Absolute frequency is a statistical term describing the number of times a given value appears during a trial.

Absolute frequency is a statistical term describing the number of times a particular piece of data or a particular value appears during a trial or set of trials. When the absolute frequency of values is tracked over the entire trial, the absolute frequency for a particular value can then be divided by the total number of values for that variable throughout the trial to get the relative frequency. Number of Alcoholic Drinks Per Week
Number of Drinks per Week
Example table showing the absolute frequency of responses
There are a few observations you can make from the table displaying absolute frequency: more accountants drink some amount of alcohol than no alcohol. While absolute frequency is a very basic form of statistical analysis, it can be used as an input for more advanced statistical analysis, such as relative frequency. Absolute frequency is a statistical term describing the number of times a particular piece of data or a particular value appears during a trial or set of trials.

Absolute Frequency

The adjusted mean accounts for outlines and anomalies in a data set, thereby offering a more accurate mean average.

FASB Initiates Projects to Improve Measurement and Disclosure of Fair Value Estimates

The adjusted mean arises when statistical averages must be corrected to compensate for data imbalances and large variances. When sex is taken into account, it turns out that male accountants drink slightly less than accountants did 50 years ago, but the bulk of the change is the growth in the total number of female accountants. Outliers present in data sets will often be removed in order to determine the adjusted mean because they can have a large impact on the calculated means of small populations. However, doing further analysis of this data may suggest that one integrated mean isn't the best way to present this data. A simpler way to calculate an adjusted mean is to add categorical variables that separate the data more finely, such as gender.

Adjusted Mean

The algebraic method refers to various means of solving a pair of linear equations, including graphing, substitution, and elimination.

The algebraic method refers to various methods of solving a pair of linear equations, including graphing, substitution and elimination. For example, to solve:
8 x \+ 6 y \= 1 6 − 8 x − 4 y \= − 8 \\begin{aligned} &8x+6y=16\\\\ &{-8}x-4y=-8\\\\ \\end{aligned} ​8x+6y\=16−8x−4y\=−8​
First, use the second equation to express x in terms of y:
− 8 x \= − 8 \+ 4 y x \= − 8 \+ 4 y − 8 x \= 1 − 0 . 5 y {-8}x=-8+4yx=\\frac{-8+4y}{{-8}x}=1-0.5y −8x\=−8+4yx\=−8x−8+4y​\=1−0.5y
Then substitute 1 - 0.5y for x in the first equation:
8 ( 1 − 0 . 5 y ) \+ 6 y \= 1 6 8 − 4 y \+ 6 y \= 1 6 8 \+ 2 y \= 1 6 2 y \= 8 y \= 4 \\begin{aligned} &8\\left(1-0.5y\\right)+6y=16\\\\ &8-4y+6y=16\\\\ &8+2y=16\\\\ &2y=8\\\\ &y=4\\\\ \\end{aligned} ​8(1−0.5y)+6y\=168−4y+6y\=168+2y\=162y\=8y\=4​
Then replace y in the second equation with 4 to solve for x:
8 x \+ 6 ( 4 ) \= 1 6 8 x \+ 2 4 \= 1 6 8 x \= − 8 x \= − 1 \\begin{aligned} &8x+6\\left(4\\right)=16\\\\ &8x+24=16\\\\ &8x=-8\\\\ &x=-1\\\\ \\end{aligned} ​8x+6(4)\=168x+24\=168x\=−8x\=−1​
The second method is the elimination method. In the case of these two equations, we can add them together to eliminate x:
8 x \+ 6 y \= 1 6 − 8 x − 4 y \= − 8 0 \+ 2 y \= 8 y \= 4 \\begin{aligned} &8x+6y=16\\\\ &{-8}x-4y=-8\\\\ &0+2y=8\\\\ &y=4\\\\ \\end{aligned} ​8x+6y\=16−8x−4y\=−80+2y\=8y\=4​
Now, to solve for x, substitute the value for y in either equation:
8 x \+ 6 y \= 1 6 8 x \+ 6 ( 4 ) \= 1 6 8 x \+ 2 4 \= 1 6 8 x \+ 2 4 − 2 4 \= 1 6 − 2 4 8 x \= − 8 x \= − 1 \\begin{aligned} &8x+6y=16\\\\ &8x+6\\left(4\\right)=16\\\\ &8x+24=16\\\\ &8x+24-24=16-24\\\\ &8x=-8\\\\ &x=-1\\\\ \\end{aligned} ​8x+6y\=168x+6(4)\=168x+24\=168x+24−24\=16−248x\=−8x\=−1​
The algebraic method is a collection of several methods used to solve a pair of linear equations with two variables. With the substitution method, rearrange the equations to express the value of variables, x or y, in terms of another variable. The intersection of the two lines will be an x,y coordinate, which is the solution.

Algebraic Method

Anomaly is when the actual result under a given set of assumptions is different from the expected result.

In economics and finance, an anomaly is when the actual result under a given set of assumptions is different from the expected result predicted by a model. Common market anomalies include the small cap effect and the January effect. Here is a short list of superstitious market indicators:
The Super Bowl Indicator: When a team from the old American Football League wins the game, the market will close lower for the year. Like the September effect before it, the October effect is a market anomaly rather than an event with a causal relationship. The Aspirin Indicator: Stock prices and aspirin production are inversely related. This indicator suggests that when the market is rising, fewer people need aspirin to heal market-induced headaches.

Anomaly

A priori probability is a likelihood of occurrence that can be deduced logically by examining existing information. 

A priori probability refers to the likelihood of an event occurring when there is a finite amount of outcomes and each is equally likely to occur. The formula for calculating a priori probability is very straightforward:
A Priori Probability = Desired Outcome(s)/The Total Number of Outcomes
So the a priori probability of rolling a six on a six-sided die is one (the desired outcome of six) divided by six. In finance, people more commonly use empirical or subjective probability as opposed to classical probability. The formula for calculating the probability becomes much more complex as your chances are based on the combination of numbers on the ticket being randomly selected in the correct order, and you can buy multiple tickets with multiple number combinations. Outside of discouraging people from putting their financial fate in the hands of the lottery, most outcomes that people in finance care about do not have a finite number of outcomes.

A Priori Probability  & Example

Asymmetrical distribution often occurs during volatile markets when the distribution of an asset's investment returns exhibits a skewed pattern.

Asymmetrical distribution is a situation in which the values of variables occur at irregular frequencies and the mean, median, and mode occur at different points. Asymmetrical distribution is the opposite of symmetrical distribution, which is when investment returns follow a regular pattern often depicted as a bell curve. Asymmetrical distribution occurs when the distribution of an asset's investment returns exhibits a distorted or skewed pattern. In contrast to asymmetrical distribution, symmetrical distribution occurs when the values of variables appear at predictable frequencies and the mean, median, and mode occur at the same points. A core concept in technical trading, symmetrical distribution assumes that over time the price action of an asset will fit this distribution curve.

Asymmetrical Distribution

Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables.

Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables. The Boolean variables are represented as binary numbers to represent truths: 1 = true and 0 = false.
Elementary algebra deals with numerical operations whereas Boolean algebra deals with logistical operations. Boolean algebra is different from elementary algebra as the latter deals with numerical operations and the former deals with logical operations. Elementary algebra is expressed using basic mathematical functions, such as addition, subtraction, multiplication, and division, whereas Boolean algebra deals with conjunction, disjunction, and negation. Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables.

Boolean Algebra

A chi-square (χ2) statistic is a test that measures how expectations compare to actual observed data (or model results).

A chi-square (_χ_2) statistic is a test that measures how a model compares to actual observed data. If there is no relationship between sex and course selection (that is, if they are independent), then the actual frequencies at which male and female students select each offered course should be expected to be approximately equal, or conversely, the proportion of male and female students in any selected course should be approximately equal to the proportion of male and female students in the sample. Chi-square is a statistical test used to examine the differences between categorical variables from a random sample in order to judge goodness of fit between expected and observed results. The chi-square statistic compares the size of any discrepancies between the expected results and the actual results, given the size of the sample and the number of variables in the relationship. A chi-square test is appropriate for this when the data being analyzed is from a random sample, and when the variable in question is a categorical variable.

Chi-Square (χ2) Statistic

Copula or probability theory is a statistical technique used for multivariate uniform distributions modeling of options and derivatives.

The copula is a probability model that represent﻿s a multivariate uniform distribution, which examines the association or dependence between many variables. The copula, therefore, has been applied to areas of finance such as option pricing and portfolio value-at-risk to deal with skewed or asymmetric distributions. Advancements in Monte Carlo simulation methods and copula functions offer an enhancement to the pricing of bivariate contingent claims, such as derivatives with embedded options. However, correlation works best with normal distributions, while distributions in financial markets are often non-normal in nature﻿. Option theory, particularly options pricing is a highly specialized area of finance.

Copula

To detrend a forecasting model is to remove the effects of accumulating data sets from a trend to show only the absolute changes in values.

A detrend involves removing the effects of trend from a data set to show only the differences in values from the trend; it allows cyclical and other patterns to be identified. A few of the alternative options are quadratic detrending, using the Baxter-King filter (for moving average trend lines only), and using the Hodrick-Prescott filter (for cyclical components of a particular time series only).
Which method is the best for the project and data at hand will depend on numerous individual factors, including the particular field of study and whether or not the data is linearly correlated. A detrend involves removing the effects of trend from a data set to show only the differences in values from the trend; it allows cyclical and other patterns to be identified. Detrending shows a different aspect of time series data by removing deterministic and stochastic trends. The deterministic trends consistently decrease or increase, and the stochastic trends inconsistently decrease or increase.

Detrend

A discrete distribution is a statistical distribution that shows the probabilities of outcomes with finite values.

Table of Contents
What Is Discrete Distribution?
Understanding Discrete Distribution
Examples of Discrete Distribution
Discrete Distribution FAQs Unlike the normal distribution, which is continuous and accounts for any possible outcome along the number line, a discrete distribution is constructed from data that can only follow a finite or discrete set of outcomes. A discrete distribution is a probability distribution that depicts the occurrence of discrete (individually countable) outcomes, such as 1, 2, 3... or zero vs. one. A discrete probability model is a statistical tool that takes data following a discrete distribution and tries to predict or model some outcome, such as an options contract price, or how likely a market shock will be in the next 5 years. For a cumulative distribution, the probability of each discrete observation must be between 0 and​ 1; and the sum of the probabilities must equal one (100%).
If there are only a set array of possible outcomes (e.g. only zero or one, or only integers), then the data are discrete.

Discrete Distribution

Excess kurtosis describes a probability distribution with fat fails, indicating an outlier event has a higher than average chance of occurring.

The term excess kurtosis refers to a metric used in statistics and probability theory comparing the kurtosis coefficient with that of a normal distribution. Excess kurtosis can be positive (leptokurtic distribution), negative (platykurtic distribution), and at or near zero (mesokurtic distribution). Excess kurtosis means the distribution of event outcomes have lots of instances of outlier results, causing fat tails on the bell-shaped distribution curve. The term excess kurtosis refers to a metric used in statistics and probability theory comparing the kurtosis coefficient with that of a normal distribution. Since normal distributions have a kurtosis of three, excess kurtosis can be calculated by subtracting kurtosis by three.

Excess Kurtosis

A growth curve is a visual depiction of the growth of a phenomenon, with the x-axis typically representing time and the y-axis growth.

A growth curve is a graphical representation that shows the course of a phenomenon over time. This curve follows the general formula: V = S \(1 + R)t
The current value, V, of an initial starting point subject to exponential growth can be determined by multiplying the starting value, S, by the sum of one plus the rate of interest, R, raised to the power of t, or the number of periods that have elapsed.
! In the image below, the growth curve displayed represents the growth of a population in millions over a span of decades. The shape of this growth curve indicates exponential growth. Growth curves are typically displayed on a set of axes where the x-axis is time and the y-axis shows an amount of growth.

Growth Curve

Homoskedastic refers to a condition in which the variance of the error term in a regression model is constant. 

Homoskedastic (also spelled "homoscedastic") refers to a condition in which the variance of the residual, or error term, in a regression model is constant. A plot of the error term data may show a large amount of study time corresponded very closely with high test scores but that low study time test scores varied widely and even included some very high scores. With these two variables, more of the variance of the test scores would be explained and the variance of the error term might then be homoskedastic, suggesting that the model was well-defined. If, for example, some students had seen the answers ahead of time, the regression model would then have two explanatory variables: time studying, and whether the student had prior knowledge of the answers. The error term would show the amount of variance in the test scores that was not explained by the amount of time studying.

Homoskedastic

An inverse correlation is a relationship between two variables such that when one variable is high the other is low and vice versa.

Trade Weighted U.S. Dollar Index vs. Gold Fixing Price

An inverse correlation, also known as negative correlation, is a contrary relationship between two variables such that when the value of one variable is high then the value of the other variable is probably low. The same must be done for the Y values:
SUM ( X 2 ) \= ( 5 5 2 ) \+ ( 3 7 2 ) \+ ( 10 0 2 ) \+ … \+ ( 8 8 2 ) \= 28 , 623 \\text{SUM}(X^2) = (55^2) + (37^2) + (100^2) + \\dotso + (88^2) = 28,623 SUM(X2)\=(552)+(372)+(1002)+…+(882)\=28,623
SUM ( Y 2 ) \= ( 9 1 2 ) \+ ( 6 0 2 ) \+ ( 7 0 2 ) \+ … \+ ( 3 0 2 ) \= 35 , 971 \\text{SUM}(Y^2) = (91^2) + (60^2) + (70^2) + \\dotso + (30^2) = 35,971 SUM(Y2)\=(912)+(602)+(702)+…+(302)\=35,971
Noting there are seven observations, _n_, the following formula can be used to find the correlation coefficient, _r:_
r \= \[ n × ( SUM ( X , Y ) − ( SUM ( X ) × ( SUM ( Y ) ) \] \[ ( n × SUM ( X 2 ) − SUM ( X ) 2 \] × \[ n × SUM ( Y 2 ) − SUM ( Y ) 2 ) \] r = \\frac{\[n \\times (\\text{SUM}(X,Y) - (\\text{SUM}(X) \\times ( \\text{SUM}(Y) ) \]} {\\sqrt{\[(n \\times \\text{SUM}(X^2) - \\text{SUM}(X)^2 \] \\times \[n \\times \\text{SUM}(Y^2) - \\text{SUM}(Y)^2)\]}} r\=\[(n×SUM(X2)−SUM(X)2\]×\[n×SUM(Y2)−SUM(Y)2)\]​\[n×(SUM(X,Y)−(SUM(X)×(SUM(Y))\]​ First, add up all the X values to find SUM(X), add up all the Y values to find SUM(Y) and multiply each X value with its corresponding Y value and sum them to find SUM(X,Y):
SUM ( X ) \= 55 \+ 37 \+ 100 \+ 40 \+ 23 \+ 66 \+ 88 \= 409 \\begin{aligned} \\text{SUM}(X) &= 55 + 37 + 100 + 40 + 23 + 66 + 88 \\\\ &= 409 \\\\ \\end{aligned} SUM(X)​\=55+37+100+40+23+66+88\=409​
SUM ( Y ) \= 91 \+ 60 \+ 70 \+ 83 \+ 75 \+ 76 \+ 30 \= 485 \\begin{aligned} \\text{SUM}(Y) &= 91 + 60 + 70 + 83 + 75 + 76 + 30 \\\\ &= 485 \\\\ \\end{aligned} SUM(Y)​\=91+60+70+83+75+76+30\=485​
SUM ( X , Y ) \= ( 55 × 91 ) \+ ( 37 × 60 ) \+ … \+ ( 88 × 30 ) \= 26 , 926 \\begin{aligned} \\\\\\text{SUM}(X,Y) &= (55 \\times 91) + (37 \\times 60) + \\dotso + (88 \\times 30) \\\\&= 26,926 \\\\\\end{aligned} SUM(X,Y)​\=(55×91)+(37×60)+…+(88×30)\=26,926​
The next step is to take each X value, square it and sum up all these values to find SUM(x2). In this example, the correlation is:
r \= ( 7 × 26 , 926 − ( 409 × 485 ) ) ( ( 7 × 28 , 623 − 40 9 2 ) × ( 7 × 35 , 971 − 48 5 2 ) ) r = \\frac{(7 \\times 26,926 - (409 \\times 485))} {\\sqrt{((7 \\times 28,623 - 409^2) \\times (7 \\times 35,971 - 485^2))}} r\=((7×28,623−4092)×(7×35,971−4852))​(7×26,926−(409×485))​
r \= 9 , 883 ÷ 23 , 414 r = 9,883 \\div 23,414 r\=9,883÷23,414
r \= − 0.42 r = -0.42 r\=−0.42
The two data sets have a correlation of -0.42, which is called an inverse correlation because it is a negative number. Assume an analyst needs to calculate the degree of correlation between the X and Y in the following data set with seven observations on the two variables:
X: 55, 37, 100, 40, 23, 66, 88
Y: 91, 60, 70, 83, 75, 76, 30
There are three steps involved in finding the correlation.

Inverse Correlation

Look-ahead bias occurs when information or data is used in a study or simulation that would not have been known or available during the period analyzed.

Look-ahead bias occurs by using information or data in a study or simulation that would not have been known or available during the period being analyzed. If you took the top performing stocks at the end of the year and then tried to choose common data points they had at the start of the year, such as the trailing P/E ratio range, you'd be falling prey to a look-ahead bias because you'd only be looking at stocks you know enjoyed significant growth rather than at all stocks with a similar trailing P/E ratio range at that time. For example, if a trade is simulated based on information that was not available at the time of the trade—such as a quarterly earnings number that was released a month later—it will diminish the accuracy of the trading strategy's true performance and potentially bias the results in favor of the desired outcome.
Look-ahead bias is one of many biases that must be accounted for when running simulations. If an investor is backtesting the performance of a trading strategy, it is vital that they only use information that would have been available at the time of the trade to avoid a look-ahead bias. Other common biases are sample selection bias, time period bias, and survivorship bias.

Look-Ahead Bias

Mesokurtic is a statistical term describing the shape of a probability distribution. Discover more about mesokurtic distributions here.

Mesokurtic is a statistical term used to describe the outlier characteristic of a probability distribution in which extreme events (or data that are rare) is close to zero. Mesokurtic distributions have the same kurtosis as that of the normal distribution, or normal curve, also known as a bell curve. Meanwhile, platykurtic distributions, on the other hand, have lighter tails, and the probability of extreme events is lesser than that implied by the normal curve. Mesokurtic is a statistical term used to describe the outlier characteristic of a probability distribution in which extreme events (or data that are rare) is close to zero. Mesokurtic distributions are similar to normal distributions, in which extreme or outlier events are very unlikely.

Mesokurtic

The mode is a statistical term that refers to the most frequently occurring number found in a set of numbers.

The mode is the value that appears most frequently in a data set. For example, in the following list of numbers, 16 is the mode since it appears more times in the set than any other number:
3, 3, 6, 9, **16, 16, 16**, 27, 27, 37, 48
A set of numbers can have more than one mode (this is known as _bimodal_ if there are two modes) if there are multiple numbers that occur with equal frequency, and more times than the others in the set.
**3, 3, 3**, 9, **16, 16, 16**, 27, 37, 48
In the above example, both the number 3 and the number 16 are modes as they each occur three times and no other number occurs more often.
If no number in a set of numbers occurs more than once, that set has no mode:
3, 6, 9, 16, 27, 37, 48
A set of numbers with two modes is **bimodal**, a set of numbers with three modes is **trimodal**, and any set of numbers with more than one mode is **multimodal**. Other popular measures of central tendency include the mean, or the average of a set, and the median, the middle value in a set.
The mode can be the same value as the mean and/or median, but this is usually not the case. A set of data may have one mode, more than one mode, or no mode at all. This means that this value is the average value, the middle value, also the mode—the most frequently occurring value in the data.

Mode

The nominal yield spread is the difference between a Treasury security and the non-Treasury version of that same security. 

A nominal yield spread is the difference, expressed in basis points, between the Treasury and non-Treasury security of the same maturity. A nominal yield spread defines one point along the entire Treasury yield curve to determine the spread, at that particular single point, where the price of the security and the present value of the security's cash flow are equal. The nominal yield spread is considered easy to understand since it involves assessing the difference between a corporate bond yield to maturity and the value of the equivalent Treasury bond having the same timeline. A zero-volatility spread (Z-spread) measures the spread realized by the investor over the entire Treasury spot-rate curve, assuming the bond would be held until maturity. A nominal yield spread is the difference, expressed in basis points, between the Treasury and non-Treasury security of the same maturity.

Nominal Yield Spread

Nonlinear regression is a form of regression analysis in which data fit to a model is expressed as a mathematical function.

Nonlinear regression is a form of regression analysis in which data is fit to a model and then expressed as a mathematical function. Likewise, it’s possible to use algebra to transform a nonlinear equation so that mimics a linear equation—such a nonlinear equation is referred to as “intrinsically linear.”
Linear regression relates two variables with a straight line; nonlinear regression relates the variables using a curve.
One example of how nonlinear regression can be used is to predict population growth over time. Simple linear regression relates two variables (X and Y) with a straight line (y = mx + b), while nonlinear regression relates the two variables in a nonlinear (curved) relationship. Nonlinear regression is a curved function of an X variable (or variables) that is used to predict a Y variable
Nonlinear regression can show a prediction of population growth over time. A scatterplot of changing population data over time shows that there seems to be a relationship between time and population growth, but that it is a nonlinear relationship, requiring the use of a nonlinear regression model.

Defining Nonlinear Regression

Options have a high degree of nonlinearity, which may make them seem unpredictable. Learn about nonlinearity and how to manage your options trading risk.

Nonlinearity is a term used in statistics to describe a situation where there is not a straight-line or direct relationship between an independent variable and a dependent variable. In contrast to a nonlinear relationship, a linear relationship refers to a situation where there is a direct correlation between an independent variable and a dependent variable. While a linear relationship creates a straight line when plotted on a graph, a nonlinear relationship does not create a straight line but instead creates a curve. Nonlinearity is a term used in statistics to describe a situation where there is not a straight-line or direct relationship between an independent variable and a dependent variable. Nonlinearity is a mathematical term describing a situation where the relationship between an independent variable and a dependent variable is not predictable from a straight line.

Nonlinearity

Nonparametric method refers to a type of statistic that does not require that the data being analyzed meet certain assumptions or parameters. 

The nonparametric method refers to a type of statistic that does not make any assumptions about the characteristics of the sample (its parameters) or whether the observed data is quantitative or qualitative. Since nonparametric statistics makes fewer assumptions about the sample data, its application is wider in scope than parametric statistics. This is because nonparametric statistics discard some information that is available in the data, unlike parametric statistics. Nonparametric statistics do not require that the population data meet the assumptions required for parametric statistics. The nonparametric method is a branch of statistics in which the data are not assumed to come from prescribed models that are determined by a small number of parameters.

Nonparametric Method

A Poisson distribution is a statistical distribution showing the likely number of times that an event will occur within a specified period of time. 

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. Poisson Distribution Formula. C.K.Taylor
_e_ is Euler's number (_e_ = 2.71828...)
_x_ is the number of occurrences
_x_! is the factorial of _x_
λ is equal to the expected value (EV) of _x_ when that is also equal to its variance
Given data that follows a Poisson distribution, it appears graphically as:
![Poisson Distribution Example](data:image/gif;charset=utf-8;base64,R0lGODlhCgAGAPQAAD9KeNvT0Obe2+ff3Obg1Ofh1eTh3OXm2Ojh2+vj2Ojg3evi3eri3+rk2Ovl2erm2+nm3+vk3uzh3+zj3uzk2evj4Orl4e3i4Ozk4T9KeD9KeD9KeD9KeD9KeD9KeD9KeCwAAAAACgAGAEQIOQAJFAgwwIACAQgGDKAw4YLDCxMYNHSAwYLDCBgjXEiwQMIBCSBDMnwwYQIACCgvNGDAkmWFlxcCAgA7)
Poisson Distribution Example. In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals. If we further assume 100 random trials, the Poisson distribution describes the likelihood of getting a certain number of errors over some period of time, such as a single day.

Poisson Distribution

Posterior probability is the revised probability of an event occurring after taking into consideration new information. 

A posterior probability, in Bayesian statistics, is the revised or updated probability of an event occurring after taking into consideration new information. The formula to calculate a posterior probability of A occurring given that B occurred:
P ( A ∣ B ) \= P ( A ∩ B ) P ( B ) \= P ( A ) × P ( B ∣ A ) P ( B ) where: A , B \= Events P ( B ∣ A ) \= The probability of B occurring given that A is true P ( A )  and  P ( B ) \= The probabilities of A occurring and B occurring independently of each other \\begin{aligned}&P(A \\mid B) = \\frac{P(A \\cap B)}{P(B)} = \\frac{P(A) \\times P(B \\mid A)}{P(B)}\\\\&\\textbf{where:}\\\\&A, B=\\text{Events}\\\\&P(B \\mid A)=\\text{The probability of B occurring given that A}\\\\&\\text{is true}\\\\&P(A) \\text{ and }P(B)=\\text{The probabilities of A occurring}\\\\&\\text{and B occurring independently of each other}\\end{aligned} ​P(A∣B)\=P(B)P(A∩B)​\=P(B)P(A)×P(B∣A)​where:A,B\=EventsP(B∣A)\=The probability of B occurring given that Ais trueP(A) and P(B)\=The probabilities of A occurringand B occurring independently of each other​
The posterior probability is thus the resulting distribution, P(A|B).
Bayes' theorem can be used in many applications, such as medicine, finance, and economics. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred. A posterior probability, in Bayesian statistics, is the revised or updated probability of an event occurring after taking into consideration new information.

Posterior Probability

A prior probability, in Bayesian statistical inference, is the probability of an event based on established knowledge, before empirical data is collected.

Prior probability, in Bayesian statistical inference, is the probability of an event before new data is collected. Baye’s theorem is a very common and fundamental theorem used in data mining and machine learning.
P ( A ∣ B )   \=   P ( A ∩ B ) P ( B )   \=   P ( A )   ×   P ( B ∣ A ) P ( B ) where: P ( A )   \=   the prior probability of  A  occurring P ( A ∣ B ) \=   the conditional probability of  A    given that  B  occurs P ( B ∣ A )   \=   the conditional probability of  B      given that  A  occurs \\begin{aligned}&P(A\\mid B)\\ =\\ \\frac{P(A\\cap B)}{P(B)}\\ = \\ \\frac{P(A)\\ \\times\\ P(B\\mid A)}{P(B)}\\\\&\\textbf{where:}\\\\&P(A)\\ =\\ \\text{the prior probability of }A\\text{ occurring}\\\\&P(A\\mid B)=\\ \\text{the conditional probability of }A\\\\&\\qquad\\qquad\\quad\\ \\text{ given that }B\\text{ occurs}\\\\&P(B\\mid A)\\ = \\ \\text{the conditional probability of }B\\\\&\\qquad\\qquad\\quad\\ \\ \\text{ given that }A\\text{ occurs}\\\\&P(B)\\ =\\ \\text{the probability of }B\\text{ occurring}\\end{aligned} ​P(A∣B) \= P(B)P(A∩B)​ \= P(B)P(A) × P(B∣A)​where:P(A) \= the prior probability of A occurringP(A∣B)\= the conditional probability of A  given that B occursP(B∣A) \= the conditional probability of B   given that A occurs​
If we are interested in the probability of an event of which we have prior observations; we call this the prior probability. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred. If there is a second event that affects P(A), which we'll call event B, then we want to know what the probability of A is given B has occurred. But if a drilling test is conducted on acre B, and the results indicate that no oil is present at the location, then the posterior probability of oil being found on acres A and C become 0.5, as each acre has one out of two chances.

Prior Probability

Robust is a characteristic describing a model's, test's or system's ability to effectively perform while its variables or assumptions are altered.

In the world of investing, robust is a characteristic describing a model's, test's, or system's ability to perform effectively while its variables or assumptions are altered. While investors analyze a corporation’s fundamental data in order to find securities that are priced below market value and are therefore perceived to be a good investment, traders analyze a security’s price data using technical analysis to forecast price movements that result from disparities in the security’s supply and demand of the moment. Traders that use computerized trading systems to analyze and trade markets using technical analysis do so by developing, testing, and optimizing statistical models based on the application of technical indicators to the price data of a security. If a trading system does not deliver positive results over different time frames or during changing market conditions, it is not robust. However, when market conditions change, or the model is applied to another time period or the future, the model fails horribly, and losses are realized.

Robust

Traders use a runs test to determine the randomness of data by revealing any variables that might affect data patterns, such as a stock's price movement.

A runs test is a statistical procedure that examines whether a string of data is occurring randomly from a specific distribution. Two powerful ways traders can use a runs test include:
1.  Testing the randomness of distribution, by taking the data in the given order and marking with a plus (+) the data greater than the median, and with a minus (-) the data less than the median (numbers equalling the median are omitted.)
2.  Testing whether a function fits well to a data set, by marking the data exceeding the function value with + and the other data with −. A runs test, also known as the Wald–Wolfowitz runs test, was developed by mathematicians Abraham Wald and Jacob Wolfowitz. The runs test is a shortened version of the full name: the Wald–Wolfowitz runs test, so named after mathematicians Abraham Wald and Jacob Wolfowitz. Some statisticians believe another type of runs test—the Kolmogorov–Smirnov test—is a better tool than the Wald-Wolfowitz test for detecting differences between distributions.

Runs Test

Sample selection bias is a type of bias caused by using non-random data for statistical analysis. Learn ways to avoid sample selection bias.

Sample selection bias is a type of bias caused by choosing non-random data for statistical analysis. There are several types of sample selection bias, including pre-screening bias, self-selection bias, exclusion bias, and observer bias. Self-selection bias—also known as volunteer response bias—occurs when the study organizers allow participants to self-select or volunteer to participate. One way to correct sample selection bias is to assign weights to misrepresented subgroups in order to statistically correct the bias. Sample selection bias is a type of bias caused by choosing non-random data for statistical analysis.

Sample Selection Bias

A Scheffé test is a statistical test that is post-hoc test used in statistical analysis. It was named after American statistician Henry Scheffé.

A 'Mixed Model' for the Analysis of Variance

A Scheffé test is a statistical test that is a post-hoc test used in statistical analysis. While the Scheffé test has the advantage of giving the experimenter the flexibility to test any comparisons that appear interesting, the drawback of this flexibility is that the test has relatively lower statistical power than tests that are designed for pre-planned comparisons. The Scheffé test is used to make unplanned comparisons, rather than pre-planned comparisons, among group means in an analysis of variance (ANOVA) experiment. The Scheffé test is used to make unplanned comparisons, rather than pre-planned comparisons, among group means in an analysis of variance (ANOVA) experiment. A drawback of the Scheffé test is that the test has relatively lower statistical power than tests that are designed for pre-planned comparisons.

Scheffé Test

Semi-deviation is a method of evaluating the below-mean fluctuations in the returns on investment. It is used as an alternative to standard deviation.

Semi-deviation is a method of measuring the below-mean fluctuations in the returns on investment. The formula for semi-deviation is:
Semi-deviation   \=   1 n   ×   ∑ r t   <   Average n ( Average   −   r t ) 2 where: n   \=   the total number of observations below the mean r t   \=   the observed value \\begin{aligned}&\\text{Semi-deviation}\\ =\\ \\sqrt{\\frac{1}{n}\\ \\times\\ \\sum^n\_{r\_t\\ <\\ \\text{Average}}(\\text{Average}\\ -\\ r\_t)^2}\\\\&\\textbf{where:}\\\\&n\\ =\\ \\text{the total number of observations below the mean}\\\\&r\_t\\ =\\ \\text{the observed value}\\\\&\\text{average}\\ =\\text{the mean or target value of a data set}\\end{aligned} ​Semi-deviation \= n1​ × rt​ < Average∑n​(Average − rt​)2​where:n \= the total number of observations below the meanrt​ \= the observed value​
An investor's entire portfolio could be evaluated according to the semi-deviation in the performance of its assets. Harry Markowitz demonstrated how to exploit the averages, variances, and covariances of the return distributions of assets of a portfolio in order to compute an efficient frontier on which every portfolio achieves the expected return for a given variance or minimizes the variance for a given expected return. Semi-deviation is an alternative to the standard deviation for measuring an asset's degree of risk. In investing, semi-deviation is used to measure the dispersion of an asset's price from an observed mean or target value.

Semi-Deviation

In statistics, a spurious correlation, or spuriousness, refers to a connection between two variables that appears causal but is not.

What to Make of the Super Bowl Indicator.

Table of Contents
What Is Spurious Correlation?
Understanding Spurious Correlation
Spotting Spuriousness
Spurious Correlation Examples
How to Spot Spurious Correlation?
What Is an Example of Correlation but not Causation?
What Is Spurious Regression?
What Is False Causality? Spurious regression is a statistical model that shows misleading statistical evidence of a linear relationship; in other words, a spurious correlation between independent non-stationary variables.
False causality refers to the assumption made that one thing causes something else because of a relationship between them. Table of Contents
What Is Spurious Correlation?
Understanding Spurious Correlation
Spotting Spuriousness
Spurious Correlation Examples
How to Spot Spurious Correlation?
What Is an Example of Correlation but not Causation?
What Is Spurious Regression?
What Is False Causality? When two random variables track each other closely on a graph, it is easy to suspect correlation where a change in one variable causes a change in the other variable. In late January, there is often chatter about the so-called Super Bowl indicator, which suggests that a win by the American Football Conference team likely means that the stock market will go down in the coming year, whereas a victory by the National Football Conference team portends a rise in the market.

Spurious Correlation

The Texas Sharpshooter Fallacy is an analysis of outcomes that can give the illusion of causation rather than attributing the outcomes to chance. 

The Texas Sharpshooter Fallacy is when outcomes are analyzed out of context, giving the illusion of causation rather than attributing the outcomes to chance. _The Texas Sharpshooter Fallacy is a logical fallacy based on the metaphor of a gunman shooting the side of a barn, then drawing targets around the bullethole clusters to make it look like he hit the target._
_It illustrates how people look for similarities, ignoring differences, and do not account for randomness._
_The Texas Sharpshooter Fallacy is only one of many fallacies a wise investor should understand and avoid._
The Texas Sharpshooter Fallacy also called a clustering illusion, takes its name from the metaphor of a gunman who shoots at a side of a barn, and only later draws targets around a cluster of points that were hit. The Texas Sharpshooter Fallacy fails to take randomness into account when determining cause and effect, instead emphasizing how outcomes are similar rather than how they are different. The Texas Sharpshooter Fallacy is when outcomes are analyzed out of context, giving the illusion of causation rather than attributing the outcomes to chance. The Texas Sharpshooter Fallacy is only one of many fallacies a wise investor should understand and avoid.

Texas Sharpshooter Fallacy

The three-way ANOVA is used by statisticians to determine whether there is a three-way relationship among variables on an outcome.

The three-way ANOVA is used by statisticians to determine whether there is a three-way relationship among variables on an outcome. The three-way ANOVA is used by statisticians to determine whether there is a three-way relationship among variables on an outcome. A pharmaceutical company, for example, may do a three-way ANOVA to determine the effect of a drug on a medical condition. For example, a pharmaceutical company may do a three-way ANOVA to test a drug, given the different genders or ethnicities of their subjects. A three-way ANOVA tests which of three separate variables have an effect on an outcome, and the relationship between the three variables.

Three-Way ANOVA

A two-way ANOVA test is a statistical test used to determine the effect of two nominal predictor variables on a continuous outcome variable. 

A two-way ANOVA is an extension of the one-way ANOVA (analysis of variances) that reveals the results of two independent variables on a dependent variable. A two-way ANOVA test is a statistical test used to determine the effect of two nominal predictor variables on a continuous outcome variable. A two-way ANOVA test is a statistical technique that analyzes the effect of the independent variables on the expected outcome along with their relationship to the outcome itself. A two-way ANOVA test analyzes the effect of the independent variables on the expected outcome along with their relationship to the outcome itself.

Two-Way ANOVA

A type I error is a kind of error that occurs when a null hypothesis is rejected, although it is true. Discover more about the type I error.

A type I error is a kind of fault that occurs during the hypothesis testing process when a null hypothesis is rejected, even though it is accurate and should not be rejected. If—when the test is conducted—the result seems to indicate that the stimuli applied to the test subject cause a reaction, the null hypothesis stating that the stimuli do not affect the test subject would, in turn, need to be rejected. A type I error is a kind of fault that occurs during the hypothesis testing process when a null hypothesis is rejected, even though it is accurate and should not be rejected. In some cases, the null hypothesis assumes that there's no cause and effect relationship between the item being tested and the stimuli being applied to the test subject to trigger an outcome to the test. A type I error occurs during hypothesis testing when a null hypothesis is rejected, even though it is accurate and should not be rejected.

Introduction to the Type 1 Error

An unconditional probability is an independent chance that a single outcome results from a sample of possible outcomes.

An unconditional probability is the chance that a single outcome results among several possible outcomes. The unconditional probability of an event can be determined by adding up the outcomes of the event and dividing by the total number of possible outcomes.
P ( A )   \=   Number of Times ‘ A ’ Occurs Total Number of Possible Outcomes P(A)\\ =\\ \\frac{\\text{Number of Times \`}A\\text{' Occurs}}{\\text{Total Number of Possible Outcomes}} P(A) \= Total Number of Possible OutcomesNumber of Times ‘A’ Occurs​
Unconditional probability is also known as marginal probability and measures the chance of an occurrence ignoring any knowledge gained from previous or external events. The probability that snow will fall in Jackson, Wyoming, on Groundhog Day, without taking into consideration the historical weather patterns and climate data for northwestern Wyoming in early February is an example of an unconditional probability. Conditional probability, on the other hand, is the likelihood of an event or outcome occurring, but based on the occurrence of some other event or prior outcome. Unconditional probability reflects the chance that some event will occur without accounting for any other possible influences or prior outcomes.

Unconditional Probability

Uniform distribution is a type of probability distribution in which all outcomes are equally likely. Learn how to calculate uniform distribution.

Table of Contents
What Is Uniform Distribution? The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.
Uniform distributions are probability distributions with equally likely outcomes.
In a discrete uniform distribution, outcomes are discrete and have the same probability. Since there are only four suits, pulling a heart yields a probability of 1/4 or 25%.
Uniform distribution is a probability distribution that asserts that the outcomes for a discrete set of data have the same probability. The formula for a discrete uniform distribution is
P x \= 1 n where: P x \= Probability of a discrete value n \= Number of values in the range \\begin{aligned}&P\_x = \\frac{ 1 }{ n } \\\\&\\textbf{where:} \\\\&P\_x = \\text{Probability of a discrete value} \\\\&n = \\text{Number of values in the range} \\\\\\end{aligned} ​Px​\=n1​where:Px​\=Probability of a discrete valuen\=Number of values in the range​
As with the example of the die, each side contains a unique whole number. In a normal distribution, the area under the curve equals 1 and 68.27% of all data falls within 1 standard deviation_—_how dispersed the numbers are_—_from the mean; 95.45% of all data falls within 2 standard deviations from the mean, and approximately 99.73% of all data falls within 3 standard deviations from the mean.

Uniform Distribution

Variable overhead efficiency variance is the difference between actual variable overhead and standard variable overhead based on time budgeted for it.

Variable overhead efficiency variance refers to the difference between the true time it takes to manufacture a product and the time budgeted for it, as well as the impact of that difference. In numerical terms, variable overhead efficiency variance is defined as:
VOEV \= ( ALH − BLH ) × Hourly Rate where: VOEV \= Variable overhead efficiency variance ALH \= Actual labor hours BLH \= Budgeted labor hours Hourly Rate \= Rate for standard variable overhead \\begin{aligned} &\\text{VOEV} = ( \\text{ALH} - \\text{BLH} ) \\times \\text{Hourly Rate} \\\\ &\\textbf{where:} \\\\ &\\text{VOEV} = \\text{Variable overhead efficiency variance} \\\\ &\\text{ALH} = \\text{Actual labor hours} \\\\ &\\text{BLH} = \\text{Budgeted labor hours} \\\\ &\\text{Hourly Rate} = \\text{Rate for standard variable overhead} \\\\ \\end{aligned} ​VOEV\=(ALH−BLH)×Hourly Ratewhere:VOEV\=Variable overhead efficiency varianceALH\=Actual labor hoursBLH\=Budgeted labor hoursHourly Rate\=Rate for standard variable overhead​ If actual labor hours are less than the budgeted or standard amount, the variable overhead efficiency variance is favorable; if actual labor hours are more than the budgeted or standard amount, the variance is unfavorable. Variable overhead efficiency variance is one of the two components of total variable overhead variance, the other being variable overhead spending variance. In this case, the unfavorable variable overhead efficiency variance is (2,200 – 2,000) x $20 = $4,000; the variance is unfavorable because the company took more time than budgeted to produce the 1,000 widgets.

Variable Overhead Efficiency Variance

Yield variance is the difference between actual output and standard output of a production or manufacturing process, based on standard inputs of materials and labor.

Yield variance is the difference between actual output and standard output of a production or manufacturing process, based on standard inputs of materials and labor. The yield variance is:
(1.25M actual toy output - 1.5M expected toy output) \* $0.50 per unit cost = $125,000 unfavorable yield variance
Yield variance measures the difference between actual output and standard output of a production or manufacturing process. Yield Variance \= SC ∗ ( Actual Yield  −  Standard Yield ) where: SC = Standard unit cost \\begin{aligned} &\\text{Yield Variance}=\\text{SC}\*\\left(\\text{Actual Yield }-\\text{ Standard Yield}\\right)\\\\ &\\textbf{where:}\\\\ &\\text{SC = Standard unit cost}\\\\ \\end{aligned} ​Yield Variance\=SC∗(Actual Yield − Standard Yield)where:SC = If 1,000 units of a product are the standard output based on 1,000 kilograms of materials in an 8-hour production unit, and the actual output is 990 units, there is an unfavorable yield variance of 10 units (1,000 - 990). Yield variance is the difference between actual output and standard output of a production or manufacturing process, based on standard inputs of materials and labor.

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