The law of large numbers and the central limit theorem

 Let’s talk about the law of large numbers and the central limit theorem in a way that is easy to understand. As we know that the law of large numbers is a statistical concept that states that as you take more and more samples from a population, the average value of those samples will tend to get closer and closer to the true average value of the population. To put it simply, the more data you collect, the more accurate your estimate of the true population value becomes.

For example, let's say you wanted to estimate the average height of all the people in a city or within a country. You could take a sample of 10 people and calculate their average height. Then, you could take another sample of 10, 100, or more people and calculate their average height. If you keep doing this, taking more and more samples and calculating their average height, you'll start to notice that the average of all your sample averages will tend to get closer and closer to the true average height of the entire population of your target population. There is another example to help us illustrate these concepts further. Let's say you wanted to estimate the average number of hours of sleep that college students get per night. You could take a small sample of 10 students and calculate their average number of hours of sleep. However, this small sample may not be representative of the entire college student population, and you may not get an accurate estimate of the true average. If you were to take a larger sample of 100 students, you'd be more likely to get a better estimate. And if you were to take an even larger sample of 1000 students, your estimate would likely be even more accurate. This is because as you take larger samples, you are more likely to capture the diversity of the population, and your estimate becomes more reliable.

Now, let's move on to the central limit theorem. This theorem states that, regardless of the shape of the original population distribution, the distribution of sample means will tend to be normally distributed as the sample size increases. In other words, if you take a large enough sample size from any population, the distribution of sample means will start to look like a bell curve, with most of the data clustering around the mean value.

To understand this better, let's go back to our example of estimating the average height of all the people in a city. Let's say that the population distribution of heights is not normally distributed, but instead is skewed to the right (there are more people who are taller than average). If you take a small sample size, your sample average might be skewed as well. However, as you take larger and larger sample sizes, the distribution of sample means will start to look more and more like a normal distribution, with most of the sample means clustering around the true population average. Another example for the central limit theorem, let's say you wanted to estimate the average weight of peanuts in a box. If you were to weigh every peanuts in the box, you'd get a good estimate of the true average weight. However, this would be time-consuming and impractical. Instead, you could take a sample of 20 peanuts and calculate their average weight. If you were to repeat this process many times, taking different samples of 100 peanuts each time, you'd find that the distribution of sample means starts to look like a bell curve. Even if the weights of individual peanuts were not normally distributed, the distribution of sample means would tend to be normally distributed as the sample size increases, as per the central limit theorem.

Therefore, to summarize, the law of large numbers tells us that the more data we collect, the more accurate our estimates of population values become, while the central limit theorem tells us that regardless of the shape of the original population distribution, the distribution of sample means will tend to be normally distributed as the sample size increases. I hope these explanations and examples help to clarify the concepts of the law of large numbers and the central limit theorem.

Reference

Yakir, B. (2011). Introduction to statistical thinking (with R, without Calculus). The Hebrew University of Jerusalem, Department of Statistics

The Difference Between The distribution of A Sample and The Sampling Distribution

The distribution of a sample refers to the distribution of values observed in a single sample of data taken from a population. On the other hand, the sampling distribution refers to the distribution of values that would be obtained if we took many random samples from the same population and calculated a statistic( the mean or standard deviation) for each sample. 

For example, suppose we are interested in the proportion of adults in a population who own a smartphone.(Own, or Not) We randomly sample 100 adults from the population and find that 70 of them own a smartphone. The distribution of this sample of 100 adults is the binomial distribution, which describes the probability of obtaining different numbers of successes in a fixed number of trials. On the other hand, the sampling distribution for the proportion of smartphone owners would be the distribution of proportions that we would obtain if we repeated this process of sampling 100 adults and calculating the proportion of smartphone owners many times. In this case, if we assume that the true proportion of smartphone owners in the population is 0.6, then the sampling distribution would also be binomial with mean 0.6 and variance 0.24. However, the shape of the sampling distribution would be different from that of the distribution of the single sample. Specifically, the sampling distribution would be narrower and more symmetric than the distribution of the single sample, reflecting the fact that the variability due to sampling error is reduced when we take larger sample sizes.

To summarize, the distribution of a sample describes the values observed in a single sample of data, while the sampling distribution describes the distribution of a statistic that we would obtain if we took many random samples from the same population.

Critical Questions for Politicians: Analyzing Statistical Claims and Assessing the Impact of Nationwide Education Programs

 Abstract

Most of the time, when politicians make claims that we need to spend a large amount of money to achieve a goal, the claim is often made without legitimate evidence to support a claim that a given program will have a particular result. Let's say that a politician wants to implement a nation-wide education program.  The politician gave four examples of schools that used the program: scores at the schools increased 0.5, 1, 2, and 2.5 points respectively (the nation-wide average of the scores is 70).  The politician gave no additional evidence about the effectiveness of the program. What questions or comments would you have pertaining to the statistical claim made by the politician? You might inquire about the sample, the sampling methods, the full population, the sampling distribution of the mean, and whatever would be useful to more accurately or precisely describe the effectiveness of the program.  

Implementing a nation-wide education program is a complex undertaking that requires careful planning and consideration of various factors such as the educational goals, curriculum, teaching methods, assessment strategies, funding, and teacher training. To implement a program successfully, it is crucial to involve relevant stakeholders such as teachers, school administrators, parents, and education experts in the design and implementation process. Additionally, it is important to conduct a thorough needs assessment and evaluate the effectiveness of the program regularly to ensure that it is achieving its intended goals. It is also important to note that implementing a nation-wide education program can be expensive, and it may require a significant amount of funding from the government or other sources. As such, it is important to weigh the potential benefits of the program against its costs to determine whether it is a worthwhile investment. Since, the politician gave no additional evidence about the effectiveness of these programs, it is hard to say that the claim is actually helpful.

Based on the information, I think there are some questions and comments that could be relevant to the statistical claim made by the politician:

i. What was the sample size for each of the four schools, and how were the schools selected?

The sample size and sampling methods can have a significant impact on the result of any study, including this one. If the sample size is too small, the observed increase in scores may not be representative of the larger population of schools, and the results may not be statistically significant. Similarly, the sampling methods used to select the schools can also affect the results. If the sample is selected randomly, it is more likely to be representative of the population of schools, whereas biased sampling methods can introduce systematic errors and lead to inaccurate conclusions. Moreover, other factors such as the characteristics of the population of schools, the geographic location, and socio-economic status, can also affect the effectiveness of the program, and these factors should be controlled for or adjusted for in the analysis.

ii. Were the schools similar in terms of student demographics, teacher quality, or other relevant factors?

If the schools that used the program had significantly different student demographics compared to the schools that did not use the program, this could affect the effectiveness of the program. Students from different socioeconomic backgrounds, for example, may respond differently to the program. Therefore, the program may not be as effective for some groups of students. Similarly, if the schools that used the program had higher-quality teachers, more resources, or a better learning environment compared to the schools that did not use the program, this could also affect the results. The program may be more effective in schools with higher-quality teachers, more resources, or better learning environments, and the observed increase in scores may not be due solely to the program. To address these potential confounding factors, the study should carefully control for any relevant variables that may affect the outcome. One possible approach is to match the schools that used the program with similar schools that did not use the program based on relevant variables and compare the changes in scores before and after the program implementation. This is to isolate the effect of the program from other potential confounding factors and provide more reliable evidence of the program's effectiveness.

iii. What was the standard deviation of the scores in each school, and was there a significant difference between the pre- and post-program scores? Are there any possible errors for each of the reported score increases?

Obviously, the standard deviation of the scores in each school could potentially affect the results. If the standard deviation is large, this could indicate that there is a wide variation in scores within the school, and the observed increase in scores may not be statistically significant or representative of the entire population of students in the school. Alternatively, if the standard deviation is small, this may suggest that the increase in scores is more reliable and representative of the population of students in the school. Moreover, the difference between the pre- and post-program scores is also an important factor to consider. If the pre-program scores were already high, the observed increase may not be significant, or the program may not have had as much room for improvement. Alternatively, if the pre-program scores were low, the observed increase may be more significant and indicate a greater potential impact of the program. 

iv. What is the distribution of the score increases across all schools that used the program, and how does it compare to the national mean?

If the majority of schools that used the program showed a significant increase in scores, this would suggest that the program is effective and has the potential to improve student learning outcomes nationwide. However, if the increase in scores is only observed in a small number of schools or if the majority of schools show little to no improvement, this may suggest that the program is not as effective as claimed. Additionally, we can compare the distribution of score increases to the national average. For example, if the distribution of score increases across all schools that used the program is significantly higher than the national average, this would suggest that the program is having a positive impact on student learning outcomes. On the other hand, if the distribution of score increases is similar to or lower than the national average, this may suggest that the program is not as effective as expected or that other factors are contributing to the observed increase in scores.

v. Causation. Is there any evidence to suggest that the score increases were due to factors other than the education program, such as changes in curriculum or testing methods?

Causation is a critical issue that needs to be addressed when assessing the effectiveness of any program, including an education program. To establish causation, we need to demonstrate that the observed increase in scores is a direct result of the education program and not due to other factors. One way to do this is to conduct a randomized controlled trial where schools are randomly assigned to either a treatment group. This design ensures that any differences in the outcome between the two groups can be attributed to the education program and not to other factors.

vi. How long did the program last at each of the schools, and what was the frequency of the program's implementation?

If the program was implemented at each of the schools for a longer duration and with a higher frequency, it is more likely to have a greater impact on student learning outcomes. Conversely, if the program was implemented for a shorter duration and with a lower frequency, it may not have had enough time to produce a meaningful effect on student scores. For example, if the program was implemented for only a few weeks or months, it may not have been enough time for the students to fully benefit from the program. Similarly, if the program was only implemented sporadically or infrequently, it may not have had a consistent impact on student learning outcomes.

vii. Other than statistic, what is the cost of implementing the program on a nationwide scale, and how does it compare to the expected benefits in terms of improved scores?

Assessing the cost-effectiveness of the education program is essential when considering its implementation on a nationwide scale. Once we have an estimate of the total cost of the program, we can compare it with the expected benefits in terms of improved scores to assess the program's cost-effectiveness. For instance, we can estimate the potential increase in student scores across the nation and translate it into economic benefits. We can then compare these benefits to the program's cost to determine whether the program is a cost-effective investment. 

Conclusion

Based on the limited information provided by the politician, it is difficult to draw a conclusive answer regarding whether the program will increase scores nationwide. Additional evidence and analysis would be necessary to determine the true effectiveness of the program. 

Confused with the z-score and the value x in the normal distribution? Let's figure it out

 In this week's study, I found that the R functions qnorm() and pnorm() are sometimes confused with the z-score and the value x in the normal distribution. It is important to understand the differences between these concepts to use them correctly in statistical analysis. The z-score is a standardized score that represents the number of standard deviations a data point is from the mean of a normal distribution. The z-score is calculated as: z = (x - μ) / σ  , where x is the data point, μ is the mean of the distribution, and σ is the standard deviation of the distribution.

On the other hand, the qnorm() function in R is used to calculate the inverse of the cumulative distribution function of the normal distribution, known as the quantile function. The qnorm() function returns the z-score that corresponds to a given percentile or probability in a normal distribution with a specified mean and standard deviation. Similarly, the pnorm() function in R calculates the cumulative distribution function of the normal distribution. The pnorm() function returns the probability that a random variable from a normal distribution with a specified mean and standard deviation is less than or equal to a specified value x. While these concepts are related, they are not interchangeable. It is important to understand which concept you are working with and use the appropriate function or formula to calculate the desired value.

Mathematical models, Making approximations when modeling real data using the normal distribution

 Mathematical models are abstract representations of real-world phenomena that use mathematical language and symbols to describe and quantify the behavior of a system or process. A good mathematical model should be based on accurate and relevant data, incorporate the relevant variables and parameters that affect the system or process being studied, and be able to make predictions or simulate outcomes under different scenarios. Mathematical models can be used to test hypotheses, make predictions, optimize processes, and inform decision-making in a wide range of fields, including physics, biology, economics, engineering, and social sciences.

There are several reasons why researchers might make approximations when modeling real data using the normal distribution:

1. Convenience: The normal distribution is a well-known distribution with many properties that make it easy to work with. For example, it has a simple mathematical formula, and its parameters can be estimated easily from data.
2. Assumptions: Many statistical models, including the normal distribution, are based on certain assumptions about the data. For example, the normal distribution assumes that the data is continuous and that the mean and variance are the only important features of the data. While these assumptions may not always hold true in reality, they can still provide a good approximation of the data in many cases.
3. Interpretability: The normal distribution has a clear interpretation in terms of the mean and standard deviation, which can help researchers to communicate their findings to others.

However, there are situations when researchers should not use the normal distribution to model their data. For example, if the data is strongly skewed or has outliers, the normal distribution may not be appropriate. In such cases, researchers may need to use a different distribution, such as the lognormal distribution or the t-distribution, that can better capture the characteristics of the data. Another example is when the data is discrete or categorical, such as the number of people in a household or the type of flower in a field, in which case a discrete probability distribution such as the Poisson or binomial distribution may be more appropriate.

As an example, consider the distribution of incomes in a given population. While the normal distribution may provide a good approximation for many populations, it may not be appropriate for populations with a large number of extremely wealthy individuals, which can result in a highly skewed distribution. In such cases, researchers may need to use a different distribution, such as the lognormal distribution, to better model the data.