# What does scale exactly mean?

I am not so familiar with statistical concepts. I am preparing a questionnaire to measure underlying constructs that affect pedestrian's road crossing behavior. However, I came across this term called "scale". Earlier I used to think that scale simply means measurement scale (e.g. nominal, ordinal, numerical etc.). But I have seen people using the term "scale construction". Could someone me tell me what it means by constructing a scale (as asked in this question: https://www.researchgate.net/post/What_is_the_fundamental_procedure_for_Scale_construction)?

Update: I have updated the question to be more clear and precise.

• It would help a lot if you could provide a link to where you read the word that confused you. With PCA it could just mean multiplication. – Dan Jul 11 '18 at 8:56
• I mentioned PCA just to tell that how I had to use statistics. My question about "scale" was not related to PCA. I am sorry for the confusion. However, following link contains what I wanted to ask about: researchgate.net/post/… – Muhammad Abdullah Jul 11 '18 at 11:16
• That link is talking about rating scales. An appropriate English definition of this use of the word scale is a set of numbers, amounts, etc., used to measure or compare the level of something – Dan Jul 11 '18 at 11:20
• "scale" is one of those words that have multiple meanings, each in their own context. At least three come immediately to mind in statistics – Glen_b Jul 11 '18 at 13:24

This word has many meanings, e.g.

• measurement scale (nominal, ordinal, etc.) is a classification of variables in statistics in terms of mathematical operations that can be applied to them,
• in probability and statistics scaling factor of probability distribution, so $Y = bX$ is a random variable defined in terms of random variable $X$ scaled by the $b$,
• what follows, we have location-scale distributions with location $a$ and scale $b$ parameters, $Y = a + bX$ (e.g. mean and variance in normal distribution) that lead to transforming probability density function to $f_Y = f_X(\tfrac{X-a}{b})\,/\,b$, and cumulative distribution function to $F_Y = F_X(\tfrac{X-a}{b})$,
• in psychometrics people often call the questionnaire as "scale", e.g. Wechsler Adult Intelligence Scale is one of the most popular IQ tests.

What exactly it means depends on context.

• Helpful, but I think your second bullet point conflates two distinct ideas. A scale parameter measuring spread (dispersion is another common word) generally has units and dimensions that are the same as the original variable, or is reducible to such. So a normal distribution fittted to data on weights has a standard deviation with the same units and dimensions; and similarly in general. Your example $Y=bX$ has $b$ as what might often be called a multiplicative or scale factor (rather than a scale parameter). – Nick Cox Jul 11 '18 at 8:55

A guess:

"scale" in PCA could arise form question "should we standardize variables before running PCA?".

The answer is "Yes", because if we do not, a scale of variable affects results. If, for example, one of variables is in centimeters and we change it's scale to meters, we would get completely different results in PCA.

Also if scale of one of the variables is from 0 to 1 and another from 0 do 1000, the later would have greater impact on PCA results.

So, my guess is that scale here means "unit", "range", "extent" etc.

Plus, R function that performs standardization is... scale :)