I was searching to learn which traits (e.g., hair color, intelligence, weight, etc) are and which are not normally distributed. In the lecture notes for some course on a university website (unfortunately I don't have the link), a college professor had stated that complex traits are normally distributed.  But then I came across an answer on Quora to someone asking what besides height and intelligence is normally distributed     A supposed statistician had replied that "Actually, nothing is described by a normal distribution.It approximately describes heights of people so long as they are all male or all female and not a mixture of East African Negros, West African Pygmies and Europeans." He went on to say that a large sample, however, with no outliers can approximate normal distribution.  So I am confused. So can anything, if the sample is large enough, approximate normal distribution?  And yet nothing is normally distributed to begin with?
13641
I am a grad student developing an interest for statistics. I like the material over-all, but I sometimes have a hard time thinking about applications to real life. Specifically, my question is about commonly used statistical distributions (normal - beta- gamma etc.). I guess for some cases I get the particular properties that make the distribution quite nice - memoryless property of exponential for example. But for many other cases, I don't have an intuition about both the importance and application areas of the common distributions that we see in textbooks.   There are probably a lot of good sources addressing my concerns, I would be glad if you could share those. I would be a lot more motivated into the material if I could associate it with real-life examples.
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Take as an example the well-known . In the original paper, the regression model was used to model the amount people were willing to pay for improved air quality.  But the 'people' whose behaviour was being modelled was never specified. Presumably, it was not limited to 'residents of Greater Boston'. Only implausibly would it be 'humans anywhere on the planet'. Nor, 'all US residents'.   So what inference are we invited by the authors to draw about the scope they intend for their theory? Perhaps 'all residents of large US metro areas'. But - what about other First World metro areas? The first two sections of the Boston paper scrupulously avoid any reference which might suggest that they had a particular 'population' to which their 'sample' was to apply.  There is also the dimension of time - the data came from the 70s, whereas the theory is has no time limitations spelt out.  Plus - going back to first principles - how could the Boston data be treated as a sample of a population including, say, the Chicago metro area when no data from the Chicago area was considered?   Besides, lots of the covariates in the Boston regression weren't samples at all, but totals - so, for example, the proportion of blacks was supplied from census stats.  My question: how can we conceive the Boston data as supporting the geographically unlimited propositions in the paper? Is there a mathematical way of showing equivalence or approximation of the Boston data analysis to other cities? Or is it just assumed as a matter of common sense - such that the paper's authors would be amazed to think anyone could have any doubts on the matter?
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Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$.  Is $XY$ also normally distributed?  Is the answer any different if we know that $X$ and $Y$ are independent?
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What is the distribution of the square of a non-standard normal random variable (i.e., the mean is not equal to 0 and the variance is not equal to 1)?
123566
Given that $a$~$\mathcal{N}(\mu, \Sigma)$ and $\mu$~$\mathcal{N}(0, \sigma^2 I)$, Is $p(a|\Sigma, \sigma)=\int p(a|\mu, \Sigma)p(\mu|\sigma)d\mu$ again a Normal distribution? And how to estimate the mean and variance respectively? Thanks a lot!
20010
I have labored under the belief that the sample median is more robust measure of central tendency than the sample mean, since it ignores outliers.  I was therefore surprised to learn (in the answer to ) that for samples drawn from a normal distribution, the variance of the sample mean is less than the variance of the sample median (at least for large $n$).  I understand mathematically why this is true.  Is there a "philosophical" way of looking at this that that would help with intuition about when to use the median rather than the mean for other distributions?    Are there mathematical tools that help quickly answer the question for a particular distribution?
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Is there any intuitive way to interpret the standard deviation regardless of the probability distribution?  So for example, for the normal distribution, I know how to interpret being within 1 standard deviation - there is a a 68% of being there. For other distributions I can calculate this probability as well.  But if I don't know the distribution, how is it a meaningful statistic? Is it only meaningful in relation to other random variables that we know follow the same distribution, but have a different standard deviation?
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Long ago I learnt that normal distribution was necessary to use a two sample T-test. Today a colleague told me that she learnt that for N>50 normal distribution was not necessary. Is that true?  If true is that because of the central limit theorem?
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Let $(X_1, X_2, \ldots, X_i, \ldots, X_k)$ be $k$ independent, normally distributed random variables with means $\mu_i$ and variances $1$. Then the random variable $$ \sum_{i=1}^k X_i^2$$ is distributed according to the noncentral chi-squared distribution with  parameters: $k$   and $ \lambda=\sum_{i=1}^k \mu_i^2$.  If $(X_1', X_2', \ldots, X_i', \ldots, X_k')$ are also independent normal distributed with variances $1$, and means $\mu'_i$'s s.t. $\sum_{i=1}^k \mu_i'^2 = \lambda$, why are $$ \sum_{i=1}^k X_i'^2$$ and $$ \sum_{i=1}^k X_i^2$$ identically distributed? Thanks.
97425
I have a question:      Does  $$N(0, x)$$ mean a normal distribution with mean $0$ and variance $x$? Or standard deviation $x$?   The notation seems ambiguous sometimes.
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I got two distribution which are not pairs. It's difficult to say that they are also from a normal distribution because they have different length.  The distributions of the vectors are, for gas: 65  65  65  65  74  74  74  74  74  77  78  81  81  83  83  83  85  85  85  85  85  87  87 89  89  90  91  91  91  91  91  91  94  94  94  95  95  95  95  98 101 101 101 102 102 102 102 102 103 103 103 103 103 104 104 104 104 104 104 106 106 106 106 107 108 108 110 110 113 113 115 115 115 118 118 118 118 119 119 121 122 122 122 125 125 125 128 128 128 128 128 129 129 134 134 134 134 134 134 137 137 137 142 145 145 148 148 148 148 150 150 150 150 150 150 150 153 153 154 154 154 158 158 161 161 161 161 161 161 161 161 164 164 168 168 168 168 168 186 188 188 192 192 194 194 197 197 231 256  while for the diesel vector distribution is:  65  91  91  93  93  93  93  94  94  95 122 128 161 161 161   I made the t.test and the wilcox.test and I got these results:  Welch Two Sample t-test  data:  diesel_type and gas_type t = -1.7104, df = 18.149, p-value = 0.1042 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -31.873874   3.256425 sample estimates: mean of x mean of y  109.0000  123.3087    and Wilcoxon test is:  data:  diesel_type and gas_type W = 843, p-value = 0.1179 alternative hypothesis: true location shift is not equal to 0   I could accept the null hypothesis in the t.test, saying that the means are equals because the p-value is over the 5%. Should I accept the null hypothesis in the Wilcoxon test too? Are the medians equals? What could I say about these results?
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How does the expected value of a continuous random variable relate to its arithmetic mean, median, etc. in a non-normal distribution (eg. skew-normal)?  I'm interested in any common/interesting distributions (eg. log-normal, simple bi/multimodal distributions, anything else weird and wonderful).  I'm looking mostly for qualitative answers, but any quantitative or formulaic answers are also welcome. I'd particularly like to see any visual representations that make it clearer.
23908
I have a problem with the normality of some data I have:  I've done a Kolmogorov test which says it isn't normal with p=.0000,  I don't understand: the skewness of my distribution =-.497, and the kurtosis =-0,024  Here is the plot of my distribution which looks very much normal ...  (I have three scores, and each one of this scores isn't normal with a significant p-value for the Kolmogorov test ... I really don't understand )
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Why did the normal distribution become such a popular (important) distribution? I know one reason is because of CLT. Can you please give more reasons?
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I think I am dealing with this issue   I have large data set(10k data points) that slightly diverges from normal, and I get p-value of 0. I am interested in having perhaps a more crude test that tells me if data is extremely divergent from normal, versus looking somewhat normally distributed. I am currently trying Kolmogorov-Smirnoff, and in both cases I just get p-value of 0. Any alternatives?  I also looked at this:   So, taking all this into consideration, is there any kind of test I can perform that distinguishes between data that's roughly normal, and not normally distributed at all?  I am using scipy.
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I am not even remotely an expert in this field; I just got curious, so help me.   First of all, let us remove all human sentimental attributes from this question (such as connotations of  or emotional reactions thereof) and ask it from a purely scientific point of view. The typical facial feature of a West/Central African person (the so-called ) exhibits a wide nose, a lack of nasal bridge projection (similar to many Asian noses) and a prognathous skull shape that resembles more closely to our ancestor Homo Heidelbergensis than many other indigenous phenotypes from other parts of the world do.  I agree that this claim sounds anecdotal, and is certainly not true of all Africans (for example, Ethiopians), but one cannot dismiss it simply on grounds of ethical policing. I cannot help but notice that many of my African friends remind me of our shared ancestry (with no disrespect meant whatsoever), as is clear from the photographs below, one of a supposed Homo Heidelbergensis and one of a random African Homo Sapiens of today.     My question is the following. What is the reason behind this seeming resemblance? Is this appearance an illusion and possibly a construct of a prejudiced mind, or does this have a valid scientific answer? For example, are most Africans closer to our ancestors than the rest of humanity is? How does evolution answer this question?  Thank you for your help.
22448
Prove that t-distribution is close to normal distribution when degree of freedom is large.   $$f(x)=\frac{\Gamma[(n+1)/2]}{\sqrt{n\pi}\Gamma(n/2)}\left(1+\frac{x^2}{n}\right)^{-(n+1)/2}$$  using limit of density and Law of large number.  Anyone can help me with this question please? Thank you and appreciate your help!