If you are a skeptic, you are wondering how can GPAs and the exact diameter of holes drilled by some machine have the same distribution—they are not even measured with the same units.
In order to see that so many things have the same normal shape, all must be measured in the same units or have the units eliminated —they must all be standardized.
Statisticians standardize many measures by using the standard deviation. All normal distributions have the same shape because they all have the same relative frequency distribution when the values for their members are measured in standard deviations above or below the mean. Using the customary Canadian system of measurement, if the weight of pet dogs is normally distributed with a mean of Any normally distributed population will have the same proportion of its members between the mean and one standard deviation below the mean.
Converting the values of the members of a normal population so that each is now expressed in terms of standard deviations from the mean makes the populations all the same.
This process is known as standardization , and it makes all normal populations have the same location and shape. This standardization process is accomplished by computing a z-score for every member of the normal population. The z-score is found by:. This converts the original value, in its original units, into a standardized value in units of standard deviations from the mean.
Look at the formula. It can be measured in centimeters, or points, or whatever. If the numerator is 15 cm and the standard deviation is 10 cm, then the z will be 1. This particular member of the population, one with a diameter 15 cm greater than the mean diameter of the population, has a z-value of 1.
We could convert the value of every member of any normal population into a z-score. If we did that for any normal population and arranged those z-scores into a relative frequency distribution, they would all be the same.
Each and every one of those standardized normal distributions would have a mean of zero and the same shape. There are many tables that show what proportion of any normal population will have a z-score less than a certain value. Because the standard normal distribution is symmetric with a mean of zero, the same proportion of the population that is less than some positive z is also greater than the same negative z.
Some values from a standard normal table appear in Table 2. You can also use the interactive cumulative standard normal distributions illustrated in the Excel template in Figure 2. The graph on the top calculates the z-value if any probability value is entered in the yellow cell. The graph on the bottom computes the probability of z for any given z-value in the yellow cell.
In either case, the plot of the appropriate standard normal distribution will be shown with the cumulative probabilities in yellow or purple. Figure 2. Kevin sees that leaving 2. He assumes that the pack weights are normally distributed, a reasonable assumption for a machine-made product, and consulting a standard normal table, he sees that.
Solving for x , Kevin finds that the upper limit is He finds that the lower limit is If this was a statistics course for math majors, you would probably have to prove this theorem. Because this text is designed for business and other non-math students, you will only have to learn to understand what the theorem says and why it is important.
To understand what it says, it helps to understand why it works. Here is an explanation of why it works. The theorem is about sampling distributions and the relationship between the location and shape of a population and the location and shape of a sampling distribution generated from that population. Specifically, the central limit theorem explains the relationship between a population and the distribution of sample means found by taking all of the possible samples of a certain size from the original population, finding the mean of each sample, and arranging them into a distribution.
The sampling distribution of means is an easy concept. Then take another sample of the same size, n , and find its x. Do this over and over until you have chosen all possible samples of size n. Arrange this population into a distribution, and you have the sampling distribution of means.
You could find the sampling distribution of medians, or variances, or some other sample statistic by collecting all of the possible samples of some size, n , finding the median, variance, or other statistic about each sample, and arranging them into a distribution. The central limit theorem is about the sampling distribution of means. It tells us that:. This makes sense when you stop and think about it. It means that only a small portion of the samples have means that are far from the population mean.
These come from the same basic reasoning as 2 , but would require a formal proof since normal distribution is a mathematical concept. While it is a difficult to see why this exact formula holds without going through a formal proof, the basic idea that larger samples yield sampling distributions with smaller standard deviations can be understood intuitively. If the mean volume of soft drink in a population of mL cans is mL with a variance of 5 and a standard deviation of 2. You can also use the interactive Excel template in Figure 2.
Do not try to change the formula in these yellow cells. 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.
We talked about that here: stats. If he were a physicist he'd see it differently. Show 2 more comments. Somebody here secretly hating Poincare?
Gabriel Lippmann was a Nobel Prize winner in Physics. He was also, e. I edited to clarify. Metzger: Most of what we measure is in fact the sum of many r. Aksakal Aksakal 53k 5 5 gold badges 84 84 silver badges bronze badges. The quote is reasonable, however one can note that the measured length cannot be negative i.
It is always an approximation. You mean like Dr. Frankenstein's unseemly experiments? No idea how to translate it though. Maybe Russians can correct me. A good way it was related to me is the following: Roll a single die, and you have an equal likelihood of rolling each number , and hence, the PDF is constant. Hope that helps. If the data does not resemble a bell curve researchers may have to use a less powerful type of statistical test, called non-parametric statistics.
We can standardized the values raw scores of a normal distribution by converting them into z-scores. This procedure allows researchers to determine the proportion of the values that fall within a specified number of standard deviations from the mean i. The empirical rule in statistics allows researchers to determine the proportion of values that fall within certain distances from the mean. The empirical rule is often referred to as the three-sigma rule or the The empirical rule allows researchers to calculate the probability of randomly obtaining a score from a normal distribution.
This means there is a Statistical software such as SPSS can be used to check if your dataset is normally distributed by calculating the three measures of central tendency. If the mean, median and mode are very similar values there is a good chance that the data follows a bell-shaped distribution SPSS command here. Normal distributions become more apparent i. You can also calculate coefficients which tell us about the size of the distribution tails in relation to the bump in the middle of the bell curve.
McLeod, S. Introduction to the normal distribution bell curve. Toggle navigation.
0コメント