# What does the expression “scale the numbers” mean?

I was reading an article about web design and had problems understanding the meaning of the phrase "scale the numbers". Does it mean the results were sorted from 0 to 1?

Context:

After creating a bunch of these algorithms, I ran it on some sample data, scaled the numbers between 0 & 1, and plotted the results. Full text

• It simply means that he adjusted all the original input values so they fell within the range 0 to 1. But apparently his didn't, since several "distances" fall outside that range. – FumbleFingers May 15 '17 at 19:00
• Thinking about it, he probably meant he scaled/adjusted/normalised the numbers so all values were between 0.n and 1.n (where 'n' = any sequence of digits), but if that's the case he hasn't expressed himself correctly. – FumbleFingers May 15 '17 at 19:06

To be honest, I have no idea what he meant since he says he, "scaled the numbers between 0 & 1" yet some of the numbers are over 1. So what I believe he meant to say is that he normalized the numbers relative to some other number -- so that the results can be expressed as a ratio of that number, rather than just raw data.

For example, suppose I take a survey asking people about their favorite ice cream flavors, and these are the results:

• 44 vanilla
• 35 chocolate
• 22 strawberry
• 12 mint
• 7 other

The total number of people surveyed is 120, so I can divide each by 120 to get the following normalized ratios:

• 0.37 vanilla
• 0.29 chocolate
• 0.18 strawberry
• 0.10 mint
• 0.06 other

Alternately, I could normalize the values relative to the top score (44) and thereby show ratios relative to the post popular flavor:

• 1.00 vanilla
• 0.80 chocolate
• 0.50 strawberry
• 0.27 mint
• 0.16 other

Alternately I could normalize relative to the lowest value, or the mean, or some other significant number. Which I choose depends on the point I want to make.

When you scale something, you adjust the total size or amount but maintain the original proportions or relative sizes or amounts of the components. Think of "scale" in the sense of a ruler divided into certain units.

One way to think of it is changing the unit of measurement without changing the values. A map, for example, might have a notation like 1" = 1 mile (in the US, anyway), meaning units of inches on the map are equivalent to units of miles in real life. You change the overall size or amount just by changing the units you measure with. Another example of scaling would be a recipe, where you want to make a lesser or greater total amount of something and need to keep the ingredients in the same proportion.

If you keep the unit of measurement the same when you scale, you adjust the values so they are in proportion. Use the recipe analogy. In volume measurement, 4 cups equals one quart. Say a recipe calls for 1 quart of X and 4 quarts of Y, and you want to make 1/4 as much. You could change the unit of measure from quart to cup, making it 1 cup of X and 4 cups of Y. Or, you could keep quarts as the unit of measure and adjust the values to 1/4 quart of X and 1 quart of Y.

The second alternative would be scaling the values (or numbers), between 0 and 1. When you scale numbers to fit in a certain range, you pick a value that is within the range for the largest example, and then make the other values proportional to their original relative size.

What he has done in the calculation is rescaled/renormalized the values for the 3 factors: xHeight, contrast, and formality. The assumption is that these 3 dimensions accurately describe a font face and should be the least correlated with each other and other factors that he calculated. This forms his basis for measuring fonts. Notice that he is using a 3 dimensional space based on these 3 factors (instead of the standard x,y,z).

He then normalized the numbers so that the maximum value is 1 and the minimum is 0 which creates a (0,1) interval or unit-vector for that dimension. Scaling or renormalizing in this way ensures the individual unit vectors (one for each font) can be compared with one another (gets rid of outliers).

By using unit vectors, he can then calculate the distance (Euclidian), which he is using as a measure of similarity between each vector (font face). You can confirm the distance calculation using this 3D calculator. It is also why the distance calculation is greater than 1, since it is not rescaled/renormalized.