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Q: Can a standard deviation of a sample be less than a standard deviation of a population?

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A sample of size 100.

No. But a small sample will be a less accurate predictor of the standard deviation of the population due to its size. Another way of saying this: Small samples have more variability of results, sometimes estimates are too high and other times too low. As the sample size gets larger, there's a better chance that your sample will be close to the actual standard deviation of the population.

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.

t-test for means

Yes, a standard deviation can be less than one.

In general, a mean can be greater or less than the standard deviation.

Yes. The standard deviation and mean would be less. How much less would depend on the sample size, the distribution that the sample was taken from (parent distribution) and the parameters of the parent distribution. The affect on the sampling distribution of the mean and standard deviation could easily be identified by Monte Carlo simulation.

No, if the standard deviation is small the data is less dispersed.

There is 1) standard deviation, 2) mean deviation and 3) mean absolute deviation. The standard deviation is calculated most of the time. If our objective is to estimate the variance of the overall population from a representative random sample, then it has been shown theoretically that the standard deviation is the best estimate (most efficient). The mean deviation is calculated by first calculating the mean of the data and then calculating the deviation (value - mean) for each value. If we then sum these deviations, we calculate the mean deviation which will always be zero. So this statistic has little value. The individual deviations may however be of interest. See related link. To obtain the means absolute deviation (MAD), we sum the absolute value of the individual deviations. We will obtain a value that is similar to the standard deviation, a measure of dispersal of the data values. The MAD may be transformed to a standard deviation, if the distribution is known. The MAD has been shown to be less efficient in estimating the standard deviation, but a more robust estimator (not as influenced by erroneous data) as the standard deviation. See related link. Most of the time we use the standard deviation to provide the best estimate of the variance of the population.

Various answers to this question are possible.The mean absolute deviation (MAD) is a measure of the dispersion or spread of a sample or a population. So one of its purposes is as a measure.As such it's an alternative to the standard deviation that is said to be more robust in the sense that the sample MAD can be used to provide more accurate estimates of the population dispersion because it is less sensitive to outliers.Beyond this, some distributions that have no standard deviations do have MADs; for example, the Cauchy. This means that the dispersions of virtually all distributions can be compared in terms of their MADs.Please see the link.

A Chip Company claims that there is 32 oz in every bag of chips with a specified population standard deviation of 1.5. A sample of 40 bags where weighted with an sample mean of 31.4. A consumer feels that this less than what the company claims.H1: μ

An acceptable standard deviation depends entirely on the study and person asking for the study. The smaller the standard deviation, the more acceptable it will be because the less likely there are to be errors.

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