A normal distribution is a common probability distribution. It has a shape often referred to as a "bell curve. Many everyday data sets typically follow a normal distribution: for example, the heights of adult humans, the scores on a test given to a large class, errors in measurements. The standard deviation is the measure of how spread out a normally distributed set of data is.
It is a statistic that tells you how closely all of the examples are gathered around the mean in a data set. The shape of a normal distribution is determined by the mean and the standard deviation. The steeper the bell curve, the smaller the standard deviation.
If the examples are spread far apart, the bell curve will be much flatter, meaning the standard deviation is large. In the figure below, this corresponds to the region shaded pink. A set of data is normally distributed with a mean of 5. What percent of the data is less than 5? A normal distribution is symmetric about the mean. So, half of the data will be less than the mean and half of the data will be greater than the mean.
The life of a fully-charged cell phone battery is normally distributed with a mean of 14 hours with a standard deviation of 1 hour. What is the probability that a battery lasts at least 13 hours? The mean is 14 and the standard deviation is 1. Your Practice.
Popular Courses. Fundamental Analysis Tools for Fundamental Analysis. What is Normal Distribution? Key Takeaways A normal distribution is the proper term for a probability bell curve. In a normal distribution the mean is zero and the standard deviation is 1. It has zero skew and a kurtosis of 3. Normal distributions are symmetrical, but not all symmetrical distributions are normal. In reality, most pricing distributions are not perfectly normal. Compare Accounts. The offers that appear in this table are from partnerships from which Investopedia receives compensation.
This compensation may impact how and where listings appear. Investopedia does not include all offers available in the marketplace. Related Terms Kurtosis Kurtosis is a statistical measure used to describe the distribution of observed data around the mean. It is sometimes referred to as the "volatility of volatility. Asymmetrical Distribution Asymmetrical distribution often occurs during volatile markets when the distribution of an asset's investment returns exhibits a skewed pattern.
What Is a Bell Curve? A bell curve describes the shape of data conforming to a normal distribution. Mesokurtic Mesokurtic is a statistical term describing the shape of a probability distribution. Discover more about mesokurtic distributions here.
What Is Symmetrical Distribution? Symmetrical distribution is evident when values of variables occur at a regular interval. In addition, the mean, median and mode occur at the same point. What Is Excess Kurtosis? Excess kurtosis describes a probability distribution with fat fails, indicating an outlier event has a higher than average chance of occurring.
Partner Links. If the shocks are truly independent and infinitesimal, then inevitably the distribution of an observable path will have normal distribution due to CLT, see e. He didn't even bother to call it by its today's name "Gaussian" or "normal". Another example is quantum mechanics. Hence, don't be surprised to get very different reactions to Gaussian distribution use from researchers in different fields. In some fields such as physics, certain phenomena are expected to be linked naturally to Gaussian distribution based on very solid theory backed by enormous amount of observations.
In other fields, Normal distribution is used for its technical convenience, handy mathematical properties or other questionable reasons.
Roll a single die, and you have an equal likelihood of rolling each number , and hence, the PDF is constant. Roll two dice and sum the results together, and the PDF is no longer constant. This is because there are 36 combinations, and the summative range is 2 to Now, looking at 7, there are multiple combinations, i. As you work away from the mid-value i. This example does not result in a clear normal distribution, but the more die you add, and the more samples you take, then the result will tend towards a normal distribution.
Therefore, as you sum a range of independent variables subject to random variation which each can have their own PDFs , the more the resulting output will tend to normality. This in Six Sigma terms give us what we call the 'Voice of the Process'.
This is what we call the result of 'common-cause variation' of a system, and hence, if the output is tending towards normality, then we call this system 'in statistical process control'.
Where the output is non-normal skewed or shifted , then we say the system is subject to 'special cause variation' in which there has been some 'signal' that has biased the outcome in some way. However, rearranging the problem a little, there is good reason to assume that is, to model a continuous quantity that you believe to have a fixed mean and variance with a Normal distribution. That's because the Normal distribution is the result of maximizing entropy subject to those moment constraints.
Since, roughly speaking, entropy is a measure of uncertainty, that makes the Normal the most non-commital or maximally uncertain choice of distributional form. Now, the idea that one should choose a distribution by maximizing its entropy subject to known constraints really does have some physics backing in terms of the number of possible ways to fulfill them. Jaynes on statistical mechanics is the standard reference here. Note that while maximum entropy motivates Normal distributions in this case, different sorts of constraints can be shown to lead to different distributional families, e.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Is there an explanation for why there are so many natural phenomena that follow normal distribution?
Ask Question. Asked 5 years, 7 months ago. Active 1 year, 2 months ago. Viewed 16k times. Improve this question. There are many phenomena and behaviors which are extreme valued, heavy-tailed or describe power law functions.
Gabaix documents many of the economic and financial variants of this distributional class in his paper Power Laws in Economics: An Introduction , ungated here Quoting the findings, "In just one case—the distribution of the frequencies of occurrence of words in English text—the power law appears to be truly convincing in the sense that it is an excellent fit to the data and none of the alternatives carries any weight.
That said, they do make the case for many distributions being heavy-tailed and, as Gabaix points out, these distributions are ubiquitous. Show 4 more comments. Active Oldest Votes. Improve this answer. Your argument begins to suggest--quite plausibly, in my view--that there may be a psychological answer to the question, such as groupthink: when everybody in your field sees normal distributions, who are you to say otherwise?
This would go especially for fields of inquiry where statistical procedures are viewed as pedestrian tools, necessary perhaps to sanctify a paper for publication, but otherwise of little inherent value or interest.
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