Statistical Simulation: Introduction and Issues (2012)

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In this article, you will learn about statistical simulation introduction, use in various fields, and issues.

Simulation is used before an existing system is altered or a new system is built, to reduce the chances of failure to meet specifications, eliminate unforeseen bottlenecks, prevent under or over-utilization of resources, and optimize system performance. Simulation is used in many contexts, such as simulation of technology for performance optimization, safety engineering, training testing, education, and video games. Often, computer experiments are used to study simulation models. Models are simulated versions/results.

Uses of Statistical Simulations

Statistical simulations are widely used in many fields:

  • Science: Scientists use statistical simulations to model complex systems, such as the climate or the spread of disease.
  • Business: Businesses use statistical simulations to forecast sales, evaluate the risks of new investments, and design logistics networks.
  • Government: Governments use simulations to model the effects of economic policies, assess the risks of natural disasters, and plan for future events.
  • Gambling: Casinos use simulations to design games that are fair and profitable.

Statistical Simulation depends on unknown (or external/ impositions/ factors) parameters and statistical tools depend on estimates. In statistics, simulation is used to assess the performance of a method, typically when there is a lack of theoretical background. With simulations, the statistician knows and controls the truth.

Monte Carlo Simulation Application: Statistical Simulations

Statistical Assumptions about Simulated Data

In simulation, data is generated artificially to test out a hypothesis or statistical method. Whenever a new statistical method is developed (or used), some assumptions need to be tested and verified (or confirmed). Statisticians use simulated data to test these assumptions.

  • The simulation follows finite sample properties (have to specify $n$)
  • The reasoning of statistical simulation can’t be proofed mathematically)
  • Simulation is used to illustrate things.
  • Simulation is used to check the validity of methods.
  • Simulation is a technique of representing the real world via a computer program.
  • A simulation is an act of initiating the behavior of some situation or some process utilizing something suitably analogous. (especially for study or some personal training)
  • A simulation is a representation of something (usually on a smaller scale).
  • Simulation is the act of giving a false/artificial appearance.

In summary, statistical simulation is a technique used to imitate the behavior of a system or process under various conditions. It involves creating a computer model of the system and running the model repeatedly with different inputs. The outputs of the model are then analyzed to learn about the behavior of the real system.

Statistical Simulation

Issues In Statistical Simulation

  • What distribution does the random variable have?
  • How do we generate these random variables for simulation?
  • How do we analyze the output of simulations?
  • How many simulation runs do we need?
  • How do we improve the efficiency of the simulation?

FAQS about Statistical Simulations

  1. What is meant by simulation in statistics?
  2. What random data is generated using simulation?
  3. What are the uses of simulations?
  4. What are the issues in Statistical simulations?
  5. What are statistical assumptions about generated data?

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Matlab as a Calculator

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MATLAB stands for “Matrix Laboratory” and is an interactive, matrix-based system and fourth-generation programming language from MathWorks Inc., which is a mathematics software company. Matlab helps to perform statistical analysis and gives the user complete freedom to implement specific algorithms and perform complex, custom-tailored operations.

Matlab has a command-driven approach. Commands with appropriate arguments are written after the MATLAB command prompt >>. The MATLAB program provides the user with a convenient environment for performing many types of calculations. This introduction to MATLAB will help users understand its importance and variety of applications in different scientific fields.

Matlab as a Calculator

Matlab has three primary windows.

1) Command windows
2) Graphics Windows
3) Edit Windows used to write M-Files

The common way to operate MATLAB is to enter commands in the command window.

Matlab as a Calculator

>> 55 – 16
ans = 39
>> ans + 11
ans =50

Matlab assigns the results ans whenever you do not explicitly assign the calculations to a chosen variable.

>> a = 4                   % assigns a scalar quantity to a
>> a                         % Prints the scalar quantity in command windows
>> a = 4                   % suppressed echo printing
>> a =4; A=6; x=1;  % multiple variable definition

Note: MATLAB treats names as Case-Sensitive.

>> format long
>> pi
>> format short
>> pi
Matlab as Calculator

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Measures of Central Tendency

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The median is one of the three main measures of central tendency, alongside the mean and mode. It represents the middle value of an ordered dataset. It is a powerful and reliable summary statistic and widely used, especially in real-life scenarios where data is skewed or contains outliers. Unlike the mean, the median is not affected by extreme values, which makes it incredibly useful in various fields. For the formula of the median, read the post: formula of median and definition.

When the Median is Preferred over the Mean

Question: What is a measure of central tendency, and what are the common measures of central tendency? Also, when is the median preferred over the mean?

A measure of central tendency is the single numerical value considered most typical of the values of a quantitative variable.

The most common measure of central tendency is the mode (i.e., the most frequently occurring number)

The median (i.e., the middle point or fiftieth percentile), and the mean (i.e., the arithmetic average).

The median is preferred over the mean when the numbers are highly skewed (i.e., non-normally distributed).

Measures of Central Tendency

Importance of Measures of Central Tendencies

Since measures of central tendency condense a bunch of information into a single, digestible value that represents the center of the data, this makes measures of central tendencies important for several reasons:

  • Summarizing data: Instead of listing every data point, one can use a central tendency measure to get a quick idea of what is typical in the data set.
  • Comparisons: By computing central tendency measures for different groups or datasets, one can easily compare them to see if there are any differences.
  • Decision making: Central tendency measures can help to make wise decisions. For instance, knowing the average income in an area can help set prices. Imagine an organization is analyzing customer purchases. Knowing the average amount spent can help them tailor promotions or target specific customer groups.
  • Identifying trends: Measures of central tendencies may help in observing the trend over time. This can be done by using different visualizations to see if there are any trends, like a rise in average house prices.

However, it is very important to understand these Measures of Central Tendency (mean, median, mode). Each measure of central tendency has its strengths and weaknesses. Choosing the right measure of central tendency depends on the kind of data and what one’s interest is to extract from and try to understand.

Real-Life Examples and Uses of Median

  • Income & Salaries: The Median is used to represent the average income of a population more accurately. It is because A few ultra-rich individuals can skew the mean income upward. The median gives a more realistic picture of what a typical person earns. Example: If most people earn around $40,000–$60,000, but a few CEOs earn $10 million or more, the median income might be $55,000 while the mean income could be $95,000 — misleading!
  • Education (Test/ Exame Scores): The median can be used to summarize exam results or performance data. A few very low or very high scores can distort the mean. For example, if most students score between 70 and 90, but a few score 10 or 100, the measure of central tendency, the median score, gives a better sense of central performance.
  • Real Estate (Home Prices): Reporting the median home price is common in real estate. Why Median? It avoids distortion from a few very expensive or very cheap homes. For example, A city may have a median home price of $350,000, even if some luxury homes cost $5 million.
  • Sports (Player Performance): To report median stats like race times, goals scored, or batting averages. To avoid skewed data from one amazing or terrible performance. For example, a runner’s median race time over 10 races can better reflect consistency.
  • Healthcare (Medical Test Results): Reporting the median wait time in hospitals or median survival time in clinical trials may be beneficial. This is because medical data often contains outliers or skewed distributions. For example, if most patients wait 30 minutes, but a few wait 5 hours, the measure of central tendency, the median wait time, might be 35 minutes, while the mean could be misleadingly high.
  • Customer Feedback (Review Rating): Median star rating for products or services. Filters out extremely negative or overly positive outliers. For example, if ratings are 1, 5, 5, 5, and 1, the mean is 3.4 but the median is 5, better reflecting the typical rating.
  • Transportation (Travel Times): Apps like Google Maps or Waze often use median travel times to reflect a more realistic average, ignoring rare traffic jams or super fast times. For example, the median commute time may be 25 minutes, even if a few people experience 60-minute delays.

Summary

Scenario/ Use CaseVariableWhy Median should be used
Income reportsSalaryAvoids distortion by billionaires
House pricesReal estate valuesNeutralizes luxury properties
ER performancePatient wait timesFilters extreme delays
Test scoresExam performanceReduces skew from outliers
Travel timesCommute estimatesReflects normal travel conditions
Product reviewsUser ratingsBalances biased reviews
Statistics Help measures of central tendency

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