correct understanding/statement?

Thanks

Oliver

## 1 Answer

• A standard deviation does not give you information about minimum and maximum values, because it is a measure of dispersion, not range. If the consistency requirement is minimum and maximum values, you should look at the range or you will make a mistake. For example, the standard deviation of a normal distribution is one and it's mean zero, yet it's range is from minus infinity to plus infinity. To assess the requirement you formulated, I would determine cutoff values (mean plus and minus x percent), and look at the distribution of your data to determine how many cases fall outside of the cutoff values.
• In absolute numbers, the amount of failures will (obviously) be higher with more cars, simply because the variance is greater than zero. In relative terms, the amount of failures depends on the underlying distribution and your criterion for success. Assuming a normal distribution, with no significant skew or kurtosis and constant variance, the percentage of cars that fall outside of a specified range (containing the mean) actually decreases. This is because the distribution is bell-shaped: it is more probable that a new event will be close to the mean than far away from it.