What Happens To The Mean When The Sample Size Increases
vii.2: Using the Cardinal Limit Theorem
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Examples of the Central Limit Theorem
Police force of Large Numbers
The police force of big numbers says that if you take samples of larger and larger size from any population, so the mean of the sampling distribution, \(\mu_{\overline 10}\) tends to get closer and closer to the true population mean, \(\mu\). From the Central Limit Theorem, we know that as \(due north\) gets larger and larger, the sample ways follow a normal distribution. The larger northward gets, the smaller the standard divergence of the sampling distribution gets. (Remember that the standard deviation for the sampling distribution of \(\overline 10\) is \(\frac{\sigma}{\sqrt{n}}\).) This means that the sample hateful \(\overline ten\) must be closer to the population mean \(\mu\) as \(n\) increases. We tin can say that \(\mu\) is the value that the sample means arroyo every bit n gets larger. The Central Limit Theorem illustrates the police of large numbers.
This concept is so important and plays such a critical office in what follows it deserves to exist developed further. Indeed, at that place are two disquisitional issues that flow from the Central Limit Theorem and the application of the Law of Large numbers to it. These are
- The probability density function of the sampling distribution of means is commonly distributed regardless of the underlying distribution of the population observations and
- standard deviation of the sampling distribution decreases as the size of the samples that were used to summate the ways for the sampling distribution increases.
Taking these in order. It would seem counterintuitive that the population may have any distribution and the distribution of means coming from it would be commonly distributed. With the use of computers, experiments tin can be false that show the procedure by which the sampling distribution changes as the sample size is increased. These simulations evidence visually the results of the mathematical proof of the Central Limit Theorem.
Hither are three examples of very different population distributions and the evolution of the sampling distribution to a normal distribution as the sample size increases. The top panel in these cases represents the histogram for the original data. The three panels testify the histograms for ane,000 randomly drawn samples for different sample sizes: \(n=10\), \(due north= 25\) and \(n=50\). Equally the sample size increases, and the number of samples taken remains abiding, the distribution of the 1,000 sample means becomes closer to the smooth line that represents the normal distribution.
Figure \(\PageIndex{3}\) is for a normal distribution of private observations and we would await the sampling distribution to converge on the normal quickly. The results show this and show that even at a very small sample size the distribution is close to the normal distribution.
Figure \(\PageIndex{iv}\) is a uniform distribution which, a fleck amazingly, speedily approached the normal distribution even with merely a sample of ten.
Effigy \(\PageIndex{5}\) is a skewed distribution. This concluding 1 could exist an exponential, geometric, or binomial with a small probability of success creating the skew in the distribution. For skewed distributions our intuition would say that this volition take larger sample sizes to motion to a normal distribution and indeed that is what nosotros notice from the simulation. Yet, at a sample size of l, not considered a very large sample, the distribution of sample ways has very incomparably gained the shape of the normal distribution.
The Primal Limit Theorem provides more than the proof that the sampling distribution of means is normally distributed. It also provides united states of america with the mean and standard difference of this distribution. Further, every bit discussed in a higher place, the expected value of the mean, \(\mu_{\overline{x}}\), is equal to the hateful of the population of the original data which is what we are interested in estimating from the sample we took. We have already inserted this conclusion of the Primal Limit Theorem into the formula we use for standardizing from the sampling distribution to the standard normal distribution. And finally, the Central Limit Theorem has besides provided the standard divergence of the sampling distribution, \(\sigma_{\overline{ten}}=\frac{\sigma}{\sqrt{n}}\), and this is critical to have to calculate probabilities of values of the new random variable, \(\overline x\).
Figure \(\PageIndex{half dozen}\) shows a sampling distribution. The mean has been marked on the horizontal axis of the \(\overline 10\)'southward and the standard departure has been written to the right above the distribution. Notice that the standard deviation of the sampling distribution is the original standard difference of the population, divided past the sample size. We have already seen that equally the sample size increases the sampling distribution becomes closer and closer to the normal distribution. Every bit this happens, the standard deviation of the sampling distribution changes in another fashion; the standard divergence decreases as \(n\) increases. At very very big \(n\), the standard deviation of the sampling distribution becomes very small and at infinity it collapses on top of the population mean. This is what information technology means that the expected value of \(\mu_{\overline{ten}}\) is the population hateful, \(\mu\).
At not-farthermost values of \(n\), this relationship between the standard deviation of the sampling distribution and the sample size plays a very of import role in our ability to estimate the parameters we are interested in.
Figure \(\PageIndex{7}\) shows three sampling distributions. The only change that was made is the sample size that was used to get the sample means for each distribution. Equally the sample size increases, \(n\) goes from x to 30 to fifty, the standard deviations of the respective sampling distributions decrease because the sample size is in the denominator of the standard deviations of the sampling distributions.
The implications for this are very important. Figure \(\PageIndex{8}\) shows the effect of the sample size on the conviction we will accept in our estimates. These are 2 sampling distributions from the aforementioned population. One sampling distribution was created with samples of size ten and the other with samples of size 50. All other things abiding, the sampling distribution with sample size 50 has a smaller standard deviation that causes the graph to be higher and narrower. The important effect of this is that for the same probability of 1 standard departure from the mean, this distribution covers much less of a range of possible values than the other distribution. I standard deviation is marked on the \(\overline X\) axis for each distribution. This is shown past the 2 arrows that are plus or minus one standard divergence for each distribution. If the probability that the true mean is one standard deviation away from the mean, then for the sampling distribution with the smaller sample size, the possible range of values is much greater. A simple question is, would you lot rather have a sample hateful from the narrow, tight distribution, or the flat, wide distribution every bit the guess of the population hateful? Your respond tells u.s. why people intuitively volition e'er cull data from a big sample rather than a small-scale sample. The sample mean they are getting is coming from a more compact distribution. This concept will be the foundation for what will be called level of confidence in the next unit of measurement.
What Happens To The Mean When The Sample Size Increases,
Source: https://stats.libretexts.org/Bookshelves/Applied_Statistics/Book%3A_Business_Statistics_(OpenStax)/07%3A_The_Central_Limit_Theorem/7.02%3A_Using_the_Central_Limit_Theorem
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