# Tag Info

Accepted

### Can mean be less than half of median if all numbers are non-negative?

You are correct. This is one example of a general result called Markov's inequality, which says that for a non-negative random variable $X$ and number $a$, $$P(X\geq a)\leq \frac{E[X]}{a}$$ If you ...
Accepted

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

Because Alan Turing was born after Ronald Fisher. In the old days, before computers, all this stuff had to be done by hand or, at best, with what we would now call calculators. Tests for comparing ...
• 125k
Accepted

### Can someone give the intuition behind Mean Absolute Error and the Median?

Here is an intuitive argument with light math. Let's say we have a $d$ claiming to be minimizing the MAE of points $x_i$. And, let's say we have $n_l$ and $n_r$ points on its left and right. If we ...
• 57.8k
Accepted

### Why is the Median Less Sensitive to Extreme Values Compared to the Mean?

if you write the sample mean $\bar x$ as a function of an outlier $O$, then its sensitivity to the value of an outlier is $d\bar x(O)/dO=1/n$, where $n$ is a sample size. the same for a median is zero,...
• 61.8k
Accepted

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

It's useful to take a step back and forget about the forecasting aspect for a minute. Let's consider just any distribution $F$ and assume we wish to summarize it using a single number. You learn very ...
• 128k

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

As @whuber and others have said, the statement is not true in general. And if you’re willing to be more intuitive — I can’t keep up with the deep math geeks around here — you might look at other ways ...
• 21.4k

### Is there more than one "median" formula?

What @Sycorax says. As a matter of fact, there are surprisingly many definitions of general quantiles, so in particular also of medians. Hyndman & Fan (1996, The American Statistician) give an ...
• 128k

### How Well Does the Mean Describe a Multimodal Probability Distribution?

The mean means what it means Whenever you compute a single real value that describes some aspect of a distribution ---whether this is the mean, mode, standard deviation, kurtosis, a particular ...
• 129k
Accepted

### Is the median preserved for any strictly monotonic mapping?

The conjecture is true and your disproof is flawed. The flaw in your steps occurs when you make a change-of-variables in the initial step but do not change the range of integration accordingly. ...
• 129k
Accepted

### Why does component-wise median not make sense in higher dimensions?

The underlying concept is that a median splits the data (or a distribution) into two halves with equal amounts in each half (by count or probability). Even in one dimension the median is problematic. ...
• 328k
Accepted

### Is there more than one "median" formula?

TL;DR - I'm not aware of specific names being given to different estimators of sample medians. Methods to estimate sample statistics from some data are rather fussy and different resources give ...
• 92.3k

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

I would like to add a third reason to the correct reasons given by Harrell and Flom. The reason is that we use Euclidean distance (or L2) and not Manhattan distance (or L1) as our standard measure ...
• 2,080

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

Stephan's answer gives you an intuitive explanation of why minimizing the absolute average error gives you the median. To answer which of the MSE, MAE or MAPE you should minimize: The MAE is robust, ...
• 571

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

Suppose you have $n$ data points from some underlying continuous distribution with mean $\mu$ and variance $\sigma^2 < \infty$. Let $f$ be the density function for this distribution and let $m$ be ...
• 129k

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

Often the mean is chosen over the median not because it's more representative, robust, or meaningful but because people confuse estimator with estimand. Put another way, some choose the population ...
• 95.8k

### Why is the Median Less Sensitive to Extreme Values Compared to the Mean?

A reasonable way to quantify the "sensitivity" of the mean/median to an outlier is to use the absolute rate-of-change of the mean/median as we change that data point. To that end, consider ...
• 129k
Accepted

### Is there a version of the correlation coefficient that is less-sensitive to outliers?

I think you want a rank correlation. Those are generally more robust to outliers, although it's worth recognizing that they are measuring the monotonic association, not the straight line association. ...

### How to find median value for five given elements based on the max min and sum of the elements

If you only know the min, max, and sum of the 5 numbers, you cannot determine the median. E.g. median(1, 2, 3, 4, 5)=3 median(1, 2.1, 2.8, 4.1, 5)=2.8. But both have (min, max, sum) = (1, 5, 15).
• 11k

### Is the median preserved for any strictly monotonic mapping?

Here is a generalization. It is intended to reveal which properties of probability are involved in the result. It turns out that density functions are irrelevant. Any number $\mu$ determines two ...
• 328k

### Median of a set with even number of elements

Another way to define a sample median is as a minimizer of the sum of $\ell_1$ distances between a given quantity and each datapoint:  m \in \mathcal{O}=\underset{x\in\mathbb{R}}{\textrm{argmin}} \...
• 4,570
Accepted