# Mean and variance after subtracting mean and dividing by standard deviation

That is mean, variance and standard deviation.

My question is how to get from point 4 to 5 and also with the variance from point 6 to 7.

• Divide top and bottom by $n$. – Nick Cox Nov 29 '16 at 19:45
• That is the last step, I want to know how to get rid of Xi. (one step earlier) – Daniel Nov 29 '16 at 19:50
• In each case, the left-most equation defines a statistic of $z$ (i.e. $\bar{z}$, $s_z^2$). Substitute "$x$" for "$z$" to get definitions for the the corresponding $x$ statistics. The steps you are asking about seem to reduce to these definitions. – GeoMatt22 Nov 29 '16 at 20:02
• Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. – gung - Reinstate Monica Nov 29 '16 at 20:07
• Well I understand how to calculate all three of these subjects with specific numbers. Only problem I have encountered are these steps. It is about proving that variance and standard deviation are equal when 1. But i don´t understand the transition where the sum and Xi disappears. – Daniel Nov 29 '16 at 20:14

These equations represent a particularly obscure way to make some important points that everybody ought to understand. I will therefore provide an indirect answer by highlighting the fundamentals (1-4 below), demonstrating them, and then applying them in what amounts to an equivalent proof.

1. When you add a constant $a$ to all data $x_i$, the mean of the new values is $a$ plus the mean of the old values. This should be obvious, because adding $a$ to each of $n$ values adds $na$ to the sum. When the sum is divided by $n$ to get the mean, $na$ is divided by $n$ to show $na/n=a$ is added to the sum.

2. When you multiply each $x_i$ by a constant $b$, the mean of the new values is $b$ times the original mean. This truly is obvious (it's a direct application of distributive and commutative laws of arithmetic).

3. When you add a constant $a$ to all data, the variance is unchanged. This is because the variance is the average of the squared residuals, $(x_i-\bar x )^2$. By (1), $\bar x$ increases by $a$ and that exactly cancels the addition of $a$ to each $x_i$, whence the residuals are unchanged. Consequently the variance is unchanged.

4. When you multiply all data by a constant $b$, the variance is multiplied by $b^2$. Since (3) tells us each $x_i$ as well as their mean $\bar x$ are multiplied by $b$, the residuals $x_i - \bar x$ are also multiplied by $b$. Consequently the squared residuals are multiplied by $b^2$ and so (exactly as in (2)) the mean squared residual is multiplied by $b^2$.

The equations in the question attempt to demonstrate that the mean and variance of $z_i$ are zero and one, respectively, when the $z_i$ are formed by standardizing the data: that is, $-\bar x$ is first added to the data (giving the residuals) and those results are divided by the square root of the variance. Call the square root $s$, so the variance is $s^2$.

Here, then, is an alternative to the equations in the question:

By (1), the mean after the first step is $\bar x - \bar x = 0$.

By (2), the mean remains zero upon division by the square root of the variance. (This should remind you of step "5" in the question.)

By (3), the variance is unchanged after the first step.

By (4), the variance $s^2$ is divided by the square of $s$ in the second step: but that just divides the variance by itself (step "7" in the question), giving $s^2/s^2=1$, QED.

• It made the issue much more clear but I still struggle with the 6 to 7 transition. What is b in step "6" how do I end up with sx^2/sx^2? Especially how do I get the upper part of the fraction from (Xi-X)^2 to sx^2 – Daniel Nov 29 '16 at 22:01
• Your image might be rendering differently on different machines, for I see no "$b$" anywhere in the equations. Step 7, as a previous commenter has noted, merely substitutes the formula for $s_x^2=\frac{\sum_i(x_i-\bar x)^2}{n}$ in $$\frac{\color{red}{\sum_i(x_i-\bar x)^2}}{\color{red}{n} s_x^2}= \color{red}{\frac{\sum_i(x_i-\bar x)^2}{n}}\frac{1}{ s_x^2}=\frac{\color{red}{s_x^2}}{s_x^2}.$$ – whuber Nov 29 '16 at 22:15

I'm not sure if this is what you wanted, but I'll give it a shot.

First, we declare these known relationships:

$\bar{x}=\dfrac{\sum_{i=1}^n x_i}{n}$

$\sum_{i=1}^n \bar{x}=n\bar{x}$

$s_x^2=\dfrac{\sum_{i=1}^{n} (x_1-\bar{x})^2}{n}$

So, starting with (4):

$\dfrac{\sum_{i=1}^n x_i}{ns_x} - \dfrac{\sum_{i=1}^n \bar{x}}{ns_x} = \left(\dfrac{\sum_{i=1}^n x_i}{n} \cdot \dfrac{1}{s_x} \right) - \dfrac{n\bar{x}}{ns_x} = \left( \bar{x} \cdot \dfrac{1}{s_x} \right) - \dfrac{n\bar{x}}{ns_x} = \dfrac{\bar{x}}{s_x} - \dfrac{n\bar{x}}{ns_x}$.

We get (5). QED.

Starting with (6):

$\dfrac{\sum_{i=1}^{n} (x_1-\bar{x})^2}{ns_x^2} = \left( \dfrac{\sum_{i=1}^{n} (x_1-\bar{x})^2}{n} \cdot \dfrac{1}{s_x^2} \right) = \left( s_x^2 \cdot \dfrac{1}{s_x^2} \right) = \dfrac{s_x^2}{s_x^2}$.

We get (7). QED.

• Welcome to the site, @ElmerVillanueva. Please be cautious about answering [self-study] questions. Our policy is to provide hints to help the OP get unstuck, not to do their homework for them. You can find the full statement of our policies here. – gung - Reinstate Monica Nov 29 '16 at 22:42