# Why is Standard Deviation used to evaluate the effect of a change in an independent variable on the dependent one?

I'm reading some economics papers about the relationship between inequality and growth and some of them have sentences like these:

an increase of 0.07 (one standard deviation in the sample) in the income share of the top 20 percent lowers the average annual growth rate just below half a percentage point

and

the estimated coefficients imply that an increase in, say, the land Gini coefficient by one standard deviation (an increase of 0.16 in the Gini index) would lead to a reduction in growth of 0.8 percentage points per year

Why is "one standard deviation" used? Why is it preferred to a unitary change? Thanks

• You cannot meaningfully increase the income share of the top $20$ percent by $1$, since it starts off above $0.2$ (i.e. $20\%$) and cannot exceed $1$ (i.e. $100\%$). Similarly the Gini coefficient lies in the range $[0,1]$ – Henry Oct 31 '13 at 18:41
• It's clear, but the Gini index could even have been expressed as a number from 0 to 100. It's not about the Gini index. It's about: why standard deviation and not 0.01? – Luigi Oct 31 '13 at 22:16
• In many cases, such descriptions should be in terms of some convenient unit, rather than in terms of standard deviations. One advantage of working with standardized coefficients is they're scale free, and they do at least give a way of getting an amount of change that you can find in the population, since a standard deviation is a kind of typical deviation from the mean... but other people (I can think of several) are no doubt better placed to explain the benefits than me. – Glen_b Oct 31 '13 at 22:45

## 2 Answers

I know this is an old question now, but coming from an engineering / policy analysis background I think about these things a bit. So, I think changing an independent variable by one standard deviation is one method to both display the Magnitude of the change and the Likelihood of the change.

For example, people commonly perform sensitivity analyses to determine how some model reacts to changes in a given independent variable. Often, the changes show some degree of sensitivity (e.g., a 10% change in X leads to a 10% change in Y) but such analyses do not provide any indication of how likely a 10% change in X is. You can see how this would become an issue:

Imagine a 10% change in A changes Y by 10% and a 10% change in B also changes Y by 10%. One would think the sensitivity of the model to both parameters is the same (and maybe technically it is) but this ignores the real world. Maybe A is the price of gasoline (which changes frequently and sometimes greatly) and B is the time it takes the earth to rotate around the sun (not going to change by 10%...hopefully). Once we know a 10% change in A is more likely, and thus the model results are more likely to change due to the result of A, we would be more likely to try and improve the model by reducing the uncertainty associated with A.

Practically, the value of 1 may have very different meaninings in different contexts, so the standard deviation would put the sensitivity in terms of a "typical" deviation from the current value.

Theoretically, if a normal approximation is being used to model the variables, then 1 standard deviation is a convenient metirc for converting deviations into probabilities via the Z-score.

• The standard deviation is a somewhat arbitrary choice, and does not work correctly when the distribution is not symmetric and smooth. Even more importantly, the approach does not handle nonlinearities. I prefer to use original units, accounting for non-linearity. – Frank Harrell Mar 4 '14 at 13:06
• @FrankHarrell I think the poster was asking why would one use the SD. The main justification is asymptotic or based on knowledge of the symmetry of the underlying distribution – user31668 Mar 4 '14 at 20:56
• I can't see what asymptotics have to do with this, or why I would assume symmetry. – Frank Harrell Mar 4 '14 at 22:43
• @Frank Harrell I guess I'm not sure what you are getting at with your comments on my post. The poster wanted to know about why standard dev is used in the cases cited, which are economic papers.... The data in question may be well approximated by a normal distribution or the standard deviation may have intrinsic theoretical meaning OR it may just happen that the standard deviation is a commonly used measure in economics and has empirical value ... – user31668 Mar 5 '14 at 0:18
• @Frank Harrell what I gave we're reasons that the standard deviation may be a useful measure of change. Your comments pointed out situations where it may not be relevant... I am not disagreeing with you nor am I advocating for the standard deviation as the ideal metric- all I was offering was plausible reasons why one may use a standard deviation as a metric. To the extent that the Gini index is discontinuous or highly skewed the standard dev would not be the ideal measure although chebyshevs inequality always provides some intuitive, albeit coarse, interpretation of a standard deviation – user31668 Mar 5 '14 at 13:27