The Graduate Record Examinations (GRE) is a standardized test that is an admissions requirement for most graduate schools in the United States. It measures a candidate's skills in analytical writing, verbal reasoning, and quantitative reasoning. Understanding how GRE scores are calculated is vital in helping test-takers strategize and prepare effectively. Let's delve into the process in detail.
1. Overview of GRE Score Calculation
The GRE consists of three scored sections: Verbal Reasoning (130-170 score scale), Quantitative Reasoning (130-170 score scale), and Analytical Writing (0-6 score scale). The Verbal Reasoning and Quantitative Reasoning sections follow a computer-adaptive by section (CABS) model, meaning the performance in the first section of each dictates the difficulty level of the second section.
2. Verbal and Quantitative Reasoning Scores
The Verbal Reasoning and Quantitative Reasoning sections each consist of two sections. The raw score, which is simply the number of questions you answered correctly, is calculated for each of these sections.
The first section of both the Verbal and Quantitative Reasoning sections is of medium difficulty. Based on your performance on this first section, you are then directed to a second section of a difficulty level that matches your ability as determined from your performance on the first section.
Scores for the Verbal and Quantitative sections are determined by a raw score, which is the total number of questions answered correctly across the two sections. This raw score is then converted into a scaled score through a process called equating, which accounts for minor variations in difficulty between different test editions. The scaled score is what you receive in your final GRE score report.
3. Analytical Writing Scores
The Analytical Writing section consists of two essays: an "Analyze an Issue" task and an "Analyze an Argument" task. Each essay is scored by at least two readers on a 0-6 point scale in half-point increments.
For each essay, if the two scores differ by less than one point, the scores are averaged to obtain the final score. If the scores differ by more than one point, the essay goes to a third reader, and the two scores that are closest are averaged to determine the final score.
4. Experimental Section
The GRE also contains an additional unscored Experimental section, which can be either Verbal or Quantitative. This section does not count toward your final score; it's used by ETS (the organization that administers the GRE) to test out questions for use in future tests.
5. Final Score Reports
Your official GRE score report includes your scores for the Verbal Reasoning, Quantitative Reasoning, and Analytical Writing sections. It also includes percentile ranks, which indicate the percentage of test takers who scored lower than you.
GRE Scoring Example
A concrete example might make the GRE scoring process clearer. Let's imagine a test-taker named Jane, who is taking the GRE General Test.
Verbal and Quantitative Reasoning Scores
Jane answers 15 out of 20 questions correctly on the first Verbal Reasoning section. Given this performance, the GRE software decides to give her a second Verbal Reasoning section of medium difficulty. On this section, she gets 16 out of 20 correct.
Her raw score for Verbal Reasoning is thus 31 (15 from the first section + 16 from the second section). This raw score is then converted into a scaled score through equating. Equating is a process that accounts for minor differences in difficulty between test versions. So, Jane might end up with a final Verbal Reasoning score of, say, 155.
The same process applies to the Quantitative Reasoning section. If Jane answers 17 out of 20 questions correctly in the first section and 18 out of 20 in the second section, her raw Quantitative score would be 35. Let's say this converts to a scaled score of 160.
Analytical Writing Scores
For the Analytical Writing section, Jane writes two essays.
Her first essay (Analyze an Issue) receives scores of 5.0 and 4.5 from the two initial readers. Since the two scores are within one point of each other, they are averaged to give a final score of 4.75 for this essay.
For her second essay (Analyze an Argument), one reader gives a score of 5.0, but the other gives a 3.5. The difference is more than one point, so a third reader reviews the essay and gives a score of 4.0. The scores closest together (4.0 and 3.5) are averaged, yielding a final score of 3.75 for this essay.
The scores from the two essays are then averaged. In this case, the average of 4.75 and 3.75 is 4.25, which is rounded to the nearest half-point, yielding a final Analytical Writing score of 4.5.
Final Score Report
On her final GRE score report, Jane receives a Verbal Reasoning score of 155, a Quantitative Reasoning score of 160, and an Analytical Writing score of 4.5. For each score, she will also see a percentile rank, showing the percentage of test takers who scored lower than she did.
Remember, Jane's scores are hypothetical. The raw to scaled score conversion can vary from test to test due to the equating process used by ETS.
Understanding how GRE scores are calculated is essential for effective test preparation. It can help you focus on areas that need improvement and strategize how to answer questions to maximize your score. With this knowledge, you can approach your GRE test date with confidence.
Remember, preparation is key to achieving a high score on the GRE. Utilize this understanding of the scoring system to guide your study plan and manage your time effectively during the test. Take our GRE practice questions and test your knowledge.
The Graduate Management Admission Test, more commonly known as the GMAT, is a critical component of the application process for many top business schools worldwide. But the way in which the GMAT score is calculated can be somewhat complex and often misunderstood. In this article, we'll break down the formula for calculating your GMAT score and provide an overview of how the scoring system works. If after reading this post, you want to try out a GMAT mock text try to answer these GMAT practice questions.
Overview of the GMAT
The GMAT consists of four sections:
The Scoring Breakdown
Let's now look at each of the GMAT sections and how their scores are calculated.
1. Analytical Writing Assessment (AWA)
In this section, you are asked to write an essay. This essay is then scored on a scale of 0 to 6, in half-point increments. Your essay is scored twice: once by a computerized system and once by a human grader. The average of these two scores is your final AWA score.
2. Integrated Reasoning (IR)
In the IR section, you're required to interpret and analyze information presented in various formats. Your IR score ranges from 1 to 8, in one-point increments. This score is based solely on the number of questions you answer correctly; there's no penalty for wrong answers.
3. Quantitative and Verbal Reasoning
Both the Quant and Verbal sections are computer-adaptive, meaning the difficulty of questions adjusts based on your performance. In other words, the better you perform, the more challenging the questions become. Your scores for these sections range from 6 to 51.
The Total GMAT ScoreThe total GMAT score is a scaled combination of the Quant and Verbal scores and ranges from 200 to 800. This score does not include the AWA and IR sections. About two-thirds of test-takers score between 400 and 600.
The Quant and Verbal scores are scaled to account for differences in overall difficulty from one test to another. Therefore, the raw score (the number of questions answered correctly) isn't enough to determine your scaled score.
The scaled scores for the Quant and Verbal sections are then combined and converted into a total score. This score is presented along with a percentile ranking, which tells you what percentage of test takers you performed better than.
For example, a 700 GMAT score would mean you performed better than 88% of test-takers. This percentile ranking is crucial for business schools as it gives them a clear idea of where you stand among other candidates. Check out our mock GMAT 10 question test to get a better sense of how GMAT tests are structured.
Understanding how the GMAT score is calculated can help you build an effective study strategy. Remember, each section requires different skills, and therefore, you should tailor your preparation accordingly. Also, while your total GMAT score is critical, don't underestimate the importance of the AWA and IR sections. Many schools use these scores to assess your analytical writing and integrated reasoning skills, which are essential for success in the business world.
Overall, achieving a high GMAT score is about more than just knowing the right answers. It's about demonstrating a range of skills and capabilities that business schools value. The more you understand about the test and its scoring system, the better equipped you'll be to excel.
Take our GMAT practice questions with explanations to test your knowledge.
AIC calculation, or Akaike Information Criterion calculation, is a helpful tool for comparing statistical models and determining which one is the most appropriate for a given dataset. Essentially, AIC calculates the relative quality of each model by balancing its accuracy against its complexity.
When analyzing a dataset, statisticians often have multiple models to choose from that could fit the data. However, not all models are created equal; some may be too simple and fail to capture key patterns, while others may be too complex and overfit the data. This is where AIC comes in - it considers both the goodness of fit and the simplicity of a model to give a single number for each model, allowing statisticians to objectively compare them.
The goal of AIC calculation is to find the model with the lowest AIC value, indicating the best balance of accuracy and simplicity. While it's not a perfect method, AIC has become a widely accepted approach for model selection that can be applied to a variety of statistical techniques. With the help of AIC, we can make confident decisions about which models are most appropriate, providing insights that could inform future studies or applications.
Understanding AIC Calculation
AIC, or Akaike Information Criterion, is a statistical measurement used to determine which model best fits a given set of data. Essentially, it provides a quantitative way to compare different models and select the one that is most appropriate for the data.
There are two main components to the AIC formula: the likelihood function and a penalty term. The likelihood function measures how well the model fits the data, while the penalty term adjusts for the number of parameters included in the model. The idea behind this penalty term is that more complex models, with more parameters, are likely to overfit the data and ultimately perform worse on new data than simpler models.
AIC values are calculated for each model under consideration, and the model with the lowest AIC value is considered the best fit for the data. This means that lower AIC values indicate a better model fit.
Let's take a look at an example. Say we are trying to predict the price of a house based on its square footage and number of bedrooms. We have two models we are considering: one that includes only square footage, and another that includes both square footage and number of bedrooms.
Using AIC, we can calculate the AIC values for both models and see which one is a better fit for the data. Let's say the AIC values for the two models are 500 and 550, respectively. This means that the model with only square footage has a lower AIC value and is therefore a better fit for the data.
It's important to note that AIC is just one tool in the data modeling toolbox, and should not be the only factor in selecting a model. Other considerations, such as interpretability and domain knowledge, should also be taken into account. However, AIC can be a helpful starting point in the model selection process.
When conducting statistical analysis, it's crucial to select the best fitting model to describe the relationship among variables. The Akaike information criterion (AIC) is a powerful tool for model selection that helps in identifying the model that best represents the data. AIC is an estimator of the relative quality of statistical models for a given set of data. The AIC value of a model depends on the number of model parameters and the goodness of fit. The lower the AIC value, the better the model.
There are several reasons why AIC is important in statistical analysis:
Steps for Performing AIC Calculation
When it comes to model selection, the Akaike Information Criterion (AIC) is one of the most commonly used techniques. AIC is a measure that helps to determine how well a statistical model fits the data while taking into account the complexity of the model. Here are the steps involved in performing AIC calculation:
Step 1: Select your statistical model
To begin with, you'll need to choose a statistical model that fits your data. This model can be selected based on:
- Expert knowledge
- Residual plots
Step 2: Estimate the model parameters
Once you've selected your model, you'll need to estimate its parameters. This can be done using maximum likelihood estimation (MLE). The MLE method seeks to find the parameter values that maximize the likelihood function of the model, given the data.
Step 3: Calculate the AIC value
Next, you'll need to calculate the AIC value for your model. The AIC value takes into account the complexity of the model and the goodness-of-fit:
- Calculate the log-likelihood of the model
- Add the product of the number of model parameters and 2 to the log-likelihood score
- This gives you the AIC value, where lower values indicate better-fitting models
Step 4: Compare AIC values
After obtaining AIC values for each model of interest, you can compare them to identify the best-fitting model. The model with the lowest AIC value is preferred as it strikes a balance between fit and complexity.
Interpreting AIC Results
When interpreting the results of Akaike Information Criterion (AIC), there are a few important factors to consider. AIC is used to compare models and select the one that fits best with the data. It is important to note that a lower AIC value indicates a better fit.
Here are some key points to keep in mind when interpreting AIC results:
To better understand the significance of AIC, let's take an example of a linear regression model with two predictors. We fit two models- one with both predictors and the other with only one predictor. The AIC values for the models are 100 and 105, respectively. According to the threshold of 2 or more difference in AIC values, we can confidently say that the model with both predictors has a significantly better fit, as compared to the one with only one predictor.
On the other hand, if we compare two models with AIC values of 500 and 502, respectively, we cannot confidently say that the model with the lower AIC value is a better fit, given the small difference of just 2 units. It's important to note that the interpretation of AIC values requires critical thinking and domain knowledge.
Advantages and Limitations of AIC Calculation
When it comes to statistical model selection, AIC (Akaike Information Criterion) is a widely-adopted approach due to its simple implementation and robustness in many scenarios. However, there are several advantages and limitations of AIC calculation that we should take into consideration.
Advantages of AIC Calculation
Limitations of AIC Calculation
AIC has several advantages over other model selection methods, such as its simplicity and computational efficiency. However, AIC's limitations should also be taken into account, such as its assumption of the error distribution and its suitability for small sample sizes.
The AIC Formula Explained
The AIC value is calculated using the following formula:
AIC = 2k - 2ln(L)
For example, suppose we have two models with AIC values of 100 and 105. Here, we can say that the model with an AIC value of 100 is a better model compared to the model with an AIC value of 105.
There are a few things to keep in mind while using AIC for model selection:
Delta Scores and Akaike Weights
Delta scores measure the difference in fit between a model and a baseline model, while Akaike weights provide a way to rank models based on their relative quality of fit. These measures are commonly used in various fields, including economics, ecology, and biology.
Delta scores are calculated by subtracting the Akaike information criterion (AIC) of one model from another. AIC is a measure of the quality of a statistical model, taking into account both the goodness of fit and the complexity of the model. The model with the smaller AIC is considered to be the better fit. Delta scores can be used to compare different models and determine which one fits the data better.
Akaike weights, on the other hand, provide a way to rank models based on their relative quality of fit. These weights are derived from the AIC of each model and represent the probability that a given model is the best fitting model among the set of candidate models. This allows researchers to compare not only the fit of different models but also the likelihood of each model being the best fit for the data.
Understanding Akaike weights
It's important to note that Akaike weights are calculated using a delta score. The delta score is the difference between the number of parameters in two models and their corresponding AIC values. AIC stands for Akaike Information Criteria, which is a statistical measure used to evaluate the quality and fitness of a model.
The Akaike weight of a given model can range from 0 to 1, with 0 indicating that the model is not a good fit for the data, and 1 indicating that the model is the superior fit. If two models have similar weights, it may indicate that they are both a good fit for the data.
Calculating Akaike weights involves comparing several models and their respective delta scores. The formula for calculating delta scores is as follows:
delta i= AICi - min(AIC)
The formula for calculating Akaike weights is as follows:
wi = (exp(-0.5*delta i))/sum(exp(-0.5*delta i))
Where wi is the Akaike weight, delta i is the delta score for the ith model, and sum(exp(-0.5*delta i)) is the sum of the exponentials of the delta scores for each model.
Akaike weights are an important tool for evaluating model accuracy and selecting the best model for a specific dataset. By using delta scores to compare different model fits, we can calculate the likelihood of each model to minimize prediction error and identify which model has the best fit for the data.
Summing Up Akaike's Information Criterion (AIC) Calculation
In conclusion, calculating Akaike's Information Criterion (AIC) allows us to determine the best statistical model for a given dataset. Through this process, we can compare the performance of different models and select the best one based on the AIC score.
After performing AIC calculations on our dataset, we have determined that the model with the lowest AIC score is the most appropriate for our data. This indicates that this model has the best balance between goodness-of-fit and parsimony.
In addition, we have also found that AIC values can be used to compare models with different numbers of parameters. By using the AICc correction, we can adjust for small sample sizes and obtain more accurate model comparisons.
Overall, AIC calculation is a valuable tool for model selection and provides a useful framework for making informed decisions in statistics.
For more helpful math and statistics resources check out z-table.com.
Variability Definition in Statistics: Understanding Variability and Its Importance in Data Analysis
Statistics is a part of mathematics that focuses on the collection, analysis, interpretation, and presentation of numbers. One of the fundamental concepts in statistics is variability, which refers to the degree of spread or dispersion of a set of data. In this post, we will explore the variability definition in statistics and its importance in data analysis.
What is Variability in Statistics?
Variability, also known as dispersion, is a measure of how spread out a set of data is. It refers to the differences or variations that exist among the values in a data set. Variability can be observed in various statistical measures, such as range, variance, standard deviation, and coefficient of variation.
The concept of variability is essential in statistics because it provides valuable information about the characteristics of the data set. For example, a data set with high variability indicates that the values are widely spread out and may have extreme values, while a data set with low variability indicates that the values are closely clustered around the mean or average.
Measures of Variability in Statistics
As mentioned earlier, variability can be measured using different statistical measures. Let's discuss some of these measures:
The range is the difference between the highest and lowest values in a data set. It is a simple measure of variability that describes how dispersed the data is. However, it has limitations as it only considers the two extreme values and does not provide information about the distribution of the data.
Variance is a measure of how far the data is spread out from its mean. It is calculated by taking the sum of the squared deviations of each data point from the mean and dividing it by the total number of observations minus one. The formula for variance is:
Variance = Σ (xi - μ)2 / (n - 1)
Where Σ is the sum, xi is the data value, μ is the mean, and n is the sample size.
The variance is useful in identifying how much variation exists in the data set. A high variance indicates that the data points are far away from the mean, while a low variance indicates that the data points are close to the mean.
Standard deviation is the square root of variance. It is a widely used measure of variability that provides information about the dispersion of the data points around the mean. The formula for standard deviation is:
Standard Deviation = √(Σ (xi - μ)2 / (n - 1))
The standard deviation is often used in statistics because it is easy to interpret and has useful properties, such as the empirical rule.
Coefficient of Variation
The coefficient of variation is a measure of relative variability that is useful when comparing data sets with different means and units. It is calculated by dividing the standard deviation by the mean and multiplying the result by 100. The formula for the coefficient of variation is:
Coefficient of Variation = (Standard Deviation / Mean) x 100
The coefficient of variation provides a way of comparing the degree of variability between data sets of different scales.
Importance of Variability in Data Analysis
Variability is an essential concept in data analysis as it provides valuable insights into the nature of the data set. Here are some of the reasons why variability is important in data analysis:
Outliers are data points that lie far away from the other values in the data set. Variability measures, such as standard deviation and variance, can help identify outliers by indicating the degree of spread or dispersion in the data.
Variability measures are essential in making statistical inferences about the population based on sample data. The variability measures, such as standard deviation, variance, and coefficient of variation, provide information about how closely the sample data represents the population. A low variability indicates that the sample data is more representative of the population, while a high variability indicates that the sample data may not be representative.
Evaluating Data Quality
Variability measures are useful in evaluating the quality of data. A high variability may indicate errors in data collection or measurement, while a low variability may indicate a lack of diversity or insufficient sample size.
Monitoring Process Stability
Variability measures are often used in process control to monitor the stability of a process. A stable process is one that produces consistent results with low variability, while an unstable process produces inconsistent results with high variability.
Comparing Data Sets
Variability measures, such as the coefficient of variation, are useful in comparing data sets with different means and units. The coefficient of variation provides a standardized measure of variability that can be used to compare the degree of variation between data sets.
To summarize, variability is a fundamental concept in statistics that refers to the degree of spread or dispersion of a set of data. Variability measures, such as range, variance, standard deviation, and coefficient of variation, provide valuable information about the characteristics of the data set. Variability is important in data analysis as it helps identify outliers, make inferences, evaluate data quality, monitor process stability, and compare data sets. By understanding variability, we can gain a deeper insight into the nature of the data and make more informed decisions based on the data analysis.
Z-scores are a common statistical tool used to compare data points in a distribution. They measure how many standard deviations a data point is from the mean of the distribution. While calculating z-scores is a fairly straightforward process, rounding them can be a bit more complex. In this article, we'll discuss the concept of z-score rounding and whether it's necessary in statistical analysis.
What is Z-Score Rounding?
Z-score rounding is the process of reducing a calculated z-score to a specified number of decimal places. In some cases, it may be necessary to round z-scores for easier interpretation or to comply with reporting guidelines. However, rounding can also result in a loss of precision and may lead to incorrect conclusions.
When Should You Round Z-Scores?
There are no hard and fast rules for when to round z scores. In general, rounding is more common when reporting z-scores to a wider audience, such as in scientific publications or business reports. In these cases, rounding to two or three decimal places is often recommended.
For example, let's say you're comparing the test scores of two different classes. The mean score for Class A is 75, with a standard deviation of 10, while the mean score for Class B is 85, with a standard deviation of 15. You calculate the z-scores for both classes and get the following results:
Class A: z = (80 - 75) / 10 = 0.5 Class B: z = (80 - 85) / 15 = -0.33
In this case, you might choose to round the z-scores to two decimal places to make them easier to interpret:
Class A: z = 0.50 Class B: z = -0.33
However, it's important to note that rounding z-scores can lead to inaccurate conclusions if the rounding is not performed correctly. For example, rounding the z-score for Class B to one decimal place would give a value of -0.3, which could lead to the incorrect conclusion that Class B performed better than Class A.
In general, it's best to avoid rounding z-scores unless it's necessary for reporting purposes. If you need to compare z-scores, it's generally better to use the unrounded values to ensure maximum precision and accuracy.
Tools for Calculating Z-Scores
To calculate z-scores and determine whether rounding is necessary, you can use a z-score calculator. This calculator allows you to input the raw score, mean, and standard deviation to calculate the z-score.
If you need to determine the probability associated with a specific z-score, you can use a z-score probability calculator. This calculator allows you to input the z-score to find the probability associated with that score.
Finally, if you need to round z-scores to a specific decimal place, you can use a rounding calculator. This calculator allows you to input the z-score and specify the number of decimal places to round to.
Z-score rounding can be a useful tool for making z-scores easier to interpret or comply with reporting guidelines. However, it's important to use caution when rounding z-scores to avoid inaccuracies that could lead to incorrect conclusions. In most cases, it's best to use unrounded z-scores for maximum precision and accuracy in statistical analysis.
Usually, when you're doing research about groups of people or objects, you will use both descriptive and inferential statistics.
Descriptive statistics refer to the analysis of the data that will help you describe, summarize, or show the data in a way that some patterns might emerge. However, you need to be aware that you shouldn't withdraw conclusions besides the data analyzed. You should be simply describing the data you got.
Despite this might not seem important, it really has a crucial part in the process since it allows you to visualize huge data in a simple and effective way.
Imagine that you wanted to analyze the performance on a test of 100 students. You might be interested in seeing the overall performance or you might be interested in looking at the spread or distribution of their marks.
When you use the descriptive statistics, you should present your data by starting with a table that summarizes the group data, followed by charts and graphs. Finally, at the end, you should add the statistical commentary like the discussion of the results.
There are many occasions when you want to analyze a specific group but you simply can't have a sample of the entire population. Unlike on the previous example, you wanted to analyze the performance of 100 students, in this case, you might want to measure the performance of all the students in a country. Since it's not doable to collect all the data, you need to choose a smaller sample of students, which will represent all the students in that country.
And this is where the inferential statistics have their crucial role. They refer to the techniques that you use that allow you to use the samples to make generalized comments regarding the entire population. So, as you understand, it's very important to be careful when selecting the sample that represents the population. It needs to be as accurate as it can or the results won't represent the truth.
The descriptive and inferential statistics have one thing in common: they both rely on the same data. However, while the descriptive statistics only relies on this particular data, the inferential statistics relies on this data to make general conclusions about a larger population.
In order to analyze the data, both descriptive and inferential statistics need to rely on some functions of the data. In the case of the descriptive statistics, it tends to rely on some classic statistics like the mean, standard deviation, min, max, skew, median, and kurtosis. In the case of the inferential statistics, they tend to use some classic statistics like the z score, t score, F-ratio, among others.
So, which one should you choose to use? You may need to use both types of statistics and the answer depends on the purpose of your research. For example, when a company is trying to show if a new medicine will be able to help patients in the future, it's in their best interest that they use inferential statistics. If they decide to use descriptive statistics, they won't be able to withdraw any conclusions regarding the population in general but simply regarding the patients that participated in the study.
The z score, also known as the standard score, allows you not only to compare two scores from different normal distributions as it also allows you to calculate the probability of one specific score to occur within the normal distribution.
The z score, by taking into account the standard deviation (SD) of the group, standardizes a score with respect to the other scores in the group. When you convert a raw score into a z-score, you'll get a number between 0 (the mean) and 1 (the standard deviation). Basically, what you will be doing is that you'll be defining each score you have in terms of how far away each one is from the group mean.
Main Advantages Of Using Z-Scores:
There are mainly two advantages why so many different organizations use the z-scores:
-- Clarity: At a single glance, you can easily see how bad or good a score is when compared to the entire group.
-- Comparison: Being able to compare scores that are measured on different scales is definitely a huge advantage.
Results From The Z-Scores:
When you are looking at a z-score table, you have three distinct areas:
-- The Area Under The Curve: Looking at this part of the table will allow you to see the different properties of the normal distribution. So, you'll be able to see how many scores are under pre-determined limits as well as you'll be able to calculate the probability of a specific score to occur.
-- The Area Between The Mean And The Z: This area corresponds to the area under the curve, and that is between the mean and the z-scores. In this area, you can see the proportion of scores that are located between the mean and any given z-score.
-- The Area Beyond Z: In this area, you'll see the proportion of the scores that are greater than any given z-score.
Calculating The Z-Scores:
The easiest way to calculate a z-score is to use a z score calculator. However, in case you're interested in knowing how to calculate it by hand, you just need to find the difference between the score and the mean score, which tells you how far away the score is from the average score, and divide it by the standard deviation. Then you can use a z score table to find a probability.
There's no question that is far more effective and time-saving using a z score calculator.
Finding Z scores and the area under the bell curve with ti-84
Descriptive Statistics and Measures of Central Tendency