In probability theory, the central limit theorem (CLT) states that the distribution of a sample variable approximates a normal distribution (i.e., a “bell curve”) as the sample size becomes larger, assuming that all samples are identical in size, and regardless of the population's actual distribution shape. Put another way, CLT is a statistical premise that, given a sufficiently large sample size from a population with a finite level of variance, the mean of all sampled variables from the same population will be approximately equal to the mean of the whole population. Furthermore, these samples approximate a normal distribution, with their variances being approximately equal to the variance of the population as the sample size gets larger, according to the law of large numbers. Although this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1930, when noted Hungarian mathematician George Pólya dubbed it the central limit theorem.
According to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question, as the sample size increases, notwithstanding the actual distribution of the data. In other words, the data is accurate whether the distribution is normal or aberrant. As a general rule, sample sizes of around 30-50 are deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. Therefore, the more samples one takes, the more the graphed results take the shape of a normal distribution. Note, however, that the central limit theorem will still be approximated in many cases for much smaller sample sizes, such as n=8 or n=5. The central limit theorem is often used in conjunction with the law of large numbers, which states that the average of the sample means and standard deviations will come closer to equaling the population mean and standard deviation as the sample size grows, which is extremely useful in accurately predicting the characteristics of populations. Investopedia / Sabrina Jiang The central limit theorem is comprised of several key characteristics. These characteristics largely revolve around samples, sample sizes, and the population of data.
The CLT is useful when examining the returns of an individual stock or broader indices, because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors of all types rely on the CLT to analyze stock returns, construct portfolios, and manage risk. Say, for example, an investor wishes to analyze the overall return for a stock index that comprises 1,000 equities. In this scenario, that investor may simply study a random sample of stocks to cultivate estimated returns of the total index. To be safe, at least 30-50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold. Furthermore, previously selected stocks must be swapped out with different names to help eliminate bias. The central limit theorem is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally-distributed in most cases. This allows for easier statistical analysis and inference. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over a period of time. A sample size of 30 is fairly common across statistics. A sample size of 30 often increases the confidence interval of your population data set enough to warrant assertions against your findings. The higher your sample size, the more likely the sample will be representative of your population set. The central limit theorem doesn't have its own formula, but it relies on sample mean and standard deviation. As sample means are gathered from the population, standard deviation is used to distribute the data across a probability distribution curve. CO-6: Apply basic concepts of probability, random variation, and commonly used statistical probability distributions. LO 6.21: Apply the sampling distribution of the sample proportion (when appropriate). In particular, be able to identify unusual samples from a given population. Approximately 60% of all part-time college students in the United States are female. (In other words, the population proportion of females among part-time college students is p = 0.6.) What would you expect to see in terms of the behavior of a sample proportion of females (p-hat) if random samples of size 100 were taken from the population of all part-time college students? As we saw before, due to sampling variability, sample proportion in random samples of size 100 will take numerical values which vary according to the laws of chance: in other words, sample proportion is a random variable. To summarize the behavior of any random variable, we focus on three features of its distribution: the center, the spread, and the shape. Based only on our intuition, we would expect the following: Center: Some sample proportions will be on the low side — say, 0.55 or 0.58 — while others will be on the high side — say, 0.61 or 0.66. It is reasonable to expect all the sample proportions in repeated random samples to average out to the underlying population proportion, 0.6. In other words, the mean of the distribution of p-hat should be p. Spread: For samples of 100, we would expect sample proportions of females not to stray too far from the population proportion 0.6. Sample proportions lower than 0.5 or higher than 0.7 would be rather surprising. On the other hand, if we were only taking samples of size 10, we would not be at all surprised by a sample proportion of females even as low as 4/10 = 0.4, or as high as 8/10 = 0.8. Thus, sample size plays a role in the spread of the distribution of sample proportion: there should be less spread for larger samples, more spread for smaller samples. Shape: Sample proportions closest to 0.6 would be most common, and sample proportions far from 0.6 in either direction would be progressively less likely. In other words, the shape of the distribution of sample proportion should bulge in the middle and taper at the ends: it should be somewhat normal. Comment:
The purpose of the next video and activity is to check whether our intuition about the center, spread and shape of the sampling distribution of p-hat was correct via simulations. Video: Simulation #1 (p-hat) (4:13) Did I Get This?: Simulation #1 (p-hat) At this point, we have a good sense of what happens as we take random samples from a population. Our simulation suggests that our initial intuition about the shape and center of the sampling distribution is correct. If the population has a proportion of p, then random samples of the same size drawn from the population will have sample proportions close to p. More specifically, the distribution of sample proportions will have a mean of p. We also observed that for this situation, the sample proportions are approximately normal. We will see later that this is not always the case. But if sample proportions are normally distributed, then the distribution is centered at p. Now we want to use simulation to help us think more about the variability we expect to see in the sample proportions. Our intuition tells us that larger samples will better approximate the population, so we might expect less variability in large samples. In the next walk-through we will use simulations to investigate this idea. After that walk-through, we will tie these ideas to more formal theory. Video: Simulation #2 (p-hat) (4:55) Did I Get This?: Simulation #2 (p-hat) The simulations reinforced what makes sense to our intuition. Larger random samples will better approximate the population proportion. When the sample size is large, sample proportions will be closer to p. In other words, the sampling distribution for large samples has less variability. Advanced probability theory confirms our observations and gives a more precise way to describe the standard deviation of the sample proportions. This is described next. The Sampling Distribution of the Sample ProportionIf repeated random samples of a given size n are taken from a population of values for a categorical variable, where the proportion in the category of interest is p, then the mean of all sample proportions (p-hat) is the population proportion (p). As for the spread of all sample proportions, theory dictates the behavior much more precisely than saying that there is less spread for larger samples. In fact, the standard deviation of all sample proportions is directly related to the sample size, n as indicated below. Since the sample size n appears in the denominator of the square root, the standard deviation does decrease as sample size increases. Finally, the shape of the distribution of p-hat will be approximately normal as long as the sample size n is large enough. The convention is to require both np and n(1 – p) to be at least 10. We can summarize all of the above by the following: Let’s apply this result to our example and see how it compares with our simulation. In our example, n = 25 (sample size) and p = 0.6. Note that np = 15 ≥ 10 and n(1 – p) = 10 ≥ 10. Therefore we can conclude that p-hat is approximately a normal distribution with mean p = 0.6 and standard deviation (which is very close to what we saw in our simulation). Comment:
Learn by Doing: Sampling Distribution of p-hat Did I Get This?: Sampling Distribution of p-hat If a sampling distribution is normally shaped, then we can apply the Standard Deviation Rule and use z-scores to determine probabilities. Let’s look at some examples. A random sample of 100 students is taken from the population of all part-time students in the United States, for which the overall proportion of females is 0.6. (a) There is a 95% chance that the sample proportion (p-hat) falls between what two values? First note that the distribution of p-hat has mean p = 0.6, standard deviation and a shape that is close to normal, since np = 100(0.6) = 60 and n(1 – p) = 100(0.4) = 40 are both greater than 10. The Standard Deviation Rule applies: the probability is approximately 0.95 that p-hat falls within 2 standard deviations of the mean, that is, between 0.6 – 2(0.05) and 0.6 + 2(0.05). There is roughly a 95% chance that p-hat falls in the interval (0.5, 0.7) for samples of this size. (b) What is the probability that sample proportion p-hat is less than or equal to 0.56? To find we standardize 0.56 into a z-score by subtracting the mean and dividing the result by the standard deviation. Then we can find the probability using the standard normal calculator or table. To see the impact of the sample size on these probability calculations, consider the following variation of our example. A random sample of 2500 students is taken from the population of all part-time students in the United States, for which the overall proportion of females is 0.6. (a) There is a 95% chance that the sample proportion (p-hat) falls between what two values? First note that the distribution of p-hat has mean p = 0.6, standard deviation and a shape that is close to normal, since np = 2500(0.6) = 1500 and n(1 – p) = 2500(0.4) = 1000 are both greater than 10. The Standard Deviation Rule applies: the probability is approximately 0.95 that p-hat falls within 2 standard deviations of the mean, that is, between 0.6 – 2(0.01) and 0.6 + 2(0.01). There is roughly a 95% chance that p-hat falls in the interval (0.58, 0.62) for samples of this size. (b) What is the probability that sample proportion p-hat is less than or equal to 0.56? To find we standardize 0.56 to into a z-score by subtracting the mean and dividing the result by the standard deviation. Then we can find the probability using the standard normal calculator or table. Comment:
Applet: Sampling Distribution for a Sample Proportion |
What effect does the sample size have on the standard deviation of all possible sample means
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