Maybe I don't understand what you would like to do. Maybe this simple transformation of the (standardized) 1-4 Likert scores: (score - 1) * 33.33?

I want to change the scales of the new variable (I named it "religiosity") which I created through principal component analysis. As the author did in his research I attached, I also want to standardize scales of the variable "religiosity" to 0-100 scores. I would like to know the equation for this...!

I assumed "religiosity" is a Likert-style four-point agree-disagree factor score. You can standardize these and then transform them like this':

score_100 = (factor-score -1) x 33.33

e.g. for the Likert scores 1 to 4:

0 = (1-1) x 33.33

33.33 = (2-1) x 33.33

66.67 = (3-1) x 33.33

100 = (4-1) x 33.33

You can "make" the "religiosity" scale to be a Likert-style 4 point score by using the "factor-based" method (see http://www.theanalysisfactor.com/index-score-factor-analysis/ ), simply by taking the average of all 4 point Likert-style variables loading high on "religiosity". Afterwards you can standardize the scores.

Thank you very much for the kind answer. I do appreciate your generosity and information you shared with me. I read the links you attached and it helped me understand the concepts. Yet,  I have come up with practical questions on how to actually do these. I used Principal component analysis to generate a variable "religiosity". Regarding this, I have 3 questions

1) 6 items were put into for the PCA, and their scales are (1) 10 point scale (2) 4 point scale (3) 7 point (4)&(5)&(6) are Yes-No questions. 

Can I just do a PCA with all of these items to generate a one factor(religiosity), or do I have to standardize their scores before PCA like z-score?

2) When PCA generates one new variable(which I name it "religiosity"), could I just standardize the scale of this variable with the way you told me in your first answer above?

The factor scores produced by a PCA are standardized, they have a mean of 0 and a standard deviation of 1. The only thing you need to do is a linear transformation so that

-1   -> 0

0    -> 50

+1  -> 100

This can be accomplished with this formula:

y = 100*((x +1)/2) , where y is the desired transfo zand x is the standardized factor score.

For example:

x = -1 ->  y = 100 * ((-1 +1)/2) , 100 *0 = 0

x = -.50 -> y = 100 * ((-.50+1)2)), 100 * (.50/2) = 25

x = 0 -> y = 100 * ((0+1)/2), 100* (1/2) = 50

x = .50 -> y = 100* ((.50+1)/2), 100*(1.5/2) = 75

x = 1 -> y = 100*((1+1)/2), 100*(1) = 100

Sorry, my former answer is wrong. The range should be from your lowest factor score to your highest. You can contact me at (remove blancs):

jos.feys @ kuleuven.be

If your lowest factor score is -3, and your highest is 3, then use the following formula:

y = 100*(x +3)/6

for -4 to 4:

y = 100*(x + 4)/8

Thank you so much for your kind answer. 

6 variables I want to use for PCA have different scales. One has 10-point scale, one has 4-point scale, and three of them are yes-no scale. I wonder I need to standardize the scales of these 6 items before conducting PCA. 

Or it doesn't matter. I can just do PCA  without any extra work. 

Just do the PCA without extra work. The procedure yields factor scores which are standardized, with a mean = 0 and a std = 1.

To decide what the range of the factor scores (x) is, take the largest absolute (positive or negative) value of x and round to the above integer.

Examples:

largest value x = -2.61   -> round to 3;  y = 100*(x +3)/6 ,here y = 6.5

largest value x = +3.15  -> round to 4;  y = 100*(x +4)/8 ,here y = 89.375

largest value x =  +2.99 -> round to 3;  y = 100*(x +3)/6 ,here y = 99.8333

largest value x = +1.82 -> round to 2;  y = 100*(x +2)/4 ,here y = 95.5

largest value x = -3.56 -> round to 4;  y = 100*(x + 4)/8 , here y = 5.5

A more exact method would be to take the exact range, from the lowest to the highest.

Let r be the exact range. Then y = 100 * ((x + r/2) /r)

Example lowest = -1.5, highest = 1.5; then the r = 3

if x  = -1.5   , then y = 100 * ((-1.5 + 1.5)/ 3) = 0

if x  = -1.0   , then y = 100 * ((-1 + 1.5)/3) = 16.67

if x = -0.5    , then y = 100 * ((-0.5 + 1.5)/3) = 33.33

if x = 0         , then y = 100 * ((0 + 1.5)/3) = 50

if x = 0.5      , then y = 100 * ((0.5 + 1.5)/3) = 66.67

if x = 1         , then y = 100 * ((1+1.5)/3) = 83.33

if x = 1.5      , then y = 100

if x = 1.25   , then y = 100 * ((1.25+1.5)/3 = 91.67 

Thank you very much Joe for the kind and sincere explanations.

I decided to standardize variables before pca, and I have 5 variables for "Religiosity"  which have different scales as below:

1) Belief in God: No - Yes

2) Religious Person or Not: No - Yes

3) Importance of God in Life: 1 - 10 points

4) Frequency of Religious Attendance: 1 - 7 points

5) Frequency of Pray: 1 - 8 points

If I want to standardize them to 100 points

1) & 2): No → 0 , Yes → 100

3) 1 → 10 ,  10 → 100  (multiply 10)

5) 1 → 12.5, 8 → 100 (multiply 12.5)

Is this a right way? 

Questions coming up on my head:

1) Highest score is 100, equal for all variables, but the lowest score is all different. This is a problem, right?

2) I am not sure for the No-Yes question, on whether I should make No: 0 or 1. 

3) What is the appropriate way to calculate this?

By the way, thank you very much for all help Joe. I'd like to deliver my great appreciation to you. I've been frustrated on solving these problems by myself for the past months as I couldn't really find anyone who could help me with this, and materials online didn't give me full trust on the methodologies I am using. 

Eunhye, I am afraid there is a misunderstanding. What you did before the PCA is not standardizing, but rescaling. You just performed  transformations of the data. This does not influence the factor scores. Yet, factor scores are standardized, with a mean = 0 and a standard deviation = 1. See:

https://en.wikipedia.org/wiki/Standard_score

What a big misunderstanding I made..! Thank you for pointing it out and teaching me.

What Norris and Inglehart did in their book Rising Tide (2003) was they standardized scales of these variables to 100 points (it is stated in the first paragraph in the first picture) and calculated mean for each variable (table 3.2). Then, they combined all these variables and named it as "religiosity" to put into the regression analysis. 

1) What would be the appropriate way to standardize the scales of the variables as they did then? z-score?

2) When they said in the book that they combined all variables for the variable "religiosity", does it mean literally just "combined the scores", or should I generate it through pca?

3) Second picture shows what I get after pca of these variables without any extra work beforehand. Do you think I am doing it right?

I would suggest you standardize the variables to T-scores, i.e. with a mean = 50 and a standard deviation = 10.

Running the PCA on these (standardized) variables is what they mean with 'combine', I think.

The authors standardized the factor score 'religiosity'. You should standardize this variable too (mean = 50, std = 10).

To normalize a variable, e.g. Population, use this formula below, in which

min = minimum value of Population and  max = maximum value of Population.

y = ((Population - min)/ (max - min)) * 100

Thank you Joe for the answers. I appreciate it a lot. 

I tried T-scores to standardize the scales of the variable Belief_G, which has only Yes and No response. 

But the result is not what I expected.

For the variable Belief_G: 0=No, 1=Yes.

When I do the t-score, 

mean is 50 and std=10, but the min value=19 and max=53.

Mean of the variable Belief_G should be very high because the original response rate is 0(No)=9.6% and 1(Yes)=90.4%. Which means, the mean of the t-scored value should also be very high, close to 100 (also the table in the picture I attached above shows the similar results). 

Is it maybe the technical error I made with the stata syntax?

Oh, I applied the equation you explained in your answer with the example of Population and I did succeed it! Wow it feels just amazing to see the results on the output file!

Does this the equation for T-score?

Yes, do use this formula to make the scales (factor scores) 100 point:

y = ((factor - min)/ (max - min)) * 100

Forget the T-score. Sorry for the confusion.

You can transform the factor scores with the above formula.

Example:

min = -1.47203

max = 1.8796

factor1 =  0.96967

y = ((0.96967+ 1.47203)/ (1.8796  + 1.47203)) * 100 

y = 72.851

Remember that factor score are standardized (mean = 0, std = 1)

Hello Dr. Feys and Dr. Kang,

Could you please tell me what is the value for "factor" here since my factor loading has the values for 41 variables? Do I need to composite them first or use their mean score? Thank you very much.

Religiosity is a known variable and there are many studies where it has been used.Pick up the standard statements and add a likert scale 1-7 in front of each statement