How do I get the mean/average with the given standard deviation 7ml and percentile which is 10%?

Question

In the word promblem, it stated that a machine was set so that 10% of a 370ml can can contain less that 356ml.

I'm not good with word problems and I'm confused which is which.

Answer 1

I believe your "370ml can" is actually a larger capacity can, but labelled as "contains 370 ml of...". The actual amount of drink in the can will vary, and can be more or less than the nominal amount stated. It's important to understand the idea of labelling and variation, and that it's usually impossible to always put an exact amount (whether it's a drink in a can, or sugar in a bag). So a factory doesn't try to be exact, but to control the amount/degree of variation.

This variation follows a Gaussian or Normal Distribution (the famous "bell curve").

It's good if you can remember how much falls under the curve by standard deviation. Approximately 34% of the curve falls under 1 SD from the mean (or 68% both sides of the mean), and then a further 13.6% falls between 1 SD and 2 SD from the mean (or a total of 95% both sides of the mean).

The number of standard deviations from the mean can be positive (to the right) or negative (to the left). It can be an integer number of standard deviations (e.g. -1 or 2), or a decimal/fraction (e.g. 0.5 or 2.3 or -1.2). This "number of standard deviations" is known as the z-score.

The formula for calculating the z-score is z = (measured value - mean) / SD

If we know z (but not the mean), we must rearrange the equation to get mean = measured value + z x SD

As we know the measured value = 356ml and SD = 7ml, we know the mean is not 370 ml. Why? Because that would be too easy! :-) But also because 2 SD from 370 ml would be 356 ml (370 - 2*7), with only 2% less than this. Therefore we need the nornal distribution curve shifted so that we have 10% under (to the left) the curve at 356 ml, which means the mean must be less than 370 ml.

If you look at a bell curve, you can see that 10% falls between 1SD (2.2%) and 2SD (15.9%), roughly half way. So at 1.5 SD we have 10.5ml, or a mean of 356 + 10.5 or approximately 366.5 ml. This is my initial "guesstimate".

A table (Z Table or Z-Score Table) of SD versus % tells us exactly how much of the curve is included. We need 80% included (10% excluded either side), which is 1.282 SDs (our z-score). This is actually -1.282 SD as we are to the left of the mean.

Hence our actual mean is 356 + 1.282(7) = 364.97400 or 365 ml.

As 99.7% of a curve falls under 3 SD, we can say that the volume of drink in the can can vary between 365 +/- 21ml or between approximately 344ml and 386ml.