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Questions tagged [descriptive-statistics]

The area of statistics that provides descriptions of data, may it be samples or the population. This includes graphical representations and numerical indicators. No information is inferred from samples about the population, as in inferential statistics.

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3answers
13k views

Intuitive Way To Understand Principal Component Analysis

I know that this is meant to explain variance but the description on Wikipedia stinks and it is not clear how you can explain variance using this technique Can anyone explain it in a simple way?
12
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10answers
7k views

Why is variance squared?

The mean absolute deviation is: $$\dfrac{\sum_{i=1}^{n}|x_i-\bar x|}{n}$$ The variance is: $$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$ So the mean deviation and the variance are ...
8
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3answers
498 views

How should a mathematically-inclined person learn descriptive statistics?

I am interested in learning descriptive statistics. But I am completely baffled, that there seem to be no mathematically rigorous books on this subject, as far as I know at least. The Wikipedia page ...
6
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4answers
97 views

$n$ vs $n-1$ for the standard deviation

Suppose that I went to Tasmania a few years before the "Tazie Tiger" (thylacine) became extinct. I sample say, $100$ thylacines and make some biometric measurements. To make the discussion concrete, ...
6
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2answers
1k views

How did Target figure out a teen girl was pregnant before her father did?

First of all I do not have a mathematics degree only a B.S. in finance so please take that into account when writing an answer. Generally what type of mathematics is involved here? And specifically ...
6
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2answers
414 views

Canonical probability distribution associated with the “harmonic mean”

Is there a canonical continuous probability distribution, the center of which is best characterized with the harmonic mean, given by $$ \mathrm{HM}(X) = n \cdot \left( \sum\limits_{k=1}^{n} x_k^{-1} ...
5
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1answer
81 views

What would one call the mean of the mean and median?

If we wanted to take the mean of the mean and median, what would we call that value? I'm considering this as a useful summary statistic with a heavily skewed distribution. For instance, if the mean ...
5
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2answers
144 views

How can the median number of sexual partners be 6.1 and 3.6?

http://www.cdc.gov/nchs/nsfg/key_statistics/n.htm Median number of female sexual partners in lifetime, for men 25-44 years of age, 2006-2008: 6.1 Wouldn't it only be able to be x.5 or x.0?
4
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2answers
267 views

What does a $4$-dimensional bell curve look like graphically?

I have some $4$D data in $(w,x,y,z)$ quaternion format that I'd like to graph to discern whether or not it follows a Gaussian function. I am using MATLAB and a slider to emulate the fourth dimension. ...
4
votes
1answer
9k views

Is $50$th percentile equal to median?

Consider we have the $100$ distinct integers between $1$ and $100$ inclusive. The median and fiftyth percentile can be calculated as below. Ordering: $1,2,3 ..... ,98, 99, 100$ The median is $(50+...
4
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2answers
51 views

The “correct” standard deviation

This may end up being a question more about scientific best practice than anything else, but I think this is the right community to ask it in to get the insight I'm looking for. Say I have two little ...
4
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3answers
362 views

Question about English sentences in statistics?

Can somebody help me interpreting the red circled sentences in planer English? I understand "We view $y_i$ as a realization of a random variable $Y_i$ that can take the values of one and zero" but ...
4
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2answers
145 views

Real Examples of Misleading Statistics

I need to give a presentation to a group of students on Tuesday about why one needs to be careful when examining statistics or mathematical results in the media or online. In his book How Not To Be ...
4
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1answer
353 views

Estimation of the expected value of a function of a random variable

Let $\mathbf{x}\in\Bbb{R}^n$ be a multivariate normal vector, i.e., $\mathbf{x}\sim\mathcal{N}(\bar{\mathbf{x}},\Sigma)$, where the mean vector $\bar{\mathbf{x}}$ and the covariance matrix $\Sigma\in\...
4
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2answers
1k views

What is the meaning of “expected value of an estimator”?

I am studying statistics and i am having trouble understanding some proofs because i don't quite understand what the concept of "expected value of an estimator" means and what is the difference with ...
4
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1answer
305 views

Intuition for Standard Deviation

I've been studying descriptive statistics and am having a hard time understanding the actual intuition behind standard deviation. I'm trying to get a practical feeling for it and so I'm trying to draw ...
4
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2answers
645 views

Is there a measure of 'evenness' of dispersion?

I looked up on the web, but couldn't find anything helpful. I'm basically looking for a way to measure how 'evenly' a value is distributed. As in, an 'evenly' distributed distribution like X: and an ...
4
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1answer
155 views

How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
4
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1answer
97 views

Intuitive explanation of the formula for mode (grouped data).

The formula for finding the mode in grouped data is given by: $$ mode = l + \frac {f_1 - f_0}{2 f_1 - f_0 - f_2} X h $$ where, l = the lower limit of the modal class, $ f_1 $ = the frequency of ...
4
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0answers
216 views

Generalizing the mode and mean like the quantile

The mode, median, and means of a series of number ($x_1,x_2,...,x_N$) can be roughly thought of as the points that minimize the $p$-norm of the sequence for $p\in \{0,1,2\}$. The median is $c=\min_c\...
3
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2answers
143 views

How to convert up/down votes into a single number?

I've got a website with voting system where people can vote a review either up or down. For example: “Did you find this review helpful? [Yes] or [No]?” In order to order the reviews by how much they ...
3
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1answer
73 views

Is there any meaning of this “Median-mean”

Given a data set $\{a_1,\cdots,a_n\}$ with median M Define the medimean to be the value of $x$ s.t. $$\left(\frac{1}{n}\sum_{n}{}{a_n}^x \right)^{1/x}=M$$ Is this $x$ value useful / used at all in ...
3
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2answers
389 views

what i am doing wrong in calculating quartiles

can somebody please help me to understand? Why 1st quartile in this data is equal 4.5 and 3rd quartile is equal 6.5. I am getting 4.25 and 6.25 , but not 4.5 and not 6.5. I use formula (n+1)/4 for ...
3
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3answers
2k views

Why is statistical properties of mode and median are difficult to determine?

I have read a book saying that statistical properties of the mode and the median are difficult to determine compared to the mean. I am not entirely sure why is so. And the book does not provide ...
3
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3answers
224 views

Is my understanding of Likelihood Ratio Tests and Likelihood Functions correct?

I'm trying to understand Likelihood Ratio Tests, Likelihood Functions and Hypotheses Tests, which makes up a significant amount of what we're supposed to learn in my statistics for beginners course. ...
3
votes
1answer
1k views

Change in single observation of data set and how it will affect sample variance

Q/ Consider the sample of size n=10, where the sample mean is 6 and the sample variance is 12. Consider a second sample that is exactly the same as the first except that there is an additional ...
3
votes
2answers
286 views

ANOVA F-test results in p-value being 1

In one of the statistics exercises I have tried to solve, I have to show that the means of the observations in four groups aren't equal. I read that a t-test can only be used to compare means for two ...
3
votes
2answers
73 views

How to pick the “best” line equation that fits data and passes through the origin?

I have some data shown in the image. Each symbol corresponds to a different dataset. As it can be seen the data initially behaves like a line and after some points it follows other behaviour. The ...
3
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1answer
75 views

The Largest Gap

If I have an X amount of randomly generated positive numbers, what type of algorithm could I run to find the following: Precisely where the largest difference exists between two of the numbers? How ...
3
votes
2answers
65 views

Population Variation with two variables

I have a dataset with two variables. I want to treat my dataset as a population not a sample. I am wondering if I can just use the formula for population variance as below: $$\frac{\sum(X-\mu)^2}{N}$$...
3
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1answer
60 views

Survey: prove 10% of population “Don't know”?

I have a simple question that I expect is a standard situation, but I can't seem to find the right answer (maybe because it's too simple). I have a survey with the following answers: (What is your ...
3
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1answer
1k views

Pair Correlation with Uniformly Distributed Random Numbers

I am trying to reconstruct the following "jiggled" distribution from this article: American Scientist. The article states that the pair correlation function is defined as follows: "For each distance ...
3
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0answers
57 views

Pearson Correlation Coefficient for samples $X,Y$ equals $1$ iff there is $a>0, b\in \mathbb{R}$ with $x_i = ay_i + b$ for every $i=1,\dots,n$.

Let $X = (x_1,\dots,x_n)$ and $Y = (y_1,\dots,y_n)$ be outcome values. We have defined the variances $s^2_X = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$, $s^2_Y = \frac{1}{n-1} \sum_{i=1}^n (y_i - \...
3
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1answer
18 views

Ranges given instead of exact values in survey

I gave a survey wherein one data there is stated as to how often do individuals watch tv in a day. I was hoping to get exact values from the individuals but some of them put ranges like 1-5 times in a ...
3
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0answers
69 views

How do I state that a data set has a 'denser' standard deviation?

Suppose I have two algorithms that produce numerical data. The first algorithm produces { 2452, 695, 318, ... } with a mean of 1155 and a standard deviation of 1138. The second algorithm produces { ...
3
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1answer
241 views

Which statistics are sufficient?

Can anyone help explain why the following statistics are either Sufficient or not Sufficient? Mode, Mean, Median, Standard Deviation, Skewness, Kurtosis, Range, IQR
2
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2answers
149 views

What is the intuitive meaning of Expected Value?

What exactly does this number mean? I am aware on how to calculate the them given the variables. I have watched several videos and articles about it but no one seems to explain it. Maybe I am just ...
2
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2answers
251 views

Matrix calculus for statistics

I want to calculate the variance of a sum of random variables $ (X_1+X_2) $. Doing statistics, I have to learn maths starting from the end and it is quite difficult, yet very interesting (please ...
2
votes
2answers
107 views

Simple probability question - how to approach systematically, not overthink, or misinterpret?

I have been studying some probability 'story problems' and surprisingly the ones with very simple solutions tend to confuse me. Particularly, the question below. The question: Losses covered by an ...
2
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3answers
899 views

Symmetric distribution around zero and median equal to zero

Let $X$ be a random variable and suppose that it is symmetrically distributed around $0$. Is this equivalent to assume that $X$ has median equal to $0$? If not, are the two assumptions completely ...
2
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1answer
124 views

Which mean is used for standard error?

While calculating the standard error, we take the standard deviation of the sampling distribution. But for calculating the standard deviation we must consider a 'mean' to take difference about. In ...
2
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1answer
35 views

Find Expected value

Suppose, $X \sim \operatorname{Poisson}(\lambda)$, Then what is $\operatorname{E}\left(\dfrac{1}{X+1}\right)$ ? My attempt: Let $Y = \dfrac{1}{X+1}.$ Then $$ F_Y(t) = \{ Y \leq t\} = \left\{ X \...
2
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2answers
1k views

Minimal sufficient statistic of $\operatorname{Uniform}(-\theta,\theta)$

I am seeking clarification on why both the vector $(X_{(1)},X_{(n)})^T$ and $\max\{-X_{(1)},X_{(n)}\}$ are sufficient for $\operatorname{Unif}(-\theta,\theta)$, but only $\max\{-X_{(1)},X_{(n)}\}$ is ...
2
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1answer
70 views

Problem related to the exact distribution and the CLT

I'm trying to solve a pretty straight forward problem but i can't find good info on the subjects necessary to solve it so i'm terribly stuck. I'll present it as follows and later try to explain my ...
2
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3answers
45 views

$E[\frac{1}{X}]$ for $X\sim\Gamma(n,\theta)$

Consider iid random varibales $(X_i)_{1\le i\le n}$ with $X_i\sim Exp(\theta)$ for $1\le i\le n$ and $\theta\in(0,\infty)$. Then we have $$\sum_{i=1}^nX_i\sim \Gamma(n,\theta)$$ with density function ...
2
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2answers
211 views

Unbiased estimator for $\theta$

Exercise : Let $X_1, \dots, X_n$ be a random sample $(n>1)$ from the distribution with pdf $f(x) = \theta x^{-2}, \; \; 0 < \theta \leq x < \infty$, where $\theta$ an unknown parameter. ...
2
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1answer
48 views

Prove that $\log[ ( x+\sigma)^a - (x-\sigma)^a ] = (a-1) \log(x) + C$

$\sigma>0$, $a>1$ and $C$ are constant real number. Does the following holds for any $x>\sigma$? $\log[ ( x+\sigma)^a - (x-\sigma)^a ] = (a-1) \log(x) + C$. Background John Tukey, on the ...
2
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2answers
130 views

Arithmetic mean is to addition as Harmonic mean is to …?

Take $n$ real numbers $x_1,\ldots,x_n.$ The Arithmetic mean $A_n=\frac{1}{n}(x_1+\ldots+x_n)$ is the answer to the question: "Which number, when added up $n$ times, is equal to the sum of the $x_1,\...
2
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1answer
2k views

What qualifies as an infinite population?

I've been looking for a clear guideline to distinguish between a finite population and an infinite one, for example, in some places an infinite population is described as something like the number of ...
2
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1answer
231 views

Why minimising the MSE in Variance-Bias tradeoff?

As I understand the Variance-Bias tradeoff, modifying estimators to minimise bias might increase the variance of the estimator and vice-versa. For the simple case of the biased variance estimator, ...