How do you work with normal distribution pdf?

I would like to push back on "the normal distribution is one of the most common probability distributions in statistics." Though normal distributions are the most common when learning about statistics, I have often found that real-world data often doesn't fit within a normal distribution. With that said, validating normality is important whether visually through histograms or through a statistical test. This will often be my first step when reviewing a population's data as it determines my assumptions for following statistical analyses in respect to using parametric and non-parametric methods.

The first paragraph is wrong. "The normal distribution is one of the most common"LY USED "probability distributions in statistics." Note my additions in CAPS. "It describes how data values are distributed around a mean" (do you sample or population?) If population, every distribution with an expectation (population mean) does that. If sample mean, only in large samples, and then only for data drawn from populations with finite expected values. The main use for the normal distribution is in approximation of distribution of weighted averages of observations: asymptotic theory. Most data I have worked with are not normal, they are power law, or Poisson, multinomial or binomial, occasionally, truncated Cauchy or other fat tailed distribution.

While many real-world datasets are not normally distributed, most statistics computed from those datasets are asymptotically normal. Many times, asymptotic normality is the only way to draw explicit analytical results about a statistic of interest, which is why the normal distribution is so widely used. For example, it is known that maximum likelihood estimators (MLEs) are asymptotically normal with covariance equal to the inverse Fisher information. So while the exact covariance may be intractable, many times the Fisher information can be computed to give approximate uncertainties for your MLEs. This is very useful when designing experiments.

Even though "the normal distribution is one of the most common probability distributions in statistics" doesn't apply to population distribution, but it is probably correct when it comes to inference. I think there were too many people overly emphasized non-normal distribution and forgot about the central-limit theory. With sufficient sample, all the method based on asymptotic theory will be correct. Therefore, in my opinion, in the real-world practice instead of checking if the population has a normal distribution, the effort should be on figuring out if the sample size is large enough to allow CLT to work if we plan to use parametric methods. In addition, MLE follows normal distributions. EM converged to MLE under the right condition.

Methods to normalize data: 1. Logarithmic transformation 2. Square root transformation 3. Box-Cox transformation 4. Z-score transformation 5. Quantile normalization etc. It's important to note that the choice of transformation method depends on the nature of the data and the specific objectives of the analysis. It's also important to ensure that the transformed data is appropriate for the intended analysis and does not violate any assumptions.

The Shapiro-Wilk and Kolmogorov-Smirnov normality tests are subject to the same limitations as any statistical test relying on probability values. The larger the sample size, the more statistical power the tests have, and the more likely you are to have p values below .05. A p value below .05 indicates that the distribution is not normal. Histograms, boxplots, and skewness and kurtosis values are better with larger sample sizes. Skewness and Kurtosis values should be between -2 and +2, although values vary by field. High kurtosis values may be a sign that you have extreme outliers. A look at your boxplots can help determine if this is true for your data. Deleting extreme values may help normalize your distribution.

In general, testing for normality of the underlying data isn't necessary. The workhorse analytic tools we use (t-tests, OLS) assume normality of *parameter distribution* not the data itself. Non-parametric methods like kernel based analyses can certainly be useful irrespective of the distribution of data. But often these methods require some user degree of freedom (e.g., selecting the appropriate kernel or bandwidth).

The rule is widely used in various fields such as finance, engineering, and social sciences for quality control, risk management, and statistical analysis. It helps make predictions, set confidence intervals, and identify outliers in data sets. In addition, the rule is related to the Central Limit Theorem, which states that the sum or average of a large number of independent random variables, each with a finite mean and variance, will approach a normal distribution. This makes the empirical rule applicable in a broad range of situations, even when the underlying distribution is not normal.

Double check the tail behavior of your data and don’t throw away extreme observations (+/- 3 standard deviations) as “outliers” unless there is a very good reason to do so (e.g., data transcription error). If you find yourself with extreme observations, it is worth considering if your data are t-distributed (fatter tails than the normal) or Cauchy-distributed (fatter tails than the t-distribution). Ignoring extreme observations and assuming normality can lead to bad model predictions and have costly real word implications, especially in the context of financial models. For a case study, see the excellent book “When Genius Failed: The Rise and Fall of Long-Term Capital Management.”