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Yes,
Mean is given by, E(X) sum of samples / no. of samples.
Variance is Var.(X) = E(X^2) - [E(X)]^2.
It is the 1st term which makes the variation of variance independent of mean.
In other words, Variance gives a measure of how far the samples are spread out.
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Does a leptokurtic distribution have a larger variance than Mesokurtic distribution?
yes
Why is standard deviation a better measure of dispersion than variance?
Because it is in same units as the original data. For example, if you have a sample of lengths, all in centimetres, the sample variance will be in units of centrimetres2 which might be more difficult to interpret but the sample standard deviation with be in units of centimetres, which would be relatively easy to intepret with reference to the data.
Can a sample statistic ever be larger than the population parameter?
Yes.
Will margins of error for sample of size 80 be larger or smaller than those for sample size of 40?
They should be smaller for the sample size 80.
Why is the standard deviation of a distribution of means smaller than the standard deviation of the population from which it was derived?
The reason the standard deviation of a distribution of means is smaller than the standard deviation of the population from which it was derived is actually quite logical. Keep in mind that standard deviation is the square root of variance. Variance is quite simply an expression of the variation among values in the population. Each of the means within the distribution of means is comprised of a sample of values taken randomly from the population. While it is possible for a random sample of multiple values to have come from one extreme or the other of the population distribution, it is unlikely. Generally, each sample will consist of some values on the lower end of the distribution, some from the higher end, and most from near the middle. In most cases, the values (both extremes and middle values) within each sample will balance out and average out to somewhere toward the middle of the population distribution. So the mean of each sample is likely to be close to the mean of the population and unlikely to be extreme in either direction. Because the majority of the means in a distribution of means will fall closer to the population mean than many of the individual values in the population, there is less variation among the distribution of means than among individual values in the population from which it was derived. Because there is less variation, the variance is lower, and thus, the square root of the variance - the standard deviation of the distribution of means - is less than the standard deviation of the population from which it was derived.
The sample variance is always smaller than the true value of the population variance is always larger than the true value of the population variance could be smaller equal to or?
yes, it can be smaller, equal or larger to the true value of the population varience.
Is population mean always larger than sample mean?
Yes
Does a leptokurtic distribution have a larger variance than Mesokurtic distribution?
yes
How is it that a random samples gives a fairly accurate representation of public opinion?
The main point here is that the Sample Mean can be used to estimate the Population Mean. What I mean by that is that on average, the Sample Mean is a good estimator of the Population Mean. There are two reasons for this, the first is that the Bias of the estimator, in this case the Sample Mean, is zero. A Bias other than zero overestimates or underestimates the Population Mean depending on its value. Bias = Expected value of estimator - mean. This can be expressed as EX(pheta) - mu (pheta) As the Sample Mean has an expected value (what value it expects to take on average) of itself then the greek letter mu which stands for the Sample Mean will provide a Bias of 0. Bias = mu - mu = 0 Secondly as the Variance of the the Sample Mean is mu/(n-1) this leads us to believe that the Variance will fall as we increase the sample size. Variance is a measure of the dispersion of values collected from the centre of the data. Where the centre of the data is a fixed value equal to the median. Put Bias and Variance together and you get the Mean Squared Error which is the error associated with using an estimator of the Population Mean. The formula for Mean Squared Error = Bias^2 + Variance With our estimator we can see that as the Bias = zero, it has no relevance to the error and as the variance falls as the sample size increases then we can conclude that the error associated with using the sample mean will fall as the sample size increases. Conclusions: The Random Sample of public opinon will on average lead to a true representation of the Population Mean and therefore the random samle you have will represnt the public opinion to a fairly high degree of accuracy. Finally, this degree of accuracy will rise incredibly quickly as the sample size rises thus leading to a very accurate representation (on average)
How do you calculate salary variance?
I believe you are interested in calculating the variance from a set of data related to salaries. Variance = square of the standard deviation, where: s= square root[sum (xi- mean)2/(n-1)] where mean of the set is the sum of all data divided by the number in the sample. X of i is a single data point (single salary). If instead of a sample of data, you have the entire population of size N, substitute N for n-1 in the above equation. You may find more information on the interpretation of variance, by searching wikipedia under variance and standard deviation. I note that an advantage of using the standard deviation rather than variance, is because the standard deviation will be in the same units as the mean.
Why is standard deviation a better measure of dispersion than variance?
Because it is in same units as the original data. For example, if you have a sample of lengths, all in centimetres, the sample variance will be in units of centrimetres2 which might be more difficult to interpret but the sample standard deviation with be in units of centimetres, which would be relatively easy to intepret with reference to the data.
Why would having a larger sample size be a better idea than having a samll sample size when doing an experiment?
less bias and error occur when sample size is larger
Can a sample statistic ever be larger than the population parameter?
Yes.
Will margins of error for sample of size 80 be larger or smaller than those for sample size of 40?
They should be smaller for the sample size 80.
Does a random largely selected sample always give a better estimate of the population than a randomly selected sample?
A larger random sample will always give a better estimate of a population parameter than a smaller random sample.
What is the difference between negative price variance and volume variance?
Negative price variance is when the cost is less than budgeted. Volume variance is a variance in the volume produce.
Can the Variance ever be smaller than standard deviation?
Yes. If the variance is less than 1, the standard deviation will be greater that the variance. For example, if the variance is 0.5, the standard deviation is sqrt(0.5) or 0.707.
