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I have a set of data made by 2 variables (V1 and V2). For each variable, the values are between 1 and 7. Now, to calculate the summary score according to the guidelines I must apply the following formula:

TOTAL-SCORE = 0.65*[(V1 + V2 - 2)*(100/12)]+22.9

The issue is the following. If I calculate the standard deviations from the original values it will not make much sense when associated with the score because it is done on values between 1 to 7. What's the best way to calculate the standard deviations in such a case? Should I apply the same transformation?

kjetil b halvorsen
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ANameToDisplay
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    If you know the variances or standard deviations of $V_1$ and $V_2$ and their covariance or correlation then you can calculate the variance and standard deviation of "Total Score" from this – Henry Aug 31 '21 at 11:42

1 Answers1

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In general, $Var(aX + bY + c) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)$. Thus, multiply out your equation to get it in that format.

$$ Z = 0.65\times[(X + Y - 2)\times(100/12)]+22.9\\ =0.65\times [(100/12)X + (100/12)Y - (200/12)] + 22.9\\ = (65/12)X + (65/12)Y + (- (130/12) + 22.9) $$

Thus, $a = 65/12$, $b=65/12$, and $c = - (130/12) + 22.9$, but $c$ drops out.

We we assume independence (which implies zero covariance (but the reverse is false)), then the variance of your resulting variable is:

$$ \bigg(\frac{65}{12}\bigg)^2Var(X) + \bigg(\frac{65}{12}\bigg)^2Var(Y) $$

And the standard deviation is the square root.

If you do not have independence of $X$ and $Y$, then you need to calculate $Cov(X, Y)$. If you have $\rho_{XY}=corr(X, Y)$ but not $Cov(X, Y)$, you can calculate the covariance by the following equation.

$$ Cov(X, Y) = \rho_{XY}\sqrt{Var(X)}\sqrt{Var(Y)} $$

EDIT

If you do not have the correlation or covariance but also do not want to assume independence, then I think the best you can do is calculate the variance of the output variable for multiple values of correlation. I will give a simulation below.

set.seed(2021)
rhos <- seq(-1, 1, 0.05)
var1 <- 3
var2 <- 2
a <- 65/12
b <- 65/12
var_z <- a^2 * var1 + b^2 * var2 + a * b * rhos * sqrt(var1) * sqrt(var2)
plot(rhos, var_z, xlab = "correlation(X, Y)", ylab = "Variance(Z)")
Dave
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