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You have a random sample of 25 objects with mean weight of 24 grams, estimate the standard deviation of the sample.

In addition, you know it's supposed to be 25 grams with a deviation of 1 gram, but this has no relevance to the above question, right?

How is this done? Looking in my formula reference this is not enough information to give an estimate.

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  • $\begingroup$ It might be easier if you know the individual weights $\endgroup$
    – Henry
    Apr 19, 2014 at 14:59
  • $\begingroup$ Well the problem is I don't. This is all information that is presented, since it is an estimate that's asked for, can you extrapolate it from a normal distribution somehow, or is it a trick question perhaps? $\endgroup$ Apr 19, 2014 at 15:02
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    $\begingroup$ So you mean you have a sample of $25$ objects taken from a population with mean $25g$ and standard deviation $1g$ and you want to estimate the standard deviation of the sample given that the mean of the sample is $24g$ - that is very much more information than just the size of the sample and the mean weight, from which no estimate could be made - is my interpretation of the actual question correct? $\endgroup$ Apr 19, 2014 at 15:03
  • $\begingroup$ Yes, although it is not certain that the mean is 25g / the deviation is 1g of the population. The reason for taking the sample of 25 objects is to see if that assumption is correct. $\endgroup$ Apr 19, 2014 at 15:10

2 Answers 2

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A sample mean of $24$ is unlikely from a sample of twenty-five items with a population mean of $25$ and population standard deviation of $1$. So the sample casts doubt on the population parameters.

But conditioned on the data given and ignoring issues such as finite populations, a good estimate of the variance of the sample conditioned on the population mean and sample mean is the same as the population mean. $1$ in this case.

Here is an illustration of ten thousands trials in R from a normal distribution, suggesting no relationship between the sample standard deviation and the difference between the population mean and the sample mean

library(matrixStats)
samplesize <- 25
cases      <- 10000
popmean    <- 25
popsd      <- 1
set.seed(1)
matdat <- matrix(rnorm(ss*cases, mean=popmean, sd=popsd), ncol=ss)
samplemeans <- rowMeans(matdat)
samplesds <- rowSds(matdat)
plot(samplesds ~ samplemeans)

enter image description here

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Standard deviation is $0$ because all the samples are completely random.

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