Examples of Normalized variable in the following topics:
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- Indeed it is so common, that people often know it as the normal curve or normal distribution, shown in Figure 3.1.
- Variables such as SAT scores and heights of US adult males closely follow the normal distribution.
- Many variables are nearly normal, but none are exactly normal.
- Thus the normal distribution, while not perfect for any single problem, is very useful for a variety of problems.
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- A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
- If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
- Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.
- Least-squares and other statistical inference methods which are optimal for normally distributed variables often become highly unreliable.
- Thus when a random variable $x$ is distributed normally with mean $\mu$ and variance $\sigma^2$, we write $X\sim N\left ( \mu ,\sigma ^{2} \right )$
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- To calculate the probability that a variable is within a range in the normal distribution, we have to find the area under the normal curve.
- To calculate the probability that a variable is within a range in the normal distribution, we have to find the area under the normal curve.
- This tells us that there is a 69.50% percent chance that a variable is less than 0.51 sigmas above the mean.
- This table gives the cumulative probability up to the standardized normal value $z$.
- Interpret a $z$-score table to calculate the probability that a variable is within range in a normal distribution
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- Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:
- If you draw random samples of size n, then as n increases, the random variable ΣX which consists of sums tends to be normally distributed and
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- If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
- Normal distributions are extremely important in statistics, and are often used in the natural and social sciences for real-valued random variables whose distributions are not known.
- One reason for their popularity is the central limit theorem, which states that (under mild conditions) the mean of a large number of random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution.
- Another reason is that a large number of results and methods can be derived analytically, in explicit form, when the relevant variables are normally distributed.
- The simplest case of normal distribution, known as the Standard Normal Distribution, has expected value zero and variance one.
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- In order to consider a normal distribution or normal approximation, a standard scale or standard units is necessary.
- In order to consider a normal distribution or normal approximation, a standard scale or standard units is necessary.
- Some types of normalization involve only a rescaling, to arrive at values relative to some size variable.
- Standard scores are also called $z$-values, $z$-scores, normal scores, and standardized variables.
- Explain the significance of normalization of ratings and calculate this normalization
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- For example, heights in centimeters and weights in kilograms, even if both variables can be described by a normal distribution.
- This variable gives a measure of how far the variable is from the mean ($x-\mu$), then "normalizes" it by dividing by the standard deviation ($\sigma$).
- This new variable gives us a way of comparing different variables.
- Normally, this would mean we'd need to use calculus.
- Calculate the probability that a variable is within a certain range by finding its z-value and using the Normal curve
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- A random variable $x$, and its distribution, can be discrete or continuous.
- As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.
- Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers).
- Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of 100.
- The image shows the probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution.
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- The occurrence of the normal distribution in practical problems can be loosely classified into three categories: exactly normal distributions, approximately normal distributions, and distributions modeled as normal.
- This includes Binomial random variables, which are associated with binary response variables, and Poisson random variables, which are associated with rare events.
- How can we tell if data in a probability histogram are normal, or at least approximately normal?
- A normal probability plot is a graphical technique for normality testing--assessing whether or not a data set is approximately normally distributed.
- This is a sample of size 50 from a normal distribution, plotted as a normal probability plot.
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- Regression models are often used to predict a response variable $y$ from an explanatory variable $x$.
- It includes many techniques for modeling and analyzing several variables when the focus is on the relationship between a dependent variable and one or more independent variables.
- Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed.
- Regression models predict a value of the $Y$ variable, given known values of the $X$ variables.
- For any fixed value of $x$, the response $y$ varies according to a normal distribution.