# How to alter the standard deviation of an empirical distribution?

#### consuli

##### Member
For the purpose of a simulation I need to increase the standard deviation of an empirical distribution. I want to simulate what will happen if the distribution would widen. How can I do that?

I am already aware, that the empirical distribution is usually replaced by a similar analytical distribution in a simulation. However my empirical distribution is so "unusual", that I cannot find a analytical distribution pendant.

#### Dason

Multiply every value by a constant

#### consuli

##### Member
Multiply every value by a constant
Interesting, I have thought about the same transformation. Multiplying each x- median(x) by a certain factor and adding back the median afterwards. Before I have thought about multiplying each x- mean(x) by a certain factor and adding back the mean afterwards, but the median would be more appropriate for a skewed distribution, I think.

The simulation I want to do is about to simulate the risk of a portfolio out of special securities. The returns of the securities in the portfolio have unusual dsitributions. For my simulation it is necessary, that I can increase the standard deviation of each empirical return distribution by a certain proportion, e.g. by 10%.

Unfortunatley the standard deviation does not increase by 10% when multiplying each value by 1.1 as mentioned above. I need some kind of converging algorithm, that produces a transformed empirical distribution with my target standard deviation.

Is it possible to make use of the Newton–Raphson method to construct a converging algorithm that produces the desired target standard deviation within a given tolerance?

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#### Dason

Unfortunatley the standard deviation does not increase by 10% when multiplying each value by 1.1 as mentioned above.

Are you sure?

Code:
``````> j <- rexp(30)
> sd(j)
[1] 0.886905
> sd(1.1*j)
[1] 0.9755955
> sd(j)*1.1
[1] 0.9755955``````
Mathematically if you multiply all the values by a certain constant then the standard deviation will increase multiplicatively by that constant

#### consuli

##### Member
Thanks Dason. I did not know before that
multiply all the values by a certain constant then the standard deviation will increase multiplicatively by that constant
.

At the moment my favorite transformation is

xt2= (x- mean(x))* 2 + mean(x)

as it produces multiple standard deviation while conserving mean and skewness.

Code:
``````##### Setup

skew= function(x) {
n= length(x)
n/(n-1)/(n-2) * sum( ((x- mean(x))/sd(x))^3) }

#####

x= rgamma(1000, shape=3, scale = 1)

xt1= x* 2
xt2= (x- mean(x))* 2 + mean(x)

# Control
sd(x)
sd(xt1)
sd(xt2)
skew(x)
skew(xt1)
skew(xt2)
mean(x)
mean(xt1)
mean(xt2)``````