Questions tagged [mean]

The expected value of a random variable; or a location measure for a sample.

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264
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12answers
78k views

Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?

It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our ...
124
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10answers
83k views

Why does the Cauchy distribution have no mean?

From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean?
101
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4answers
106k views

Difference between standard error and standard deviation

I'm struggling to understand the difference between the standard error and the standard deviation. How are they different and why do you need to measure the standard error?
90
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8answers
30k views

If mean is so sensitive, why use it in the first place?

It is a known fact that median is resistant to outliers. If that is the case, when and why would we use the mean in the first place? One thing I can think of perhaps is to understand the presence of ...
77
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5answers
76k views

Mean absolute error OR root mean squared error?

Why use Root Mean Squared Error (RMSE) instead of Mean Absolute Error (MAE)?? Hi I've been investigating the error generated in a calculation - I initially calculated the error as a Root Mean ...
55
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2answers
90k views

How should one interpret the comparison of means from different sample sizes?

Take the case of book ratings on a website. Book A is rated by 10,000 people with an average rating of 4.25 and the variance $\sigma = 0.5$. Similarly Book B is rated by 100 people and has a rating of ...
50
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6answers
37k views

How can a distribution have infinite mean and variance?

It would be appreciated if the following examples could be given: A distribution with infinite mean and infinite variance. A distribution with infinite mean and finite variance. A distribution with ...
49
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6answers
7k views

Is Amazon's “average rating” misleading?

If I understand correctly, book ratings on a 1-5 scale are Likert scores. That is, a 3 for me may not necessarily be a 3 for someone else. It's an ordinal scale IMO. One shouldn't really average ...
48
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14answers
82k views

Why is median age a better statistic than mean age?

If you look at Wolfram Alpha Or this Wikipedia page List of countries by median age Clearly median seems to be the statistic of choice when it comes to ages. I am not able to explain to myself why ...
47
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5answers
9k views

What can we say about population mean from a sample size of 1?

I am wondering what we can say, if anything, about the population mean, $\mu$ when all I have is one measurement, $y_1$ (sample size of 1). Obviously, we'd love to have more measurements, but we can'...
37
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5answers
4k views

Will the fact that my Italian son is going to attend a primary school change the expected number of Italian children to be present in his class?

This is a question stemming from a real-life situation, for which I have been genuinely puzzled about its answer. My son is due to start primary school in London. As we are Italian, I was curious to ...
34
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5answers
111k views

What is the difference between “mean value” and “average”?

Wikipedia explains: For a data set, the mean is the sum of the values divided by the number of values. This definition however corresponds to what I call "average" (at least that's what I remember ...
33
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8answers
30k views

Replacing outliers with mean

This question was asked by my friend who is not internet savvy. I've no statistics background and I've been searching around internet for this question. The question is : is it possible to replace ...
33
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11answers
9k views

Regression to the mean vs gambler's fallacy

On the one hand, I have the regression to the mean and on the other hand I have the gambler´s fallacy. Gambler’s fallacy is defined by Miller and Sanjurjo (2019) as “the mistaken belief that random ...
33
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3answers
4k views

Does a sample version of the one-sided Chebyshev inequality exist?

I am interested in the following one-sided Cantelli's version of the Chebyshev inequality: $$ \mathbb P(X - \mathbb E (X) \geq t) \leq \frac{\mathrm{Var}(X)}{\mathrm{Var}(X) + t^2} \,. $$ Basically, ...
33
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2answers
3k views

Is there a reliable nonparametric confidence interval for the mean of a skewed distribution?

Very skewed distributions such as the log-normal do not result in accurate bootstrap confidence intervals. Here is an example showing that the left and right tail areas are far from the ideal 0.025 ...
32
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3answers
2k views

Why does basic hypothesis testing focus on the mean and not on the median?

In basic under-grad statistics courses, students are (usually?) taught hypothesis testing for the mean of a population. Why is it that the focus is on the mean and not on the median? My guess is that ...
32
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5answers
23k views

What are the relative merits of Winsorizing vs. Trimming data?

Winsorizing data means to replace the extreme values of a data set with a certain percentile value from each end, while Trimming or Truncating involves removing those extreme values. I always see ...
31
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4answers
8k views

Does mean=mode imply a symmetric distribution?

I know this question has been asked with the case mean=median, but I did not find anything related to mean=mode. If the mode equals the mean, can I always conclude this is a symmetric distribution? ...
30
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4answers
59k views

How do I calculate confidence intervals for a non-normal distribution?

I have 383 samples that have a heavy bias for some common values, how would I calculate the 95% CI for the mean? The CI that I calculated seems way off, which I assume is because my data does not ...
30
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4answers
75k views

How is finding the centroid different from finding the mean?

When performing hierarchical clustering, one can use many metrics to measure the distance between clusters. Two such metrics imply calculation of the centroids and means of data points in the clusters....
28
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4answers
26k views

How to compute the confidence interval of the ratio of two normal means

I want to derive the limits for the $100(1-\alpha)\%$ confidence interval for the ratio of two means. Suppose, $X_1 \sim N(\theta_1, \sigma^2)$ and $X_2 \sim N(\theta_2, \sigma^2)$ being independent, ...
28
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3answers
6k views

Why does minimizing the MAE lead to forecasting the median and not the mean?

From the Forecasting: Principles and Practice textbook by Rob J Hyndman and George Athanasopoulos, specifically the section on accuracy measurement: A forecast method that minimizes the MAE will ...
25
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3answers
3k views

What can one conclude about the data when arithmetic mean is very close to geometric mean?

Is there anything significant about a geometric mean and arithmetic mean that fall very close to one another, say ~0.1%? What conjectures can be made about such a data set? I've been working on ...
25
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4answers
3k views

Why does mean tend be more stable in different samples than median?

Section 1.7.2 of Discovering Statistics Using R by Andy Field, et al., while listing virtues of mean vs median, states: ... the mean tends to be stable in different samples. This after explaining ...
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6answers
6k views

What makes the mean of some distributions undefined?

Many PDFs range from minus to positive infinity, yet some means are defined and some are not. What common trait makes some computable?
23
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5answers
37k views

Can mean plus one standard deviation exceed maximum value?

I have mean 74.10 and standard deviation 33.44 for a sample that has minimum 0 and maximum 94.33. My professor asks me how can mean plus one standard deviation exceed the maximum. I showed her ...
23
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4answers
30k views

How to tell if my data distribution is symmetric?

I know that if the median and mean are approximately equal then this means there is a symmetric distribution but in this particular case I'm not certain. The mean and median are quite close (only 0....
22
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4answers
28k views

Averaging correlation values

Let's say I test how variable Y depends on variable X under different experimental conditions and obtain the following graph: ...
22
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6answers
2k views

When we calculate mean and variance, do we assume data are normally distributed?

When we calculate mean and variance using the two equations taught in school: $\mu = \frac{1}{N}\sum_{i=1}^N{x_i}$ $\sigma^2 = \frac{1}{N}\sum_{i=1}^N{(x_i-\mu)^2}$ Then do we assume, that the data ...
22
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4answers
33k views

Calculate mean of ordinal variable

I've read in a number of places that calculating the mean of an ordinal variable is inappropriate. I'm trying to get an intuition for why it might be inappropriate. I think it is because, in general, ...
21
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4answers
4k views

Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

For a uniformly distributed variable between 0 and 1 generated using rand(1,10000) this returns 10,000 random numbers between 0 and 1. If you take the mean, it ...
21
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4answers
12k views

What is the distribution of the rounded down average of Poisson random variables?

If I have random variables $X_1,X_2,\ldots,X_n$ that are Poisson distributed with parameters $\lambda_1, \lambda_2,\ldots, \lambda_n$, what is the distribution of $Y=\left\lfloor\frac{\sum_{i=1}^n X_i}...
21
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6answers
2k views

Is the median a type of mean, for some generalization of “mean”?

The concept of "mean" roams far wider than the traditional arithmetic mean; does it stretch so far as to include the median? By analogy, $$ \text{raw data} \overset{\text{id}}{\longrightarrow} \text{...
20
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4answers
62k views

Should the mean be used when data are skewed?

Often introductory applied statistics texts distinguish the mean from the median (often in the the context of descriptive statistics and motivating the summarization of central tendency using the mean,...
20
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3answers
8k views

Crash course in robust mean estimation

I have a bunch (around 1000) of estimates and they are all supposed to be estimates of long-run elasticity. A little more than half of these is estimated using method A and the rest using a method B. ...
20
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1answer
8k views

Can we use leave one out mean and standard deviation to reveal the outliers?

Suppose I have normally distributed data. For each element of the data I want to check how many SDs it is away from the mean. There might be an outlier in the data (likely only one, but might be also ...
20
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3answers
18k views

How do I calculate a confidence interval for the mean of a log-normal data set?

I've heard/seen in several places that you can transform the data set into something that is normal-distributed by taking the logarithm of each sample, calculate the confidence interval for the ...
19
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6answers
15k views

Is there an example where MLE produces a biased estimate of the mean?

Can you provide an example of an MLE estimator of the mean that is biased? I am not looking for an example that breaks MLE estimators in general by violating regularity conditions. All examples I ...
19
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4answers
56k views

Does mean = median imply that a unimodal distribution is symmetric?

For a unimodal distribution, if mean = median then is it sufficient to say that distribution is symmetric? Wikipedia says in relationship between mean and median: "If the distribution is symmetric ...
19
votes
5answers
7k views

Does a univariate random variable's mean always equal the integral of its quantile function?

I just noticed that integrating a univariate random variable's quantile function (inverse cdf) from p=0 to p=1 produces the variable's mean. I haven't heard of this relationship before now, so I'm ...
19
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2answers
40k views

Transform Data to Desired Mean and Standard Deviation

I am looking for a method to transform my dataset from its current mean and standard deviation to a target mean and a target standard deviation. Basically, I want to shrink/expand the dispersion and ...
19
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3answers
2k views

Example of distribution where large sample size is necessary for central limit theorem

Some books state a sample size of size 30 or higher is necessary for the central limit theorem to give a good approximation for $\bar{X}$. I know this isn't enough for all distributions. I wish ...
19
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2answers
2k views

Can someone give the intuition behind Mean Absolute Error and the Median? [duplicate]

I do not understand the intuition behind why the median is the best estimate if we are going to judge prediction accuracy using the Mean Absolute Error. Let's say you have a random variable $X$ and ...
19
votes
1answer
29k views

Should I use t-test on highly skewed data ? Scientific proof, please?

I have samples from a highly skewed (looking like an exponential distribution) dataset about users' participation (e.g.: number of posts), that have different sizes (but not less than 200) and I want ...
19
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4answers
2k views

Non-transitivity of correlation: correlations between gender and brain size and between brain size and IQ, but no correlation between gender and IQ

I found a following explanation on a blog and I would like to get more information about the non-transitivity of correlation: We have the following indisputable facts: On average, there is a ...
19
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4answers
25k views

Mean and Median properties

Can somebody explain me clear the mathematical logic that would link two statements (a) and (b) together? Let us have a set of values (some distribution). Now, a) Median does not depend on every ...
19
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2answers
2k views

For what (symmetric) distributions is sample mean a more efficient estimator than sample median?

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 ...
18
votes
7answers
5k views

Is median fairer than mean?

I recently read the advice that you should generally use median not mean to eliminate outliers. Example: The following article http://www.amazon.com/Forensic-Science-Introduction-Scientific-...
18
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3answers
14k views

Maximum value of coefficient of variation for bounded data set

In the discussion following a recent question about whether the standard deviation can exceed the mean, one question was raised briefly but never fully answered. So I am asking it here. Consider a ...

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