Normalizations: dividing by mean

Question

I'm reading various papers and I don't understand the meaning of three types of normalizations used. Let's say I have the number of calls $X_i(t)$ in region $i$ at time $t$. I see it normalized with:

1. Z-score: $X_i(t) = (X_i(t)-\mu(t)) / \sigma(t)$. This takes somehow the "shape" of the calls time series.
2. dividing by the mean: $X_i(t) = {X_i(t)}/{\mu(t)}$ ref. This is unknown by me ^^
3. subtracting the mean: $X_i(t) = X_i(t) - \mu(t)$.

What's the difference between 2 and 3? Why should I divide by the mean and what's its "meaning"?

Answer 1

The difference between subtracting the mean and dividing by the mean is the difference between subtraction and division; presumably you are not really asking about the mathematics. There is no mystery here, as it's no more than a statistical analogue of

• Bill is 5 cm taller than Betty (subtraction)

• Bill is twice the weight of his son Bob (division)

with the difference that the mean is used as a reference level, rather than another value. We should emphasise that

• (Bill $-$ Betty) or (value $-$ mean) preserves units of measurement

while

• (Bill / Bob) or (value / mean) is independent of units of measurement.

and that subtraction of the mean is always possible, while division by the mean usually only makes sense if the mean is guaranteed to be positive (or more widely that no two values have different signs and the mean cannot be zero).

Taking it further then (value $-$ mean) / SD is scaling by the standard deviation and so again produces a measure independent of units of measurement, and also of the variability of the variable. It's always possible so long as the SD is positive, which does not bite. (If the SD were zero then every value is the same, and detailed summary is easy without any of these devices.) This kind of rescaling is often called standardization, although it is also true that that term too is overloaded.

Note that subtraction of the mean (without or with division by SD) is just a change of units, so distribution plots and time series plots (which you ask about) look just the same before and after; the numeric axis labels will differ, but the shape is preserved. Conversely, division by the mean will change shape, except on a logarithmic scale, on which shape is preserved.

The choice is usually substantive rather than strictly statistical, so that it is question of which kind of adjustment is a helpful simplification, or indeed whether that is so.

I'll add that your question points up in reverse a point often made on this forum that asking about normalization is futile unless a precise definition is offered; in fact, that are even more meanings in use than those you mentioned.

The OP's context of space-time data is immaterial here; the principles apply regardless of whether you have temporal, spatial or spatial-temporal data.

Answer 2

If you're considering datapoints from multiple years, subtracting or dividing by year-specific means would change plots combining multiple years. Dividing by the mean might be interesting in many applications, one of which I dealt with today. For instance, if you're interested in observing how a sociodemographic group is distributed/concentrated in different equally sized neighbourhoods of a city you might simply look at the percentage of people, in each neighbourhood, that belongs to that group. However, if you're interested in observing how the concentration patterns evolve in time, you might want to net off the effect of changes in the total number of members of the group living in the city (for instance cos you're only interested in location choices within the city). If that is the case, for each t, it would be helpful to divide each neighbourhood level percentage by the share of the group in the city population in time t (that, if neighborhoods are equally sized and cover the whole city, is equal to the mean percentage). And, of course, it could make the difference!