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Statistics & Probability Letters adopts a novel and highly innovative approach to the publication of research findings in statistics and probability . It features concise articles, rapid publication and broad coverage of the statistics and probability literature.

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section " Probability and Statistics ".

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Dear Colleagues,

Statistics and probability are important domains in the scientific world, having many applications in various fields, such as engineering, reliability, medicine, biology, economics, physics, and not only, probability laws providing an estimated image of the world we live in.  This Special Volume deals targets some certain directions of the two domains as described below. 

Some applications of statistics are clustering of random variables based on simulated and real data or scan statistics, the latter being introduced in 1963 by Joseph Naus. In reliability theory, some important statistical tools are hazard rate and survival functions, order statistics, and stochastic orders. In physics, the concept of entropy is at its core, while special statistics were introduced and developed, such as statistical mechanics and Tsallis statistics.

~In economics, statistics, mathematics, and economics formed a particular domain called econometrics. ARMA models, linear regressions, income analysis, and stochastic processes are discussed and analyzed in the context of real economic processes. Other important tools are Lorenz curves and broken stick models.

~Theoretical results such as modeling of discretization of random variables and estimation of parameters of new and old statistical models are welcome, some important probability laws being heavy-tailed distributions. In recent years, many distributions along with their properties have been introduced in order to better fit the growing data available.

The purpose of this Special Issue is to provide a collection of articles that reflect the importance of statistics and probability in applied scientific domains. Papers providing theoretical methodologies and applications in statistics are welcome.

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Statistics and Probability Have Always Been Value-Laden: An Historical Ontology of Quantitative Research Methods

• Original Paper
• Published: 27 May 2019
• Volume 167 , pages 1–18, ( 2020 )

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• Michael J. Zyphur 1 &
• Dean C. Pierides 2  

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Quantitative researchers often discuss research ethics as if specific ethical problems can be reduced to abstract normative logics (e.g., virtue ethics, utilitarianism, deontology). Such approaches overlook how values are embedded in every aspect of quantitative methods, including ‘observations,’ ‘facts,’ and notions of ‘objectivity.’ We describe how quantitative research practices, concepts, discourses, and their objects/subjects of study have always been value-laden, from the invention of statistics and probability in the 1600s to their subsequent adoption as a logic made to appear as if it exists prior to, and separate from, ethics and values. This logic, which was embraced in the Academy of Management from the 1960s, casts management researchers as ethical agents who ought to know about a reality conceptualized as naturally existing in the image of statistics and probability (replete with ‘constructs’), while overlooking that S&P logic and practices, which researchers made for themselves, have an appreciable role in making the world appear this way. We introduce a different way to conceptualize reality and ethics, wherein the process of scientific inquiry itself requires an examination of its own practices and commitments. Instead of resorting to decontextualized notions of ‘rigor’ and its ‘best practices,’ quantitative researchers can adopt more purposeful ways to reason about the ethics and relevance of their methods and their science. We end by considering implications for addressing ‘post truth’ and ‘alternative facts’ problems as collective concerns, wherein it is actually the pluralistic nature of description that makes defending a collectively valuable version of reality so important and urgent.

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Acknowledgements

For reviewing previous versions of this manuscript the authors thank Adam Barsky, Andrew Gelman, Andrew Van de Ven, Barbara Lawrence, Fred Oswald, Graham Sewell, Karen Jehn, Kristopher Preacher, Maria Carla Galavotti, Ray Zammuto, Tom Lee, Zhen Zhang, and Dan Woodman. For help with historical insights, the authors thank James March, Steven Schlossman, William Starbuck, and the many independent presses that have made it possible to investigate the roots of our quantitative practices. We would also like to thank the anonymous reviewers who have read and responded to previous versions of this manuscript.

This research was supported by Australian Research Council’s Future Fellowship scheme (project FT140100629).

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Zyphur, M.J., Pierides, D.C. Statistics and Probability Have Always Been Value-Laden: An Historical Ontology of Quantitative Research Methods. J Bus Ethics 167 , 1–18 (2020). https://doi.org/10.1007/s10551-019-04187-8

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A practical overview on probability distributions

Andrea viti, alberto terzi, luca bertolaccini.

Aim of this paper is a general definition of probability, of its main mathematical features and the features it presents under particular circumstances. The behavior of probability is linked to the features of the phenomenon we would predict. This link can be defined probability distribution. Given the characteristics of phenomena (that we can also define variables), there are defined probability distribution. For categorical (or discrete) variables, the probability can be described by a binomial or Poisson distribution in the majority of cases. For continuous variables, the probability can be described by the most important distribution in statistics, the normal distribution. Distributions of probability are briefly described together with some examples for their possible application.

A short definition of probability

We can define the probability of a given event by evaluating, in previous observations, the incidence of the same event under circumstances that are as similar as possible to the circumstances we are observing [this is the frequentistic definition of probability, and is based on the relative frequency of an observed event, observed in previous circumstances ( 1 )]. In other words, probability describes the possibility of an event to occur given a series of circumstances (or under a series of pre-event factors). It is a form of inference, a way to predict what may happen, based on what happened before under the same (never exactly the same) circumstances. Probability can vary from 0 (our expected event was never observed, and should never happen) to 1 (or 100%, the event is almost sure). It is described by the following formula: if X = probability of a given x event ( Eq. [1] ):

This is one of the three axioms of probability, as described by Kolmogorov ( 2 ):

• If under some circumstances, a given number of events ( E ) could verify ( E 1 , E 2 , E 3 , …, E n ), the probability ( P ) of any E is always more than zero;
• The sum of the probabilities of E =  P ( E 1 ) +  P ( E 2 ) +  ⋯  +  P ( E n ) is 100%;
• If E 1 and E 3 are two possible events, the probability that one or the other could happen P ( E 1 or E 3 ) is equal to the sum of the probability of E 1 and the probability of E 3 ( Eq. [2] ): P ( E 1   o r   E 2 ) =  P ( E 1 ) +  P ( E 3 ) [2]

Probability could be described by a formula, a graph, in which each event is linked to its probability. This kind of description of probability is called probability distribution.

Binomial distribution

A classic example of probability distribution is the binomial distribution. It is the representation of the probability when only two events may happen, that are mutually exclusive. The typical example is when you toss a coin. You can only have two results. In this case, the probability is 50% for both events. However, binomial distribution may describe also two events that are mutually exclusive but are not equally possible (for instance that a newborn baby will be left-handed or right-handed). The probability that x individuals present a given characteristic, p , that is mutually exclusive of another one, called q , depends on the possible number of combinations of x individuals within the population, called C. If my population is composed of five5 individuals, that can be p or q , I have ten possible combinations of, for instance, three individuals with p is ( Eq. [3] ):

pppqq , pqqpp , ppqpq , ppqqp, pqpqpq, qpppq, qpqpp, qppqp, qqppp

Then p 3 q 2 will be multiplied for the number of combinations (ten times).

If, in experimental population, I had a big number of individuals ( n ), the number of combinations of x individuals within the population will be ( Eq. [4] ):

Therefore, the probability that a group of x individuals within the population of n individuals presents the characteristic p , that excludes q , will be described by the following formula ( Eq. [5] ):

that describes the binomial distribution. It follows the Kolmogorow’s rules ( Eq. [6] ):

In a given population, 30% of the people are left-handed. If we select ten individuals from this population, what is the probability that four out of ten individuals are left handed?

We can apply the binomial distribution, since we suppose that a person may be either left-handed or right-handed.

Se we can use our formula ( Eq. [7] ):

Poisson distribution

Another important distribution of probability is the Poisson distribution. It is useful to describe the probability that a given event can happen within a given period (for instance, how many thoracic traumas could need the involvement of the thoracic surgeon in a day, or a week, etc.). The events that may be described by this distribution have the following characteristics:

• The events are independent from one another;
• Within a given interval the event may present from 0 to infinite times;
• The probability of an event to happen increases when the period of observation is longer.

To predict the probability, I must know how the events behave (this data comes from previous, or historical, observations of the same event before the time I am trying to perform my analysis). This parameter, that is a mean of the events in a given interval, as derived from previous observations, is called λ .

The Poisson distribution follows the following formula ( Eq. [8] ):

where the number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828.

For example, the distribution of major thoracic traumas needing intensive care unit (ICU) recovery during a month in the last three years in a Third Level Trauma Center follows a Poisson distribution, were λ =2.75. In a future period of one month, what is the probability to have three patients with major thoracic trauma in ICU? ( Eq. [9] ):

Therefore, the probability is 22.1%.

The binomial distribution refers only to discrete variables (that present a limited number of values within a given interval). However, in nature, many variables may present an infinite distribution of values, within a given interval. These are called continuous variables ( 3 ).

Distributions of continuous variables

An example of continuous variable is the systolic blood pressure. Within a given cohort of systolic blood pressure can be presented as in Figure 1 . Each single histogram length represents an interval of the measure of interest between two intervals on the x -axis, while the histogram height represents the number of measured values within the interval. When the number of observation becomes very large (tends to infinite) and the length of the histogram becomes narrower (tends to 0), the above representation becomes more similar to a curved line ( Figure 2 ). This curve describes the distribution of probability, f (density of probability) for any given value of x , the continuous variable. The area under the curve is equal to 1 (100% of probability). We can now assume that the value of our continuous variable X depends on a very large number of other factors (in many cases beyond our possibility of direct analysis), the probability distribution of X becomes similar to a particular form of distribution, called normal distribution or Gauss distribution. The aforementioned concept is the famous Central Limit Theorem. The normal distribution represents a very important distribution of probability because f , that is the distribution of probability of our variables, can be represented by only two parameters:

Graphical description of the distribution of systolic blood pressure in a given population.

Graphical description of the normal distribution.

• µ = mean;
• σ = standard deviation.

The mean is a so-called measure of central tendency (it represents the more central value of our curve), while the standard deviation represents how dispersed are the values of probability around the central value (is a measure of dispersion).

• The main characteristics of this distribution are:
• It is symmetric around the µ ;
• The area under the curve is 1;

If we consider the area under the curve between µ ± σ , this area will cover 68% of all the possible values of X , while the area between µ ± 2 σ , it will cover 95% of all the values.

The two parameters of the distribution are linked in the formula ( Eq. [10] ):

For µ = 0, and σ = 1, the curve is called standardized normal distribution. All the possible normal distributions of x may be “normalized” by defining a derived variable called z . ( Eq. [11] ):

To calculate the probability that our variable falls within a given interval, for instance z 0 and z 1 , we should calculate the following definite integral calculus ( Eq. [12] ):

Fortunately, for the standard normalized distribution of z every possible interval has been tabulated.

In a given population of adult men, the mean weight is 70 kg, with a standard deviation of 3 kg. What is the probability that a randomly selected individual from this population would have a weight of 65 kg or less?

To “normalize” our distribution, we should calculate the value of z ( Eq. [13] ):

Then, we should calculate the area under the curve ( Eq. [14] ):

The value of our interval has been already calculated and tabulated [the tables can be easily found in any text of statistics or in the web ( 4 )]. Our probability is 0.0475 (4.75%). We may also calculate the probability to find, within the same population, someone whose weight is between 65 and 74 kg. This probability can be seen as the difference of distribution between those whose weight is 74 kg or less and those whose weight is 65 kg or less: ( Eq. [15] ):

We already know that ( Eq. [16] ):

In the table we can find also the value for ( Eq. [17] ):

Our probability is ( Eq. [18] ):

Conclusions

The probability distributions are a common way to describe, and possibly predict, the probability of an event. The main point is to define the character of the variables whose behaviour we are trying to describe, trough probability (discrete or continuous). The identification of the right category will allow a proper application of a model (for instance, the standardized normal distribution) that would easily predict the probability of a given event.

Acknowledgements

Disclosure: The authors declare no conflict of interest.

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Research topics in probability and statistics, problem solving in mathematics and statistics is inspiring and enjoyable. but are achievements in mathematics and statistics any of use in the so-called real world , researchers in the department of statistics at warwick are developing and utilising modern statistics, mathematics, and computing to solve practical problems., examples of themes for undergraduate research projects:.

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Bayesian Models of Category-Specific Emotional Brain Responses

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The work of mathematicians and statisticians often turns out useful and essential, but typically in a less concrete manner than say the work of a scientists or a physician. David Hilbert, in his now historical address to scientists and physicians, put it this way:

"The instrument that mediates between theory and practice, between thought and observation, is mathematics; it builds the connecting bridge and makes it stronger and stronger. Thus it happens that our entire present-day culture, insofar as it rests on intellectual insight into and harnessing of nature, is founded on mathematics"

Probability and Statistics in the 21st century

Almost a century after Hilbert's words, the mathematical fundations of sciences and social sciences, and the evidence based approach in medicine are often being taken for granted. In the 21st century we are facing complex big data sets with unknown structures from manifold aspecs of the 'real world' as well as fascinating discourses about objective and subjective notions of risk and uncertainty.

Probability and statistics are mathematical disciplines for modelling and analysing theoretical and practical aspects of these burning questions.

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The Beginner's Guide to Statistical Analysis | 5 Steps & Examples

Statistical analysis means investigating trends, patterns, and relationships using quantitative data . It is an important research tool used by scientists, governments, businesses, and other organizations.

To draw valid conclusions, statistical analysis requires careful planning from the very start of the research process . You need to specify your hypotheses and make decisions about your research design, sample size, and sampling procedure.

After collecting data from your sample, you can organize and summarize the data using descriptive statistics . Then, you can use inferential statistics to formally test hypotheses and make estimates about the population. Finally, you can interpret and generalize your findings.

This article is a practical introduction to statistical analysis for students and researchers. We’ll walk you through the steps using two research examples. The first investigates a potential cause-and-effect relationship, while the second investigates a potential correlation between variables.

Table of contents

Step 1: write your hypotheses and plan your research design, step 2: collect data from a sample, step 3: summarize your data with descriptive statistics, step 4: test hypotheses or make estimates with inferential statistics, step 5: interpret your results, other interesting articles.

To collect valid data for statistical analysis, you first need to specify your hypotheses and plan out your research design.

Writing statistical hypotheses

The goal of research is often to investigate a relationship between variables within a population . You start with a prediction, and use statistical analysis to test that prediction.

A statistical hypothesis is a formal way of writing a prediction about a population. Every research prediction is rephrased into null and alternative hypotheses that can be tested using sample data.

While the null hypothesis always predicts no effect or no relationship between variables, the alternative hypothesis states your research prediction of an effect or relationship.

• Null hypothesis: A 5-minute meditation exercise will have no effect on math test scores in teenagers.
• Alternative hypothesis: A 5-minute meditation exercise will improve math test scores in teenagers.
• Null hypothesis: Parental income and GPA have no relationship with each other in college students.
• Alternative hypothesis: Parental income and GPA are positively correlated in college students.

Planning your research design

A research design is your overall strategy for data collection and analysis. It determines the statistical tests you can use to test your hypothesis later on.

First, decide whether your research will use a descriptive, correlational, or experimental design. Experiments directly influence variables, whereas descriptive and correlational studies only measure variables.

• In an experimental design , you can assess a cause-and-effect relationship (e.g., the effect of meditation on test scores) using statistical tests of comparison or regression.
• In a correlational design , you can explore relationships between variables (e.g., parental income and GPA) without any assumption of causality using correlation coefficients and significance tests.
• In a descriptive design , you can study the characteristics of a population or phenomenon (e.g., the prevalence of anxiety in U.S. college students) using statistical tests to draw inferences from sample data.

Your research design also concerns whether you’ll compare participants at the group level or individual level, or both.

• In a between-subjects design , you compare the group-level outcomes of participants who have been exposed to different treatments (e.g., those who performed a meditation exercise vs those who didn’t).
• In a within-subjects design , you compare repeated measures from participants who have participated in all treatments of a study (e.g., scores from before and after performing a meditation exercise).
• In a mixed (factorial) design , one variable is altered between subjects and another is altered within subjects (e.g., pretest and posttest scores from participants who either did or didn’t do a meditation exercise).
• Experimental
• Correlational

First, you’ll take baseline test scores from participants. Then, your participants will undergo a 5-minute meditation exercise. Finally, you’ll record participants’ scores from a second math test.

In this experiment, the independent variable is the 5-minute meditation exercise, and the dependent variable is the math test score from before and after the intervention. Example: Correlational research design In a correlational study, you test whether there is a relationship between parental income and GPA in graduating college students. To collect your data, you will ask participants to fill in a survey and self-report their parents’ incomes and their own GPA.

Measuring variables

When planning a research design, you should operationalize your variables and decide exactly how you will measure them.

For statistical analysis, it’s important to consider the level of measurement of your variables, which tells you what kind of data they contain:

• Categorical data represents groupings. These may be nominal (e.g., gender) or ordinal (e.g. level of language ability).
• Quantitative data represents amounts. These may be on an interval scale (e.g. test score) or a ratio scale (e.g. age).

Many variables can be measured at different levels of precision. For example, age data can be quantitative (8 years old) or categorical (young). If a variable is coded numerically (e.g., level of agreement from 1–5), it doesn’t automatically mean that it’s quantitative instead of categorical.

Identifying the measurement level is important for choosing appropriate statistics and hypothesis tests. For example, you can calculate a mean score with quantitative data, but not with categorical data.

In a research study, along with measures of your variables of interest, you’ll often collect data on relevant participant characteristics.

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In most cases, it’s too difficult or expensive to collect data from every member of the population you’re interested in studying. Instead, you’ll collect data from a sample.

Statistical analysis allows you to apply your findings beyond your own sample as long as you use appropriate sampling procedures . You should aim for a sample that is representative of the population.

Sampling for statistical analysis

There are two main approaches to selecting a sample.

• Probability sampling: every member of the population has a chance of being selected for the study through random selection.
• Non-probability sampling: some members of the population are more likely than others to be selected for the study because of criteria such as convenience or voluntary self-selection.

In theory, for highly generalizable findings, you should use a probability sampling method. Random selection reduces several types of research bias , like sampling bias , and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.

But in practice, it’s rarely possible to gather the ideal sample. While non-probability samples are more likely to at risk for biases like self-selection bias , they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population.

If you want to use parametric tests for non-probability samples, you have to make the case that:

• your sample is representative of the population you’re generalizing your findings to.
• your sample lacks systematic bias.

Keep in mind that external validity means that you can only generalize your conclusions to others who share the characteristics of your sample. For instance, results from Western, Educated, Industrialized, Rich and Democratic samples (e.g., college students in the US) aren’t automatically applicable to all non-WEIRD populations.

If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalized in your discussion section .

Create an appropriate sampling procedure

Based on the resources available for your research, decide on how you’ll recruit participants.

• Will you have resources to advertise your study widely, including outside of your university setting?
• Will you have the means to recruit a diverse sample that represents a broad population?
• Do you have time to contact and follow up with members of hard-to-reach groups?

Your participants are self-selected by their schools. Although you’re using a non-probability sample, you aim for a diverse and representative sample. Example: Sampling (correlational study) Your main population of interest is male college students in the US. Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area.

Calculate sufficient sample size

Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics. A sample that’s too small may be unrepresentative of the sample, while a sample that’s too large will be more costly than necessary.

There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be (e.g., in clinical research). As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.

To use these calculators, you have to understand and input these key components:

• Significance level (alpha): the risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
• Statistical power : the probability of your study detecting an effect of a certain size if there is one, usually 80% or higher.
• Expected effect size : a standardized indication of how large the expected result of your study will be, usually based on other similar studies.
• Population standard deviation: an estimate of the population parameter based on a previous study or a pilot study of your own.

Once you’ve collected all of your data, you can inspect them and calculate descriptive statistics that summarize them.

Inspect your data

There are various ways to inspect your data, including the following:

• Organizing data from each variable in frequency distribution tables .
• Displaying data from a key variable in a bar chart to view the distribution of responses.
• Visualizing the relationship between two variables using a scatter plot .

By visualizing your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data.

A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.

In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions.

Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values.

Calculate measures of central tendency

Measures of central tendency describe where most of the values in a data set lie. Three main measures of central tendency are often reported:

• Mode : the most popular response or value in the data set.
• Median : the value in the exact middle of the data set when ordered from low to high.
• Mean : the sum of all values divided by the number of values.

However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all.

Calculate measures of variability

Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:

• Range : the highest value minus the lowest value of the data set.
• Interquartile range : the range of the middle half of the data set.
• Standard deviation : the average distance between each value in your data set and the mean.
• Variance : the square of the standard deviation.

Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions.

Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values? If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test.

From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics (correlational study) After collecting data from 653 students, you tabulate descriptive statistics for annual parental income and GPA.

It’s important to check whether you have a broad range of data points. If you don’t, your data may be skewed towards some groups more than others (e.g., high academic achievers), and only limited inferences can be made about a relationship.

A number that describes a sample is called a statistic , while a number describing a population is called a parameter . Using inferential statistics , you can make conclusions about population parameters based on sample statistics.

Researchers often use two main methods (simultaneously) to make inferences in statistics.

• Estimation: calculating population parameters based on sample statistics.
• Hypothesis testing: a formal process for testing research predictions about the population using samples.

You can make two types of estimates of population parameters from sample statistics:

• A point estimate : a value that represents your best guess of the exact parameter.
• An interval estimate : a range of values that represent your best guess of where the parameter lies.

If your aim is to infer and report population characteristics from sample data, it’s best to use both point and interval estimates in your paper.

You can consider a sample statistic a point estimate for the population parameter when you have a representative sample (e.g., in a wide public opinion poll, the proportion of a sample that supports the current government is taken as the population proportion of government supporters).

There’s always error involved in estimation, so you should also provide a confidence interval as an interval estimate to show the variability around a point estimate.

A confidence interval uses the standard error and the z score from the standard normal distribution to convey where you’d generally expect to find the population parameter most of the time.

Hypothesis testing

Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.

Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true. These tests give two main outputs:

• A test statistic tells you how much your data differs from the null hypothesis of the test.
• A p value tells you the likelihood of obtaining your results if the null hypothesis is actually true in the population.

Statistical tests come in three main varieties:

• Comparison tests assess group differences in outcomes.
• Regression tests assess cause-and-effect relationships between variables.
• Correlation tests assess relationships between variables without assuming causation.

Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.

Parametric tests

Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead.

A regression models the extent to which changes in a predictor variable results in changes in outcome variable(s).

• A simple linear regression includes one predictor variable and one outcome variable.
• A multiple linear regression includes two or more predictor variables and one outcome variable.

Comparison tests usually compare the means of groups. These may be the means of different groups within a sample (e.g., a treatment and control group), the means of one sample group taken at different times (e.g., pretest and posttest scores), or a sample mean and a population mean.

• A t test is for exactly 1 or 2 groups when the sample is small (30 or less).
• A z test is for exactly 1 or 2 groups when the sample is large.
• An ANOVA is for 3 or more groups.

The z and t tests have subtypes based on the number and types of samples and the hypotheses:

• If you have only one sample that you want to compare to a population mean, use a one-sample test .
• If you have paired measurements (within-subjects design), use a dependent (paired) samples test .
• If you have completely separate measurements from two unmatched groups (between-subjects design), use an independent (unpaired) samples test .
• If you expect a difference between groups in a specific direction, use a one-tailed test .
• If you don’t have any expectations for the direction of a difference between groups, use a two-tailed test .

The only parametric correlation test is Pearson’s r . The correlation coefficient ( r ) tells you the strength of a linear relationship between two quantitative variables.

However, to test whether the correlation in the sample is strong enough to be important in the population, you also need to perform a significance test of the correlation coefficient, usually a t test, to obtain a p value. This test uses your sample size to calculate how much the correlation coefficient differs from zero in the population.

You use a dependent-samples, one-tailed t test to assess whether the meditation exercise significantly improved math test scores. The test gives you:

• a t value (test statistic) of 3.00
• a p value of 0.0028

Although Pearson’s r is a test statistic, it doesn’t tell you anything about how significant the correlation is in the population. You also need to test whether this sample correlation coefficient is large enough to demonstrate a correlation in the population.

A t test can also determine how significantly a correlation coefficient differs from zero based on sample size. Since you expect a positive correlation between parental income and GPA, you use a one-sample, one-tailed t test. The t test gives you:

• a t value of 3.08
• a p value of 0.001

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The final step of statistical analysis is interpreting your results.

Statistical significance

In hypothesis testing, statistical significance is the main criterion for forming conclusions. You compare your p value to a set significance level (usually 0.05) to decide whether your results are statistically significant or non-significant.

Statistically significant results are considered unlikely to have arisen solely due to chance. There is only a very low chance of such a result occurring if the null hypothesis is true in the population.

This means that you believe the meditation intervention, rather than random factors, directly caused the increase in test scores. Example: Interpret your results (correlational study) You compare your p value of 0.001 to your significance threshold of 0.05. With a p value under this threshold, you can reject the null hypothesis. This indicates a statistically significant correlation between parental income and GPA in male college students.

Note that correlation doesn’t always mean causation, because there are often many underlying factors contributing to a complex variable like GPA. Even if one variable is related to another, this may be because of a third variable influencing both of them, or indirect links between the two variables.

Effect size

A statistically significant result doesn’t necessarily mean that there are important real life applications or clinical outcomes for a finding.

In contrast, the effect size indicates the practical significance of your results. It’s important to report effect sizes along with your inferential statistics for a complete picture of your results. You should also report interval estimates of effect sizes if you’re writing an APA style paper .

With a Cohen’s d of 0.72, there’s medium to high practical significance to your finding that the meditation exercise improved test scores. Example: Effect size (correlational study) To determine the effect size of the correlation coefficient, you compare your Pearson’s r value to Cohen’s effect size criteria.

Decision errors

Type I and Type II errors are mistakes made in research conclusions. A Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s false.

You can aim to minimize the risk of these errors by selecting an optimal significance level and ensuring high power . However, there’s a trade-off between the two errors, so a fine balance is necessary.

Frequentist versus Bayesian statistics

Traditionally, frequentist statistics emphasizes null hypothesis significance testing and always starts with the assumption of a true null hypothesis.

However, Bayesian statistics has grown in popularity as an alternative approach in the last few decades. In this approach, you use previous research to continually update your hypotheses based on your expectations and observations.

Bayes factor compares the relative strength of evidence for the null versus the alternative hypothesis rather than making a conclusion about rejecting the null hypothesis or not.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval

Methodology

• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Likert scale

Research bias

• Implicit bias
• Framing effect
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hostile attribution bias
• Affect heuristic

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Top 99+ Trending Statistics Research Topics for Students

Being a statistics student, finding the best statistics research topics is quite challenging. But not anymore; find the best statistics research topics now!!!

Statistics is one of the tough subjects because it consists of lots of formulas, equations and many more. Therefore the students need to spend their time to understand these concepts. And when it comes to finding the best statistics research project for their topics, statistics students are always looking for someone to help them. 

In this blog, we will share with you the most interesting and trending statistics research topics in 2023. It will not just help you to stand out in your class but also help you to explore more about the world.

If you face any problem regarding statistics, then don’t worry. You can get the best statistics assignment help from one of our experts.

As you know, it is always suggested that you should work on interesting topics. That is why we have mentioned the most interesting research topics for college students and high school students. Here in this blog post, we will share with you the list of 99+ awesome statistics research topics.

Why Do We Need to Have Good Statistics Research Topics?

Table of Contents

Having a good research topic will not just help you score good grades, but it will also allow you to finish your project quickly. Because whenever we work on something interesting, our productivity automatically boosts. Thus, you need not invest lots of time and effort, and you can achieve the best with minimal effort and time. 

What Are Some Interesting Research Topics?

If we talk about the interesting research topics in statistics, it can vary from student to student. But here are the key topics that are quite interesting for almost every student:-

• Literacy rate in a city.
• Abortion and pregnancy rate in the USA.
• Eating disorders in the citizens.
• Parent role in self-esteem and confidence of the student.
• Uses of AI in our daily life to business corporates.

Top 99+ Trending Statistics Research Topics For 2023

Here in this section, we will tell you more than 99 trending statistics research topics:

Sports Statistics Research Topics

• Statistical analysis for legs and head injuries in Football.
• Statistical analysis for shoulder and knee injuries in MotoGP.
• Deep statistical evaluation for the doping test in sports from the past decade.
• Statistical observation on the performance of athletes in the last Olympics.
• Role and effect of sports in the life of the student.

Psychology Research Topics for Statistics

• Deep statistical analysis of the effect of obesity on the student’s mental health in high school and college students.
• Statistical evolution to find out the suicide reason among students and adults.
• Statistics analysis to find out the effect of divorce on children in a country.
• Psychology affects women because of the gender gap in specific country areas.
• Statistics analysis to find out the cause of online bullying in students’ lives. 
• In Psychology, PTSD and descriptive tendencies are discussed.
• The function of researchers in statistical testing and probability.
• Acceptable significance and probability thresholds in clinical Psychology.
• The utilization of hypothesis and the role of P 0.05 for improved comprehension.
• What types of statistical data are typically rejected in psychology?
• The application of basic statistical principles and reasoning in psychological analysis.
• The role of correlation is when several psychological concepts are at risk.
• Actual case study learning and modeling are used to generate statistical reports.
• In psychology, naturalistic observation is used as a research sample.
• How should descriptive statistics be used to represent behavioral data sets?

Applied Statistics Research Topics

• Does education have a deep impact on the financial success of an individual?
• The investment in digital technology is having a meaningful return for corporations?
• The gap of financial wealth between rich and poor in the USA.
• A statistical approach to identify the effects of high-frequency trading in financial markets.
• Statistics analysis to determine the impact of the multi-agent model in financial markets. 

Personalized Medicine Statistics Research Topics

• Statistical analysis on the effect of methamphetamine on substance abusers.
• Deep research on the impact of the Corona vaccine on the Omnicrone variant. 
• Find out the best cancer treatment approach between orthodox therapies and alternative therapies.
• Statistics analysis to identify the role of genes in the child’s overall immunity.
• What factors help the patients to survive from Coronavirus .

Experimental Design Statistics Research Topics

• Generic vs private education is one of the best for the students and has better financial return.
• Psychology vs physiology: which leads the person not to quit their addictions?
• Effect of breastmilk vs packed milk on the infant child overall development
• Which causes more accidents: male alcoholics vs female alcoholics.
• What causes the student not to reveal the cyberbullying in front of their parents in most cases. 

Easy Statistics Research Topics

• Application of statistics in the world of data science
• Statistics for finance: how statistics is helping the company to grow their finance
• Advantages and disadvantages of Radar chart
• Minor marriages in south-east Asia and African countries.
• Discussion of ANOVA and correlation.
• What statistical methods are most effective for active sports?
• When measuring the correctness of college tests, a ranking statistical approach is used.
• Statistics play an important role in Data Mining operations.
• The practical application of heat estimation in engineering fields.
• In the field of speech recognition, statistical analysis is used.
• Estimating probiotics: how much time is necessary for an accurate statistical sample?
• How will the United States population grow in the next twenty years?
• The legislation and statistical reports deal with contentious issues.
• The application of empirical entropy approaches with online grammar checking.
• Transparency in statistical methodology and the reporting system of the United States Census Bureau.

Statistical Research Topics for High School

• Uses of statistics in chemometrics
• Statistics in business analytics and business intelligence
• Importance of statistics in physics.
• Deep discussion about multivariate statistics
• Uses of Statistics in machine learning

Survey Topics for Statistics

• Gather the data of the most qualified professionals in a specific area.
• Survey the time wasted by the students in watching Tvs or Netflix.
• Have a survey the fully vaccinated people in the USA 
• Gather information on the effect of a government survey on the life of citizens
• Survey to identify the English speakers in the world.

Statistics Research Paper Topics for Graduates

• Have a deep decision of Bayes theorems
• Discuss the Bayesian hierarchical models
• Analysis of the process of Japanese restaurants. 
• Deep analysis of Lévy’s continuity theorem
• Analysis of the principle of maximum entropy

AP Statistics Topics

• Discuss about the importance of econometrics
• Analyze the pros and cons of Probit Model
• Types of probability models and their uses
• Deep discussion of ortho stochastic matrix
• Find out the ways to get an adjacency matrix quickly

Good Statistics Research Topics 

• National income and the regulation of cryptocurrency.
• The benefits and drawbacks of regression analysis.
• How can estimate methods be used to correct statistical differences?
• Mathematical prediction models vs observation tactics.
• In sociology research, there is bias in quantitative data analysis.
• Inferential analytical approaches vs. descriptive statistics.
• How reliable are AI-based methods in statistical analysis?
• The internet news reporting and the fluctuations: statistics reports.
• The importance of estimate in modeled statistics and artificial sampling.

Business Statistics Topics

• Role of statistics in business in 2023
• Importance of business statistics and analytics
• What is the role of central tendency and dispersion in statistics
• Best process of sampling business data.
• Importance of statistics in big data.
• The characteristics of business data sampling: benefits and cons of software solutions.
• How may two different business tasks be tackled concurrently using linear regression analysis?
• In economic data relations, index numbers, random probability, and correctness are all important.
• The advantages of a dataset approach to statistics in programming statistics.
• Commercial statistics: how should the data be prepared for maximum accuracy?

Statistical Research Topics for College Students

• Evaluate the role of John Tukey’s contribution to statistics.
• The role of statistics to improve ADHD treatment.
• The uses and timeline of probability in statistics.
• Deep analysis of Gertrude Cox’s experimental design in statistics.
• Discuss about Florence Nightingale in statistics.
• What sorts of music do college students prefer?
• The Main Effect of Different Subjects on Student Performance.
• The Importance of Analytics in Statistics Research.
• The Influence of a Better Student in Class.
• Do extracurricular activities help in the transformation of personalities?
• Backbenchers’ Impact on Class Performance.
• Medication’s Importance in Class Performance.
• Are e-books better than traditional books?
• Choosing aspects of a subject in college

How To Write Good Statistics Research Topics?

So, the main question that arises here is how you can write good statistics research topics. The trick is understanding the methodology that is used to collect and interpret statistical data. However, if you are trying to pick any topic for your statistics project, you must think about it before going any further. 

As a result, it will teach you about the data types that will be researched because the sample will be chosen correctly. On the other hand, your basic outline for choosing the correct topics is as follows:

• Introduction of a problem
• Methodology explanation and choice. 
• Statistical research itself is in the main part (Body Part). 
• Samples deviations and variables. 
• Lastly, statistical interpretation is your last part (conclusion). 

Note:   Always include the sources from which you obtained the statistics data.

Top 3 Tips to Choose Good Statistics Research Topics

It can be quite easy for some students to pick a good statistics research topic without the help of an essay writer. But we know that it is not a common scenario for every student. That is why we will mention some of the best tips that will help you choose good statistics research topics for your next project. Either you are in a hurry or have enough time to explore. These tips will help you in every scenario.

1. Narrow down your research topic

We all start with many topics as we are not sure about our specific interests or niche. The initial step to picking up a good research topic for college or school students is to narrow down the research topic.

For this, you need to categorize the matter first. And then pick a specific category as per your interest. After that, brainstorm about the topic’s content and how you can make the points catchy, focused, directional, clear, and specific. 

2. Choose a topic that gives you curiosity

After categorizing the statistics research topics, it is time to pick one from the category. Don’t pick the most common topic because it will not help your grades and knowledge. Instead of it, please choose the best one, in which you have little information, or you are more likely to explore it.

In a statistics research paper, you always can explore something beyond your studies. By doing this, you will be more energetic to work on this project. And you will also feel glad to get them lots of information you were willing to have but didn’t get because of any reasons.

It will also make your professor happy to see your work. Ultimately it will affect your grades with a positive attitude.

3. Choose a manageable topic

Now you have decided on the topic, but you need to make sure that your research topic should be manageable. You will have limited time and resources to complete your project if you pick one of the deep statistics research topics with massive information.

Then you will struggle at the last moment and most probably not going to finish your project on time. Therefore, spend enough time exploring the topic and have a good idea about the time duration and resources you will use for the project. 

Statistics research topics are massive in numbers. Because statistics operations can be performed on anything from our psychology to our fitness. Therefore there are lots more statistics research topics to explore. But if you are not finding it challenging, then you can take the help of our statistics experts . They will help you to pick the most interesting and trending statistics research topics for your projects. 

With this help, you can also save your precious time to invest it in something else. You can also come up with a plethora of topics of your choice and we will help you to pick the best one among them. Apart from that, if you are working on a project and you are not sure whether that is the topic that excites you to work on it or not. Then we can also help you to clear all your doubts on the statistics research topic. 

Frequently Asked Questions

Q1. what are some good topics for the statistics project.

Have a look at some good topics for statistics projects:- 1. Research the average height and physics of basketball players. 2. Birth and death rate in a specific city or country. 3. Study on the obesity rate of children and adults in the USA. 4. The growth rate of China in the past few years 5. Major causes of injury in Football

Q2. What are the topics in statistics?

Statistics has lots of topics. It is hard to cover all of them in a short answer. But here are the major ones: conditional probability, variance, random variable, probability distributions, common discrete, and many more. 

Q3. What are the top 10 research topics?

Here are the top 10 research topics that you can try in 2023:

1. Plant Science 2. Mental health 3. Nutritional Immunology 4. Mood disorders 5. Aging brains 6. Infectious disease 7. Music therapy 8. Political misinformation 9. Canine Connection 10. Sustainable agriculture

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Stochastic Stability Analysis of Ship Rolling Based on the Stochastic Averaging Method and First Passage Probability

28 Pages Posted: 19 Sep 2024

Tianjin University

Jianxing Yu

Osaka University

China Ship Scientific Research Center

Hamburg University of Technology

Prediction of dynamics stability of ships under complex sea state is a major scientific issue in the field of safe navigation. Stability discrimination methods based on deterministic waves cannot accurately identify the instability domain in random waves. In this paper, the stability characteristics of nonlinear roll motion in random waves are investigated based on stochastic analysis method. The stability boundary of roll system with only parametric excitation is given based on the stochastic averaging method. The stochastic Hopf bifurcation phenomenon of is demonstrated. Based on the first passage theory, the first passage probability of the stochastic roll motion is calculated with given boundary conditions and initial conditions. The entire sea areas are divided into three parts, including high stability, medium stability, and low stability regions. Based on the first passage probability approach, the probabilities of the roll response exceeding 25° are calculated for different wave directions. The results of the two methods are compared and the pros and cons of using the two methods to assess the stability of large roll are analyzed. With the two methods combined, more information for stochastic stability of roll motion under different wave directions can be provided.

Keywords: Nonlinear roll, Additive excitation, Stochastic stability, Stochastic averaging method, First passage probability

Suggested Citation: Suggested Citation

Tianjin University ( email )

92, Weijin Road Nankai District Tianjin, 300072 China

Liqin Liu (Contact Author)

Osaka university ( email ).

1-1 Yamadaoka Suita Osaka, 565-0871 Japan

China Ship Scientific Research Center ( email )

Hamburg university of technology ( email ).

Hamburg Germany

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