The median minimizes the sum of absolute deviations (the $ {\ell}_{1} $ norm)

Question

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median.

This is a sample exam question of one of the exams that I need to take and I don't know how to proceed.

Answer 1

Introduction: The solution below is essentially the same as the solution given by Brian M. Scott, but it will take a lot longer. You are expected to assume that $S$ is a finite set. with say $k$ elements. Line them up in order, as $s_1<s_2<\cdots <s_k$.

The situation is a little different when $k$ is odd than when $k$ is even. In particular, if $k$ is even there are (depending on the exact definition of median) many medians. We tell the story first for $k$ odd.
Recall that $|x-s_i|$ is the distance between $x$ and $s_i$, so we are trying to minimize the sum of the distances. For example, we have $k$ people who live at various points on the $x$-axis. We want to find the point(s) $x$ such that the sum of the travel distances of the $k$ people to $x$ is a minimum.

The story: Imagine that the $s_i$ are points on the $x$-axis. For clarity, take $k=7$. Start from well to the left of all the $s_i$, and take a tiny step, say of length $\epsilon$, to the right. Then you have gotten $\epsilon$ closer to every one of the $s_i$, so the sum of the distances has decreased by $7\epsilon$.

Keep taking tiny steps to the right, each time getting a decrease of $7\epsilon$. This continues until you hit $s_1$. If you now take a tiny step to the right, then your distance from $s_1$ increases by $\epsilon$, and your distance from each of the remaining $s_i$ decreases by $\epsilon$. What has happened to the sum of the distances? There is a decrease of $6\epsilon$, and an increase of $\epsilon$, for a net decrease of $5\epsilon$ in the sum.

This continues until you hit $s_2$. Now, when you take a tiny step to the right, your distance from each of $s_1$ and $s_2$ increases by $\epsilon$, and your distance from each of the five others decreases by $\epsilon$, for a
net decrease of $3\epsilon$.

This continues until you hit $s_3$. The next tiny step gives an increase of $3\epsilon$, and a decrease of $4\epsilon$, for a net decrease of $\epsilon$.

This continues until you hit $s_4$. The next little step brings a total increase of $4\epsilon$, and a total decrease of $3\epsilon$, for an increase of $\epsilon$. Things get even worse when you travel further to the right. So the minimum sum of distances is reached at $s_4$, the median.

The situation is quite similar if $k$ is even, say $k=6$. As you travel to the right, there is a net decrease at every step, until you hit $s_3$. When you are between $s_3$ and $s_4$, a tiny step of $\epsilon$ increases your distance from each of $s_1$, $s_2$, and $s_3$ by $\epsilon$. But it decreases your distance from each of the three others, for no net gain. Thus any $x$ in the interval from $s_3$ to $s_4$, including the endpoints, minimizes the sum of the distances. In the even case, I prefer to say that any point between the two "middle" points is a median. So the conclusion is that the points that minimize the sum are the medians. But some people prefer to define the median in the even case to be the average of the two "middle" points. Then the median does minimize the sum of the distances, but some other points also do.

Answer 2

We're basically after: $$ \arg \min_{x} \sum_{i = 1}^{N} \left| {s}_{i} - x \right| $$

One should notice that $ \frac{\mathrm{d} \left | x \right | }{\mathrm{d} x} = \operatorname{sign} \left( x \right) $ (Being more rigorous would say it is a Sub Gradient of the non smooth $ {L}_{1} $ Norm function).
Hence, deriving the sum above yields $ \sum_{i = 1}^{N} \operatorname{sign} \left( {s}_{i} - x \right) $.
This equals to zero only when the number of positive items equals the number of negative which happens when $ x = \operatorname{median} \left\{ {s}_{1}, {s}_{2}, \cdots, {s}_{N} \right\} $.

Remarks

1. One should notice that the median of a discrete group is not uniquely defined.
2. The median is not necessarily an item within the group.
3. Not every set can bring the Sub Gradient to vanish. Yet employing the Sub Gradient Method is guaranteed to converge to median.
4. It is not the optimal way to calculate the Median. It is given to give intuition about what's the median.

Answer 3

Suppose that the set $S$ has $n$ elements, $s_1<s_2<\dots<s_n$. If $x<s_1$, then $$f(x)=\sum_{s\in S}|s-x|=\sum_{s\in S}(s-x)=\sum_{k=1}^n(s_k-x)\;.\tag{1}$$ As $x$ increases, each term of $(1)$ decreases until $x$ reaches $s_1$, therefore $f(s_1)<f(x)$ for all $x<s_1$.

Now suppose that $s_k\le x\le x+d\le s_{k+1}$. Then

$$\begin{align*}f(x+d)&=\sum_{i=1}^k\Big(x+d-s_i\Big)+\sum_{i=k+1}^n\Big(s_i-(x+d)\Big)\\ &=dk+\sum_{i=1}^k(x-s_i)-d(n-k)+\sum_{i=k+1}^n(s_i-x)\\ &=d(2k-n)+\sum_{i=1}^k(x-s_i)+\sum_{i=k+1}^n(s_i-x)\\ &=d(2k-n)+f(x)\;, \end{align*}$$

so $f(x+d)-f(x)=d(2k-n)$. This is negative if $2k<n$, zero if $2k=n$, and positive if $2k>n$. Thus, on the interval $[s_k,s_{k+1}]$

$$f(x)\text{ is }\begin{cases} \text{decreasing},&\text{if }2k<n\\ \text{constant},&\text{if }2k=n\\ \text{increasing},&\text{if }2k>n\;. \end{cases}$$

From here it shouldn’t be too hard to show that $f(x)$ is minimal when $x$ is the median of $S$.

Answer 4

You want the median of $n$ numbers. Say $x$ is bigger than $12$ of them and smaller than $8$ of them (so $n=20$). If $x$ increases, it's getting closer to $8$ of the numbers and farther from $12$ of them, so the sum of the distances gets greater. And if $x$ decreases, it's getting closer to $12$ of them and farther from $8$ of them, so the sum of the distances gets smaller.

A similar thing happens if $x$ is smaller than more of the $n$ numbers than $x$ is bigger than.

But if $x$ is smaller than $10$ of them and bigger than $10$ of them, then when $x$ moves, it's getting farther from $10$ of them and closer to just as many of them, so the sum of the distances is not changing.

So the sum of the distances is smallest when the number of data points less than $x$ is the same as the number of data points bigger than $x$.

Answer 5

Consider two real numbers $a<b$. Then the objective becomes

$$dist(a,b) = |x-a|+|x-b|$$

This expression is minimum when $a\leq x \leq b$. It can be proved by calculating the objective on 3 cases ($x<a, a\leq x\leq b, x>b$).

Now consider the general case where $S$ has $n$ elements. Sort them in increasing order as $S_1, S_2, \ldots, S_n$.

Pair the smallest and the largest numbers. As explained above, $dist(S_1,S_n)$ is minimum when $S_1\leq x\leq S_n$. Remove these two elements from the list and continue this procedure until there is at most one element left in the set.

If there is an element $S_i$ left, then $x=S_i$ minimizes $dist(x-S_i)$. It also lies between all the pairs.

In the case of even elements, finally the sequence will be empty. As in the case above, median lies between all the pairs.

Answer 6

Starting with $$f(x)=\sum_{i=1}^n |s_i-x|$$

Assume we rearranged our terms such that $s_1<s_2<\cdots<s_n$

We first proceed by making the following observation $$\sum_{i=1}^n |s_i-x| = \sum_{i=2}^{n-1} |s_i-x| +(s_n -s_1) \quad \text{when} \quad x \in [s_1,s_n]$$

Now suppose that $n$ is odd, then by applying the above identity repeatedly we get $$f(x)=\sum_{i=1}^n |s_i-x|=|s_{\frac{n+1}2}-x|+(s_n -s_1)+(s_{n-1}-s_2)+\cdots+(s_{\frac{n+3}2}-s_{\frac{n-1}2})$$ or in other words $$f(x)=|s_{\frac{n+1}{2}}-x|+\text{constant}$$

This is just the absolute value function with its vertex being at $(s_{\frac{n+1}{2}},\text{constant})$, the minimum of the absolute value function occurs at its vertex, therefore $s_{\frac{n+1}{2}}$(median) minmizes $f(x)$.

Now suppose $n$ is even, again by using our identity, we have $$f(x)=\sum_{i=1}^n |s_i-x|=|s_{\frac{n}2}-x|+|s_{\frac{n+2}2}-x| + \text{constant}$$

Where the minimum occurs at $f'(x)=0$(or when not defined), therefore by differentiating and setting $f'(x)$ to zero we get $$\dfrac{|s_{\frac{n}{2}}-x|}{s_{\frac{n}{2}}-x}+\dfrac{|s_{\frac{n+2}{2}}-x|}{s_{\frac{n+2}{2}}-x}=0$$

Observe that $s:=\dfrac{s_{\frac{n+2}{2}}+s_{\frac{n}{2}}}{2}$(median) satisfies the above equation, since $s$ is halfway between $s_{\frac{n}{2}}$ and $s_{\frac{n+2}{2}}$ $$s_{\frac{n}{2}}-s=-(s_{\frac{n+2}{2}}-s)$$ that is by setting $x=s$ we get $$\dfrac{|s_{\frac{n}{2}}-s|}{s_{\frac{n}{2}}-s}+\dfrac{|s_{\frac{n}{2}}-s|}{-(s_{\frac{n}{2}}-s)}=0$$

Therefore $s$ is a minimum.

Answer 7

Consider two $x_i$'s $x_1$ and $x_2$,

For $x_1\leq a\leq x_2$, $\sum_{i=1}^{2}|x_i-a|=|x_1-a|+|x_2-a|=a-x_1+x_2-a=x_2-x_1$

For $a\lt x_1$, $\sum_{i=1}^{2}|x_i-a|=x_1-a+x_2-a=x_1+x_2-2a\gt x_1+x_2-2x_1=x_2-x_1$

For $a\gt x_2$,$\sum_{i=1}^{2}|x_i-a|=-x_1+a-x_2+a=-x_1-x_2+2a\gt -x_1-x_2+2x_2=x_2-x_1$

$\implies$for any two $x_i$'s the sum of the absolute values of the deviations is minimum when $x_1\leq a\leq x_2$ or $a\in[x_1,x_2]$.

When $n$ is odd, $$ \sum_{i=1}^n|x_i-a|=|x_1-a|+|x_2-a|+\cdots+\left|x_{\tfrac{n-1}{2}}-a\right| + \left|x_{\tfrac{n+1}{2}}-a\right|+\left|x_{\tfrac{n+3}{2}}-a|+\cdots+|x_{n-1}-a\right|+|x_n-a| $$ consider the intervals $[x_1,x_n], [x_2,x_{n-1}], [x_3,x_{n-2}], \ldots, \left[x_{\tfrac{n-1}{2}}, x_{\tfrac{n+3}{2}}\right]$. If $a$ is a member of all these intervals. i.e, $\left[x_{\tfrac{n-1}{2}},x_{\tfrac{n+3}{2}}\right],$

using the above theorem, we can say that all the terms in the sum except $\left|x_{\tfrac{n+1}{2}}-a\right|$ are minimized. So $$ \sum_{i=1}^n|x_i-a|=(x_n-x_1)+(x_{n-1}-x_2)+(x_{n-2}-x_3)+\cdots + \left(x_{\tfrac{n+3}{2}}-x_{\tfrac{n-1}{2}}\right) + \left|x_{\tfrac{n+1}{2}}-a\right| = \left|x_{\tfrac{n+1}{2}}-a \right|+\text{costant} $$ Now since the derivative of modulus function is signum function, $f'(a)=\operatorname{sgn}\left(x_{\tfrac{n+1}{2}}-a\right)=0$ for $a=x_{\tfrac{n+1}{2}}=\text{Median}$

$\implies$ When $n$ is odd,the median minimizes the sum of absolute values of the deviations.

When $n$ is even, $$ \sum_{i=1}^n|x_i-a|=|x_1-a|+|x_2-a|+\cdots+|x_{\tfrac{n}{2}}-a|+|x_{\tfrac{n}{2}+1}-a|+\cdots+|x_{n-1}-a|+|x_n-a|\\ $$ If $a$ is a member of all the intervals $[x_1,x_n], [x_2,x_{n-1}], [x_3,x_{n-2}], \ldots, \left[x_{\tfrac{n}{2}},x_{\tfrac{n}{2}+1}\right]$, i.e, $a\in\left[x_{\tfrac{n}{2}},x_{\tfrac{n}{2}+1}\right]$,

$$ \sum_{i=1}^n|x_i-a|=(x_n-x_1)+(x_{n-1}-x_2)+(x_{n-2}-x_3)+\cdots + \left(x_{\tfrac{n}{2}+1}-x_{\tfrac{n}{2}}\right) $$

$\implies$ When $n$ is even, any number in the interval $[x_{\tfrac{n}{2}},x_{\tfrac{n}{2}+1}]$, i.e, including the median, minimizes the sum of absolute values of the deviations. For example consider the series:$2, 4, 5, 10$, median, $M=4.5$.

$$ \sum_{i=1}^4|x_i-M|=2.5+0.5+0.5+5.5=9 $$ If you take any other value in the interval $\left[x_{\tfrac{n}{2}},x_{\tfrac{n}{2} + 1} \right] =[4,5]$, say $4.1$ $$ \sum_{i=1}^4|x_i-4.1|=2.1+0.1+0.9+5.9=9 $$ For any value outside the interval $\left[x_{\tfrac{n}{2}},x_{\tfrac{n}{2}+1}\right]=[4,5]$, say $5.2$ $$ \sum_{i=1}^4|x_i-5.2|=3.2+1.2+0.2+4.8=9.4 $$

Answer 8

Let $S=\{X_1,X_2,\ldots X_n\}$, w.l.o.g. assume $X_1\leq X_2 \leq \ldots \leq X_n$

$\implies D=\sum\limits_{k=1}^{n}|X_k-x|=|X_1-x|+|X_2-x|+\ldots +|X_{n-1}-x|+|X_n-x|$

Let $n \in \mathbb{Z}_{odd}^{+}$, i.e., $n=2m+1$ for some $m \in \mathbb{Z}^{+}$, for $m=0$ it's trivial.

Also, let's notice that $D=|X_{m+1}-x|+\sum\limits_{k=1}^{m}\left(|X_k-x| + |X_{n-k+1}-x|\right)$

Now, let's consider the sum $D_1=|X_1-x|+|X_n-x|$ and consider the following cases:

$(1) \quad x \leq X_1 \implies D_1=X_1+X_n-2x \geq X_n-X_1$

$(2) \quad x \geq X_n \implies D_1=2x-X_1-X_n \geq X_n-X_1$

$(3) \quad X_1 \leq x \leq X_n \implies D_1=X_n-X_1$

$\implies D_1$ is minimum when $X_1\leq x \leq X_n$

Similarly, $D_2=|X_2-x|+|X_{n-1}-x|$ is minimum when $X_2 \leq x \leq X_{n-1}$

$\ldots$

Likewise, $D_m=|X_m-x|+|X_{m+2}-x|$ is minimum when $X_m \leq x \leq X_{m+2}$

Finally, $D_{m+1}=|X_{m+1}-x|$ is minimum when $x=X_{m+1}$

Hence, $D$ is minimum when $\sum\limits_{i=1}^{m+1}D_i$ is minimum, when each $D_i$ has minimum possible value for $i=1,2,\ldots m+1$

$\implies x=X_{m+1}$, i.e., at the median value $D$ is minimum.

Similarly, we can show the same result for $n \in \mathbb{Z_{even}^{+}}$ too.

Answer 9

Suppose $S$ is finite (with cardinal $s$), without repetitions, and ordered. Then the sum of absolute values is continuous (sum of continuous functions), and piecewise linear (hence differentiable), with left-most slope $-s$. By induction, the slope increases by 2 for each interval from left to right, with right-most slope $+s$. Hence the piece-wise slope first reaches either $-1$ or $0$ at index $\left\lfloor \frac{s+1}{2}\right\rfloor$, and $0$ or $+1$ at index $\left\lceil \frac{s+1}{2}\right\rceil$.

Hence the function attains its minima in the interval $\left[\left\lfloor \frac{s+1}{2}\right\rfloor, \left\lceil \frac{s+1}{2}\right\rceil\right]$, which reduces to a singleton when $s$ is odd.

The notion of median for continuous functions is detailed in Sunny Garlang Noah, The Median of a Continuous Function, Real Anal. Exchange, 2007

Answer 10

As a start, i will define the median of a set with an even cardinality to be one of the two elements in the mid, for example {1,2,3,4} the median is either 2 or 3. and for a set with odd cardinality the median is the middle element.

Suppose that the set has elements, and 1<2<⋯<, we will start by showing that the median gets min sum for sets of cardinality (size) 1,2, and that any problem can be reduced to a set of cardinality 1 or 2.

For set {}, the median is , and the sum is zero.
For set {1,2}, the median is either 1,2, and the sum is |1 - 2| always.
Obvioulsy its easy to see and prove for sets of sizes 1,2 or any other size, that if x is not one of the set elements then the sum bigger than than if x was an element in the set.

So we have proved that the median works for sets of sizes 1,2. Now lets consider set of size 3 where its sorted: {1,2,3}, to get the minimum its easy to see and prove, that we have to pick x such that x is between s1,s3. but for such an x, |s1 - x| + |s3 - x| is always the same and equals |1 - 3| and so x that acheives min for set {1,2,3} is the same for set {s1} which is s1, which is the median.

Its also not hard to show that the same logic applies for set of four elements, the element that achieves min for set of 4 elements is the same element that achieves min for a set of 2 elements which is the median. And using the same logic reduce the problem from a set of 7 elements to set of 5 elements, and from a set of 6 elements to a set of 4 elements,and so on.

Answer 11

To find the minimum of

$$f(x) = \sum_{i = 1}^{N} \left| {s}_{i} - x \right|$$

we can just follow the usual road and demand $f'(x) \overset{!}{=} 0$ . So let's calculate the derivative $$ f'(x) = \sum_{i = 1}^{N} \frac{{s}_{i} - x}{\left| {s}_{i} - x \right|} = \sum_{i = 1}^{N} \text{sgn}( {s}_{i} - x ) $$ The sign-function can be expressed using Iverson brackets as $\text{sgn}(x) = [x > 0] - [x < 0]$ so we get¹ $$ f'(x) = \sum_{i = 1}^{N} [{s}_{i} > x] - \sum_{i = 1}^{N} [{s}_{i} < x] \overset{!}{=} 0 $$ Iverson brackets evaluate to 1 if the statement is true and zero otherwise. So one can already see here that the derivative can only be zero if the same number of true statements exists for each sum. This means the number of $s_i$ smaller than $x$ must be the same as the number of $s_i$ bigger than $x$.

To make this even more explizit write out the sums assuming the $s_i$ are realisations of a set/random variable $S$ for an arbitrary distribution $$ f'(x) = (1 + 1 + 0 + 1 + \ldots) - (0 + 0 + 1 + 0 + \ldots) = 0 $$ If the divide these sums by their length $N$ $$ f'(x) = \frac{1}{N}(1 + 1 +0 + 1 + \ldots ) - \frac{1}{N}(0 + 0 + 1 + 0 + \ldots) = 0 $$ we see that each (normalized) sum gives the probability for $s_i > x$ and $s_i < x$. So we can rewrite the statement in terms of cumulative probability distributions like $$ f'(x) = P(S > x) - P(S < x) = 0 $$ In addition we know from the conversation of probability that $$ P(S > x) + P(S < x) = 1 $$ Combining both statements gives $$ P(S < x) = P(S > x) = \frac{1}{2} $$ But this is precisely the definition of $x$ being the median of the distribution of the set $S$.

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_Footnotes

_{1. https://byorgey.wordpress.com/2021/06/08/average-and-median-via-optimization/}