Question 1134399: Hi,Please help
The following results were obtained in the replicate analysis of a blood sample for its lead(Pb) content 0.752, 0.756, 0.752, 0.751 and 0.760 parts per million(ppm) Pb.
Calculate the (i) mean, (ii) spread, (iii) standard deviation, (iv) variance, (v) relative standard deviation in parts per thousand (vi) coefficient of variation for the data set.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the data elements are:
.752, .756, .752, .751, .760.
add them all up and you get a total of 3.771.
divide by the number of elements to get 3.771 / 5 = .7542.
that's the mean, or the average of all scores.
you divided by 5 because that's the number of data elements in the data set.
now take each individual element and subtract it from the mean and then square it.
you will get:
.752 - .7542 = -.0022 squared = 4.84 * 10^-6 = .00000484
.756 - .7542 = .0018 squared = 3.24 * 10^-6 = .00000324
.752 - .7542 = -.0022 squared = 4.84 * 10^-6 = .00000484
.751 - .7542 = -.0032 squared = 1.024 * 10^-5 = .00001024
.760 - .7542 = .0058 squared = 3.364 * 10^-5 = .00003364
add them all up and you get a total of 5.68 * 10^-5 = .0000568
that's the sum of squares.
divide that by the total number of elements minus 1 to get .0000568 / 4 = 1.42 * 10^-5 = .0000142.
that's the sample variance.
take the square root of the sample variance to get .0037682887.
that's the sample standard deviation.
the spread of the data is the largest element minus the smallest element.
that would be equal to .760 - .751 = .09
the coefficient of variation for the sample is the sample standard deviation multiplied by 100 and then divided by the sample mean.
for this data set, that would be equal to .0037682887 * 100 / .7542 = .4996405113%.
the relative standard deviation sample data set is the sample standard deviation multiplied by 100 and then divided by the absolute value of the sample mean.
for this data set, that would be equal to .0037682887 * 100 / |.7542| = .4996405113%.
the relative standard deviation and the coefficient of variation are the same if the mean is positive and different if the mean is negative.
the relative standard deviation is always positive while the coefficient of variation can be negative or positive.
that's reflected in the formula since the relativfe standard deviation is divided by the absolute value of the mean while the coefficient of variation is divided by the mean.
the sum of squares is the wsame whether you're dealing with a population or a sample.
the variance for the sample is created by dividing the sum of squares by 1 less than the number of elements in the data set.
the population variance is created by dividing the sum of squares by the number of elements in the population.
the reason for that is that it was discovered that the standard deviation of samples with smaller number of data elements in the data set were understating the standard deviation of the population.
dividing by 1 less than the number of elements makes the standard deviation of the small samples larger.
as the number of elements in the sample data set gets larger, the relative effect of dividing by 1 less than the number of elements versus dividing by the number of elements gets smaller and so the difference is not noted when the samples contain larger number of data elements.
here's a reference on relative standard deviation and coefficient of variation.
https://www.statisticshowto.datasciencecentral.com/relative-standard-deviation/
here's a reference on variance and standard deviation.
http://davidmlane.com/hyperstat/A16252.html
there is an alternate formula to derive the sum of suares.
the regular formula is sum of squares equals sum of (x - m)^2
the alternate formula is sum of squares equals sum of x^2 - (sum of x)^2 / n
recall that sum of (x - m)^2 = .0000568
the sum of x^2 is equal to the sum of:
.752^2 = .565504
.756^2 = .571536
.752^2 = .565504
.751^2 = .564001
.760^2 = .5776
which is equal to 2.844145
(sum of x)^2 / n is equal to 3.771^2 / 5 = 2.8440882
sum of (x^2) - (sum of x)^2 / n is equal to 2.844145 - 2.8440882 = .0000568.
that's the same as we got before using sum of (x-m)^2.
bottomline is that the sum of squares can be derived from the following two formulas that give you the same answer.
sum of squares equals sum of (x-m)^2
sum of squares equals sum of x^2 minus (sum of x)^2 / n
the advantage of the second formula is that you don't have to calculate the mean in order to calculate the sum of squares.
|
|
|
