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Apologies im sure this is a very simple math question ...

How do I represent the sum of three means in mathematically notation.

$p =$ Mean$(x_1,x_2,x_3)$ + Mean$(y_1,y_2,y_3)$ + Mean$(z_1,z_2,z_3)$

Here is my best effort

$\displaystyle\sum_{i=1}^{3} M(x,y,z) $

Aiden Chow
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Bat
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    Your sum does not depend on $i$. – Dietrich Burde Feb 12 '20 at 15:56
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    I now think I understand what you are saying. When you say Mean(x1,2,3)$ you mean $\frac 13(x_1+x_2+x_3)$ . What you wrote is quite confusing. I'll have to get back. – Ross Millikan Feb 12 '20 at 17:12
  • @RossMillikan apologies yes I probably didnt explain it very well. I want to represent the mean of x1,x2,x3 and the mean of y1,y2,y3 and finally the mean of z1,z2,z2. Then represent the sum of the three means – Bat Feb 12 '20 at 17:24

2 Answers2

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If you view $x_1,x_2,x_3$ as a sequence $x$ you can just define and talk of $\operatorname{Mean}(x)$ or $\overline x$. Then the sum of means would be $\operatorname{Mean}(x)+\operatorname{Mean}(y)+\operatorname{Mean}(z)$ or $\overline x+ \overline y + \overline z$. I don't think there is a useful way to use a sigma to represent the sum because there is not a way to iterate over $x,y,z$

Ross Millikan
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  • thank you. The mean of x,y,z all have a maximum of three values each also i.e. y1,y2,y3 and z1,z2,z3. Is the notation you supplied still valid ? thanks – Bat Feb 12 '20 at 16:58
  • It would work for a sequence of any length. Please note that it is not the mean that has a maximum of three values but the sequence. The mean of one sequence is just one value. That is part of what confused me. – Ross Millikan Feb 12 '20 at 20:53
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It's a little unclear what the three means are. I understand that the point of the question is get some help in making it clear, so let me go through some possibilities of what you meant and how you might state things more clearly.

$$\begin{align*} \text{If }p &= \text{Mean}(x_1,2,3)+\text{Mean}(y_1,2,3)+\text{Mean}(z_1,2,3)\\ \\ &=\frac{x_1+2+3}{3}+\frac{y_1+2+3}{3}+\frac{z_1+2+3}{3} \end{align*}$$

Then, this can be written more simply as

$$\begin{align*} p &= \frac{x_1+y_1+z_1}{3}+\frac{2+2+2}{3}+\frac{3+3+3}{3}\\ \\ &= \frac{x_1+y_1+z_1}{3}+2+3\\ \\ &= \text{Mean}(x_1,y_1,z_1)+5 \end{align*}$$

On the other hand,

$$\begin{align*} \text{if }p &= \text{Mean}(x_1,x_2,x_3)+\text{Mean}(y_1,y_2,y_3)+\text{Mean}(z_1,z_2,z_3)\\ \\ &=\frac{x_1+x_2+x_3}{3}+\frac{y_1+y_2+y_3}{3}+\frac{z_1+z_2+z_3}{3} \end{align*}$$

Then I don't think you could write this more simply without choosing different notation. One way to do this would be to let

$$a_1=x_1\quad a_2=x_2\quad a_3=x_3$$ $$a_4=y_1\quad a_5=y_2\quad a_6=y_3$$ $$a_7=z_1\quad a_8=z_8\quad a_9=z_3$$

Then we have that

$$\begin{align*} p &=\frac{x_1+x_2+x_3}{3}+\frac{y_1+y_2+y_3}{3}+\frac{z_1+z_2+z_3}{3}\\ \\ &= \frac{a_1+a_2+a_3}{3}+\frac{a_4+a_5+a_6}{3}+\frac{a_7+a_8+a_9}{3}\\ \\ &= \frac{1}{3}\cdot(a_1+a_2+\ldots+a_9)\\ \\ &= \frac{1}{3}\sum_{i=1}^9a_i \end{align*}$$

You could also represent the sum of the three means more compactly using a double summation. We have that

$$\begin{align*} p &= \text{Mean}(x_1,x_2,x_3)+\text{Mean}(y_1,y_2,y_3)+\text{Mean}(z_1,z_2,z_3)\\ \\ &=\frac{x_1+x_2+x_3}{3}+\frac{y_1+y_2+y_3}{3}+\frac{z_1+z_2+z_3}{3} \end{align*}$$

Let's introduce the following notation:

$$a_{11}=x_1\quad a_{12}=x_2\quad a_{13}=x_3$$ $$a_{21}=y_1\quad a_{22}=y_2\quad a_{23}=y_3$$ $$a_{31}=z_1\quad a_{32}=z_8\quad a_{33}=z_3$$

Then we have that

$$\begin{align*} p &= \frac{x_1+x_2+x_3}{3}+\frac{y_1+y_2+y_3}{3}+\frac{z_1+z_2+z_3}{3} \\ \\ &= \frac{a_{11}+a_{12}+a_{13}}{3}+\frac{a_{21}+a_{22}+a_{23}}{3}+\frac{a_{31}+a_{32}+a_{33}}{3} \\ \\ &=\frac{1}{3}\sum_{j=1}^3a_{1j}+\frac{1}{3}\sum_{j=1}^3a_{2j}+\frac{1}{3}\sum_{j=1}^3a_{3j}\\ \\ &= \frac{1}{3}\sum_{i=1}^3\sum_{j=1}^3a_{ij} \end{align*}$$

I hope this helped. If you had something else in mind when you defined $p$, or if you had any questions about what I've wrote, then let me know.

user729424
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  • wow thanks Andrew for taking the time to write this out. Sorry my math is very weak. To answer your questions yes what I meant to put was p = Mean(x1,x2,x3) + Mean(y1,y2,y3) + Mean(z1,z2,z3) I just wondered if this was the best to represent it or if there was a better way (as in my feable attempt). The later examples you provide are interesting but it looks like I would possibly be making it more difficult that it needs to be ? So p = Mean(x1,x2,x3) + Mean(y1,y2,y3) + Mean(z1,z2,z3) is possibly the simplest way to represent it ? thanks – Bat Feb 12 '20 at 19:09
  • I agree. I think $p=\text{Mean}(x_1,x_2,x_3)+\text{Mean}(y_1,y_2,y_3)+\text{Mean}(z_1,z_2,z_3)$ is probably the simplest. I'm glad what I wrote was helpful. – user729424 Feb 13 '20 at 00:35