Calculation of Z Scores with Sample Standard Deviation

[Evan Harrington, Ph.D.]

I just realized that the SPSS function for calculation of Z scores (using the “Save Standardized Values as Variables” function) utilizes the sample standard deviation instead of population standard deviation.

 

Am I mistaken in thinking that the Z formula ought to have population standard deviation in the denominator, rather than the sample version as used by this SPSS function?

 

Evan Harrington, Ph.D.

IRB Committee Chair, Chicago Campus

Associate Professor, Department of Forensic Psychology

The Chicago School of Professional Psychology

325 N. Wells Street

Chicago, IL 60654

 

[phone]

 

Practicable: able to be done or put into practice successfully.

 

[Matt Freeman]

Are you assuming that the standardized scores are z-scores rather than t-
scores?

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[Evan Harrington, Ph.D.]

They are "Standardized scores" yet they use sample standard deviation to determine the value of "Z"

But I want Z scores -- I'm assuming the group of scores that I have is actually a population. Thus, I was dismayed to see that SPSS lacks a choice to use population instead of sample values. Now I need to create my own transformations.


Evan Harrington, Ph.D.
IRB Committee Chair, Chicago Campus
Associate Professor, Department of Forensic Psychology
The Chicago School of Professional Psychology
325 N. Wells Street
Chicago, IL 60654

[phone]

Practicable: able to be done or put into practice successfully.


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of Matt Freeman
Sent: Thursday, January 24, 2013 4:25 PM
To: [hidden email]
Subject: Re: Calculation of Z Scores with Sample Standard Deviation

Are you assuming that the standardized scores are z-scores rather than t- scores?

=====================
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=====================
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[Baker, Harley]

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Yes, it should and it does. The use of  "N" - 1 in the denominator (it is actually W, the sum of weights) of the SD rather than N provides the unbiased estimate of the population standard deviation. That is why we use N-1 in the denominator for inferential statistics. The use of N in the denominator calculates the standard deviation of the scores as if it was a population rather than being a sample from a population about which one wants to draw inferences. SPSS uses the unbiased estimate of the population standard deviation (N - 1) as it should. BTW, the same formulas are used by virtually all of the major statistical packages (e.g., SAS, Statistica, SYSTAT.)

Dr. Harley Baker

Professor of Psychology

Madera Hall 2413

California State University Channel Islands

One University Drive

Camarillo, CA 93012

 

805.437.8997 (p)

805.437.8951 (f)

 

[hidden email]

---

From: SPSSX(r) Discussion [[hidden email]] on behalf of Evan Harrington [[hidden email]]
Sent: Thursday, January 24, 2013 2:07 PM
To: [hidden email]
Subject: Calculation of Z Scores with Sample Standard Deviation

I just realized that the SPSS function for calculation of Z scores (using the “Save Standardized Values as Variables” function) utilizes the sample standard deviation instead of population standard deviation.

 

Am I mistaken in thinking that the Z formula ought to have population standard deviation in the denominator, rather than the sample version as used by this SPSS function?

 

Evan Harrington, Ph.D.

IRB Committee Chair, Chicago Campus

Associate Professor, Department of Forensic Psychology

The Chicago School of Professional Psychology

325 N. Wells Street

Chicago, IL 60654

 

[phone]

 

Practicable: able to be done or put into practice successfully.

 

[Bruce Weaver]

The FM calls them z-scores.  E.g.,

DESCRIPTIVES VARIABLES=NTCSAL NTCPUR (PURCHZ) NTCPRI (PRICEZ).

* DESCRIPTIVES creates z-score variables named PURCHZ and PRICEZ for NTCPUR and NTCPRI, respectively. No z-score variable is created for NTCSAL.

SAVE Subcommand

SAVE creates a z-score variable for each variable specified on the VARIABLES subcommand. The new variables are added to the active dataset.

AFAIK, the FM doesn't say that the sample SD is used when computing these "z"-scores.  

Matt Freeman wrote

Are you assuming that the standardized scores are z-scores rather than t-
scores?

=====================
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[Poes, Matthew Joseph]

It uses the sample because it has no way of calculating the population SD.  You have to manually calculate a true z-score based on population data.  The example I personally run into with this is making a standardized score for state student achievement data.  The typical method is to grab the state student achievement technical manual, use the state SD, and then create a compute command to calculate the true population Z score. 

 

Matthew J Poes

Research Data Specialist

Center for Prevention Research and Development

University of Illinois

510 Devonshire Dr.

Champaign, IL 61820

Phone: [phone]

email: [hidden email]

 

 

From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of Evan Harrington
Sent: Thursday, January 24, 2013 4:07 PM
To: [hidden email]
Subject: Calculation of Z Scores with Sample Standard Deviation

 

I just realized that the SPSS function for calculation of Z scores (using the “Save Standardized Values as Variables” function) utilizes the sample standard deviation instead of population standard deviation.

 

Am I mistaken in thinking that the Z formula ought to have population standard deviation in the denominator, rather than the sample version as used by this SPSS function?

 

Evan Harrington, Ph.D.

IRB Committee Chair, Chicago Campus

Associate Professor, Department of Forensic Psychology

The Chicago School of Professional Psychology

325 N. Wells Street

Chicago, IL 60654

 

[phone]

 

Practicable: able to be done or put into practice successfully.

 

[Evan Harrington, Ph.D.]

Yes, I understand that. But I am operating under the assumption that the numbers I have in front of me constitute the full population, I am not conducting inferential statistics.

 

What I’m asking is whether or not there is an easier way to do this, other than computing the population value (using SQRT SS/n), and then creating my own new variable based on transformation of raw scores by (x-mean)/pop SD

 

Evan Harrington, Ph.D.

IRB Committee Chair, Chicago Campus

Associate Professor, Department of Forensic Psychology

The Chicago School of Professional Psychology

325 N. Wells Street

Chicago, IL 60654

 

[phone]

 

Practicable: able to be done or put into practice successfully.

 

From: Baker, Harley [mailto:[hidden email]]
Sent: Thursday, January 24, 2013 4:42 PM
To: Evan Harrington; [hidden email]
Subject: RE: Calculation of Z Scores with Sample Standard Deviation

 

Yes, it should and it does. The use of  "N" - 1 in the denominator (it is actually W, the sum of weights) of the SD rather than N provides the unbiased estimate of the population standard deviation. That is why we use N-1 in the denominator for inferential statistics. The use of N in the denominator calculates the standard deviation of the scores as if it was a population rather than being a sample from a population about which one wants to draw inferences. SPSS uses the unbiased estimate of the population standard deviation (N - 1) as it should. BTW, the same formulas are used by virtually all of the major statistical packages (e.g., SAS, Statistica, SYSTAT.)

 

Dr. Harley Baker

Professor of Psychology

Madera Hall 2413

California State University Channel Islands

One University Drive

Camarillo, CA 93012

 

805.437.8997 (p)

805.437.8951 (f)

 

[hidden email]

---

From: SPSSX(r) Discussion [[hidden email]] on behalf of Evan Harrington [[hidden email]]
Sent: Thursday, January 24, 2013 2:07 PM
To: [hidden email]
Subject: Calculation of Z Scores with Sample Standard Deviation

I just realized that the SPSS function for calculation of Z scores (using the “Save Standardized Values as Variables” function) utilizes the sample standard deviation instead of population standard deviation.

 

Am I mistaken in thinking that the Z formula ought to have population standard deviation in the denominator, rather than the sample version as used by this SPSS function?

 

Evan Harrington, Ph.D.

IRB Committee Chair, Chicago Campus

Associate Professor, Department of Forensic Psychology

The Chicago School of Professional Psychology

325 N. Wells Street

Chicago, IL 60654

 

[phone]

 

Practicable: able to be done or put into practice successfully.

 

[Jon K Peck]

The complete details of the Z scores computation is given in the Algorithms section.


Jon Peck (no "h") aka Kim
Senior Software Engineer, IBM
[hidden email]
new phone: [phone]




From:        Bruce Weaver <[hidden email]>
To:        [hidden email],
Date:        01/24/2013 03:48 PM
Subject:        Re: [SPSSX-L] Calculation of Z Scores with Sample Standard              Deviation
Sent by:        "SPSSX(r) Discussion" <[hidden email]>

---

The FM calls them z-scores.  E.g.,

DESCRIPTIVES VARIABLES=NTCSAL NTCPUR (PURCHZ) NTCPRI (PRICEZ).

* DESCRIPTIVES creates z-score variables named PURCHZ and PRICEZ for NTCPUR
and NTCPRI, respectively. No z-score variable is created for NTCSAL.

*SAVE Subcommand*

SAVE creates a z-score variable for each variable specified on the VARIABLES
subcommand. The new variables are added to the active dataset.

AFAIK, the FM doesn't say that the sample SD is used when computing these
"z"-scores.



Matt Freeman wrote
> Are you assuming that the standardized scores are z-scores rather than t-
> scores?
>
> =====================
> To manage your subscription to SPSSX-L, send a message to

> LISTSERV@.UGA

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-----
--
Bruce Weaver
[hidden email]
http://sites.google.com/a/lakeheadu.ca/bweaver/

"When all else fails, RTFM."

NOTE: My Hotmail account is not monitored regularly.
To send me an e-mail, please use the address shown above.

--
View this message in context: http://spssx-discussion.1045642.n5.nabble.com/Calculation-of-Z-Scores-with-Sample-Standard-Deviation-tp5717675p5717678.html
Sent from the SPSSX Discussion mailing list archive at Nabble.com.

=====================
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[Ryan]

SPSS calculates z-scores based on the sample standard deviation, as is appropriate under most circumstances. If you view your data as representing the entire population, then convert the sample z-scores to population z-scores. 

Ryan 

Sent from my iPhone

On Jan 24, 2013, at 5:07 PM, Evan Harrington <[hidden email]> wrote:

I just realized that the SPSS function for calculation of Z scores (using the “Save Standardized Values as Variables” function) utilizes the sample standard deviation instead of population standard deviation.

 

Am I mistaken in thinking that the Z formula ought to have population standard deviation in the denominator, rather than the sample version as used by this SPSS function?

 

Evan Harrington, Ph.D.

IRB Committee Chair, Chicago Campus

Associate Professor, Department of Forensic Psychology

The Chicago School of Professional Psychology

325 N. Wells Street

Chicago, IL 60654

 

[phone]

 

Practicable: able to be done or put into practice successfully.

 

[Bruce Weaver]

I may be wrong, but I suspect Evan is not treating his data as representing an entire population.  I think he is talking about situations where the population SD is known -- e.g., sigma = 15 for most IQ tests.

HTH.

Ryan Black wrote

SPSS calculates z-scores based on the sample standard deviation, as is appropriate under most circumstances. If you view your data as representing the entire population, then convert the sample z-scores to population z-scores.

Ryan

Sent from my iPhone

On Jan 24, 2013, at 5:07 PM, Evan Harrington <[hidden email]> wrote:

> I just realized that the SPSS function for calculation of Z scores (using the “Save Standardized Values as Variables” function) utilizes the sample standard deviation instead of population standard deviation.
>  
> Am I mistaken in thinking that the Z formula ought to have population standard deviation in the denominator, rather than the sample version as used by this SPSS function?
>  
> Evan Harrington, Ph.D.
> IRB Committee Chair, Chicago Campus
> Associate Professor, Department of Forensic Psychology
> The Chicago School of Professional Psychology
> 325 N. Wells Street
> Chicago, IL 60654
>  
> [phone]
>  
> Practicable: able to be done or put into practice successfully.
>

[Evan Harrington, Ph.D.]

I just want to transform a batch of raw scores into Z scores using the formula

Z = (x-mu)/sigma

Where mu = population mean and sigma = population standard deviation.

I'm doing this to create an illustration to show my psychometrics class how such a procedure can be done using SPSS, to accompany an example in a psychometrics textbook where the author of that book uses population values.

I also will be covering the importance of the unbiased statistic using n-1

I created a walkthrough showing them how to take the nasty sample SD and turn it into SS so we can backtrack and get sigma, then use the Create Variable option to create the Zscores we actually want, rather than the one (the unbiased one) SPSS gives.

Would be nice if the kind folks at SPSS could add a box to click for the option of biased versus unbiased standardized scores. Sometimes people actually want the biased scores, perhaps only for rhetorical purposes  :p

Best regards


Evan Harrington, Ph.D.
IRB Committee Chair, Chicago Campus
Associate Professor, Department of Forensic Psychology
The Chicago School of Professional Psychology
325 N. Wells Street
Chicago, IL 60654

[phone]

Practicable: able to be done or put into practice successfully.


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of Bruce Weaver
Sent: Thursday, January 24, 2013 5:41 PM
To: [hidden email]
Subject: Re: Calculation of Z Scores with Sample Standard Deviation

I may be wrong, but I suspect Evan is not treating his data as representing an entire population.  I think he is talking about situations where the population SD is known -- e.g., sigma = 15 for most IQ tests.

HTH.



Ryan Black wrote

> SPSS calculates z-scores based on the sample standard deviation, as is
> appropriate under most circumstances. If you view your data as
> representing the entire population, then convert the sample z-scores
> to population z-scores.
>
> Ryan
>
> Sent from my iPhone
>
> On Jan 24, 2013, at 5:07 PM, Evan Harrington &lt;

> EHarrington@

> &gt; wrote:
>
>> I just realized that the SPSS function for calculation of Z scores
>> (using the “Save Standardized Values as Variables” function) utilizes
>> the sample standard deviation instead of population standard deviation.
>>
>> Am I mistaken in thinking that the Z formula ought to have population
>> standard deviation in the denominator, rather than the sample version
>> as used by this SPSS function?
>>
>> Evan Harrington, Ph.D.
>> IRB Committee Chair, Chicago Campus
>> Associate Professor, Department of Forensic Psychology The Chicago
>> School of Professional Psychology
>> 325 N. Wells Street
>> Chicago, IL 60654
>>
>> [phone]
>>
>> Practicable: able to be done or put into practice successfully.
>>

-----
--
Bruce Weaver
[hidden email]
http://sites.google.com/a/lakeheadu.ca/bweaver/

"When all else fails, RTFM."

NOTE: My Hotmail account is not monitored regularly.
To send me an e-mail, please use the address shown above.

--
View this message in context: http://spssx-discussion.1045642.n5.nabble.com/Calculation-of-Z-Scores-with-Sample-Standard-Deviation-tp5717675p5717682.html
Sent from the SPSSX Discussion mailing list archive at Nabble.com.

=====================
To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD

=====================
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[Ryan]

Well, how would SPSS know the population sd? (Question obviously not directed at Bruce).

One can use COMPUTE to calculate a z-score if one knows mu and sigma.

Ryan

Sent from my iPhone

On Jan 24, 2013, at 6:40 PM, Bruce Weaver <[hidden email]> wrote:

> I may be wrong, but I suspect Evan is not treating his data as representing
> an entire population.  I think he is talking about situations where the
> population SD is known -- e.g., sigma = 15 for most IQ tests.
>
> HTH.
>
>
>
> Ryan Black wrote
>> SPSS calculates z-scores based on the sample standard deviation, as is
>> appropriate under most circumstances. If you view your data as
>> representing the entire population, then convert the sample z-scores to
>> population z-scores.
>>
>> Ryan
>>
>> Sent from my iPhone
>>
>> On Jan 24, 2013, at 5:07 PM, Evan Harrington &lt;
>
>> EHarrington@
>
>> &gt; wrote:
>>
>>> I just realized that the SPSS function for calculation of Z scores (using
>>> the “Save Standardized Values as Variables” function) utilizes the sample
>>> standard deviation instead of population standard deviation.
>>>
>>> Am I mistaken in thinking that the Z formula ought to have population
>>> standard deviation in the denominator, rather than the sample version as
>>> used by this SPSS function?
>>>
>>> Evan Harrington, Ph.D.
>>> IRB Committee Chair, Chicago Campus
>>> Associate Professor, Department of Forensic Psychology
>>> The Chicago School of Professional Psychology
>>> 325 N. Wells Street
>>> Chicago, IL 60654
>>>
>>> [phone]
>>>
>>> Practicable: able to be done or put into practice successfully.
>
>
>
>
>
> -----
> --
> Bruce Weaver
> [hidden email]
> http://sites.google.com/a/lakeheadu.ca/bweaver/
>
> "When all else fails, RTFM."
>
> NOTE: My Hotmail account is not monitored regularly.
> To send me an e-mail, please use the address shown above.
>
> --
> View this message in context: http://spssx-discussion.1045642.n5.nabble.com/Calculation-of-Z-Scores-with-Sample-Standard-Deviation-tp5717675p5717682.html
> Sent from the SPSSX Discussion mailing list archive at Nabble.com.
>
> =====================
> To manage your subscription to SPSSX-L, send a message to
> [hidden email] (not to SPSSX-L), with no body text except the
> command. To leave the list, send the command
> SIGNOFF SPSSX-L
> For a list of commands to manage subscriptions, send the command
> INFO REFCARD

=====================
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[Baker, Harley]

Hi Evan,

The easiest way would be to multiply the SPSS-generated z-scores by
sqrt(n/(n-1)). Algebraically, this would give you the z-score you are
looking for. I am assuming, of course, that the mean  for your data is not
an issue. And I think you are right - it would be nice for an option in
SPSS for treating a set of data as a population rather than always
assuming it is a sample.

Harley

Dr. Harley Baker
Professor of  Psychology
California State University Channel Islands
Madera Hall 2413
One University Drive
Camarillo, CA 93012

805.437.8997 (p)
805.437.8951 (f)






On 1/24/13 3:57 PM, "Evan Harrington" <[hidden email]>
wrote:

>I just want to transform a batch of raw scores into Z scores using the
>formula
>
>Z = (x-mu)/sigma
>
>Where mu = population mean and sigma = population standard deviation.
>
>I'm doing this to create an illustration to show my psychometrics class
>how such a procedure can be done using SPSS, to accompany an example in a
>psychometrics textbook where the author of that book uses population
>values.
>
>I also will be covering the importance of the unbiased statistic using n-1
>
>I created a walkthrough showing them how to take the nasty sample SD and
>turn it into SS so we can backtrack and get sigma, then use the Create
>Variable option to create the Zscores we actually want, rather than the
>one (the unbiased one) SPSS gives.
>
>Would be nice if the kind folks at SPSS could add a box to click for the
>option of biased versus unbiased standardized scores. Sometimes people
>actually want the biased scores, perhaps only for rhetorical purposes  :p
>
>Best regards
>
>
>Evan Harrington, Ph.D.
>IRB Committee Chair, Chicago Campus
>Associate Professor, Department of Forensic Psychology
>The Chicago School of Professional Psychology
>325 N. Wells Street
>Chicago, IL 60654
>
>[phone]
>
>Practicable: able to be done or put into practice successfully.
>
>
>-----Original Message-----
>From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
>Bruce Weaver
>Sent: Thursday, January 24, 2013 5:41 PM
>To: [hidden email]
>Subject: Re: Calculation of Z Scores with Sample Standard Deviation
>
>I may be wrong, but I suspect Evan is not treating his data as
>representing an entire population.  I think he is talking about
>situations where the population SD is known -- e.g., sigma = 15 for most
>IQ tests.
>
>HTH.
>
>
>
>Ryan Black wrote
>> SPSS calculates z-scores based on the sample standard deviation, as is
>> appropriate under most circumstances. If you view your data as
>> representing the entire population, then convert the sample z-scores
>> to population z-scores.
>>
>> Ryan
>>
>> Sent from my iPhone
>>
>> On Jan 24, 2013, at 5:07 PM, Evan Harrington &lt;
>
>> EHarrington@
>
>> &gt; wrote:
>>
>>> I just realized that the SPSS function for calculation of Z scores
>>> (using the ³Save Standardized Values as Variables² function) utilizes
>>> the sample standard deviation instead of population standard deviation.
>>>
>>> Am I mistaken in thinking that the Z formula ought to have population
>>> standard deviation in the denominator, rather than the sample version
>>> as used by this SPSS function?
>>>
>>> Evan Harrington, Ph.D.
>>> IRB Committee Chair, Chicago Campus
>>> Associate Professor, Department of Forensic Psychology The Chicago
>>> School of Professional Psychology
>>> 325 N. Wells Street
>>> Chicago, IL 60654
>>>
>>> [phone]
>>>
>>> Practicable: able to be done or put into practice successfully.
>>>
>
>
>
>
>
>-----
>--
>Bruce Weaver
>[hidden email]
>http://sites.google.com/a/lakeheadu.ca/bweaver/
>
>"When all else fails, RTFM."
>
>NOTE: My Hotmail account is not monitored regularly.
>To send me an e-mail, please use the address shown above.
>
>--
>View this message in context:
>http://spssx-discussion.1045642.n5.nabble.com/Calculation-of-Z-Scores-with
>-Sample-Standard-Deviation-tp5717675p5717682.html
>Sent from the SPSSX Discussion mailing list archive at Nabble.com.
>
>=====================
>To manage your subscription to SPSSX-L, send a message to
>[hidden email] (not to SPSSX-L), with no body text except the
>command. To leave the list, send the command SIGNOFF SPSSX-L For a list
>of commands to manage subscriptions, send the command INFO REFCARD
>
>=====================
>To manage your subscription to SPSSX-L, send a message to
>[hidden email] (not to SPSSX-L), with no body text except the
>command. To leave the list, send the command
>SIGNOFF SPSSX-L
>For a list of commands to manage subscriptions, send the command
>INFO REFCARD

=====================
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[Rich Ulrich]

If you are offering that as a suggestion for future development by SPSS,
let me cast my vote as, "Please DON'T."   Or, SPSS, if you do, make it
a special, harder-to-use option, like, "Available only through syntax";
and then present it with a warning.

It seems to me that the temptation to use it would be a smaller example
of the temptation offered to statistical newbies by "Finite Population
Correction"  for ANOVA, etc.   FPC is almost always inappropriate, but that
conclusion is not obvious until you get a bit of sophisticated education in
the theories of statistical inference.  Meanwhile, FPC seems to offer "better"
results.

--
Rich Ulrich

> Date: Thu, 24 Jan 2013 16:29:23 -0600
> From: [hidden email]
> Subject: Re: Calculation of Z Scores with Sample Standard Deviation
> To: [hidden email]
>
> They are "Standardized scores" yet they use sample standard deviation to determine the value of "Z"
>
> But I want Z scores -- I'm assuming the group of scores that I have is actually a population. Thus, I was dismayed to see that SPSS lacks a choice to use population instead of sample values. Now I need to create my own transformations.
>
>
 ...

[Ryan]

Example. Suppose one collects data (e.g., GPA, engagement in various activities) on all students currently enrolled at a University. If one plans on making inferences about future students  (which often is the case) based on these data, then the students currently enrolled at a University do NOT encompass the entire population; future students become part of the population, IMHO.

 

Do others agree with me here?

 

Ryan

On Fri, Jan 25, 2013 at 1:09 PM, Rich Ulrich <[hidden email]> wrote:

If you are offering that as a suggestion for future development by SPSS,
let me cast my vote as, "Please DON'T."   Or, SPSS, if you do, make it
a special, harder-to-use option, like, "Available only through syntax";
and then present it with a warning.

It seems to me that the temptation to use it would be a smaller example
of the temptation offered to statistical newbies by "Finite Population
Correction"  for ANOVA, etc.   FPC is almost always inappropriate, but that
conclusion is not obvious until you get a bit of sophisticated education in
the theories of statistical inference.  Meanwhile, FPC seems to offer "better"
results.

--
Rich Ulrich

> Date: Thu, 24 Jan 2013 16:29:23 -0600
> From: [hidden email]

> Subject: Re: Calculation of Z Scores with Sample Standard Deviation

> To: [hidden email]

>
> They are "Standardized scores" yet they use sample standard deviation to determine the value of "Z"
>
> But I want Z scores -- I'm assuming the group of scores that I have is actually a population. Thus, I was dismayed to see that SPSS lacks a choice to use population instead of sample values. Now I need to create my own transformations.
>
>

 ...

[Bruce Weaver]

I missed this post earlier.  Given that Evan is treating the dataset in front of him as a population (for didactic reasons), he could do something like this:

data list free / X (f5.0).
begin data
11 12 14 21 22 16 18 16 17
end data.

AGGREGATE
  /OUTFILE=* MODE=ADDVARIABLES
  /BREAK=
  /muX=MEAN(X)
  /sdX=SD(X)
  /NumCases=N.

compute #SS = (NumCases-1)*sdX**2.
compute sigmaX = SQRT(#SS/NumCases).
compute ZX.N = (X - muX) / sigmaX.

descriptives X (ZX.nminus1).
formats zx.n zx.nminus1 (f8.4).
list.

OUTPUT:

    X      muX      sdX NumCases   sigmaX     ZX.N ZX.nminus1
 
   11    16.33     3.71        9     3.50  -1.5255   -1.4383
   12    16.33     3.71        9     3.50  -1.2395   -1.1686
   14    16.33     3.71        9     3.50   -.6674    -.6293
   21    16.33     3.71        9     3.50   1.3348    1.2585
   22    16.33     3.71        9     3.50   1.6209    1.5282
   16    16.33     3.71        9     3.50   -.0953    -.0899
   18    16.33     3.71        9     3.50    .4767     .4495
   16    16.33     3.71        9     3.50   -.0953    -.0899
   17    16.33     3.71        9     3.50    .1907     .1798
 
Number of cases read:  9    Number of cases listed:  9

HTH.

Evan Harrington, Ph.D. wrote

Yes, I understand that. But I am operating under the assumption that the numbers I have in front of me constitute the full population, I am not conducting inferential statistics.

What I'm asking is whether or not there is an easier way to do this, other than computing the population value (using SQRT SS/n), and then creating my own new variable based on transformation of raw scores by (x-mean)/pop SD

Evan Harrington, Ph.D.
IRB Committee Chair, Chicago Campus
Associate Professor, Department of Forensic Psychology
The Chicago School of Professional Psychology
325 N. Wells Street
Chicago, IL 60654

[phone]

Practicable: able to be done or put into practice successfully.

From: Baker, Harley [mailto:[hidden email]]
Sent: Thursday, January 24, 2013 4:42 PM
To: Evan Harrington; [hidden email]
Subject: RE: Calculation of Z Scores with Sample Standard Deviation

Yes, it should and it does. The use of  "N" - 1 in the denominator (it is actually W, the sum of weights) of the SD rather than N provides the unbiased estimate of the population standard deviation. That is why we use N-1 in the denominator for inferential statistics. The use of N in the denominator calculates the standard deviation of the scores as if it was a population rather than being a sample from a population about which one wants to draw inferences. SPSS uses the unbiased estimate of the population standard deviation (N - 1) as it should. BTW, the same formulas are used by virtually all of the major statistical packages (e.g., SAS, Statistica, SYSTAT.)

Dr. Harley Baker
Professor of Psychology
Madera Hall 2413
California State University Channel Islands
One University Drive
Camarillo, CA 93012

805.437.8997 (p)
805.437.8951 (f)

[hidden email]<mailto:[hidden email]>
________________________________
From: SPSSX(r) Discussion [[hidden email]] on behalf of Evan Harrington [[hidden email]]
Sent: Thursday, January 24, 2013 2:07 PM
To: [hidden email]
Subject: Calculation of Z Scores with Sample Standard Deviation
I just realized that the SPSS function for calculation of Z scores (using the "Save Standardized Values as Variables" function) utilizes the sample standard deviation instead of population standard deviation.

Am I mistaken in thinking that the Z formula ought to have population standard deviation in the denominator, rather than the sample version as used by this SPSS function?

Evan Harrington, Ph.D.
IRB Committee Chair, Chicago Campus
Associate Professor, Department of Forensic Psychology
The Chicago School of Professional Psychology
325 N. Wells Street
Chicago, IL 60654

[phone]

Practicable: able to be done or put into practice successfully.

[Bruce Weaver]

I agree that it is very rare to have an entire population in hand.  Usually, one wishes to make an inference about some larger population, Ryan's future students being just one example.  

The only time I have operated as if I've had the entire population is for teaching examples meant to illustrate the central limit theorem.  (I.e., I've created a small "population" of numbers, then drawn all possible samples of size n and computed the mean for each, plotted the sample means etc.)  I gather that Evan is doing something similar to this.

Cheers,
Bruce

Ryan Black wrote

Example. Suppose one collects data (e.g., GPA, engagement in various
activities) on all students currently enrolled at a University. If one
plans on making inferences about future students  (which often is the case)
based on these data, then the students currently enrolled at a University
do NOT encompass the entire population; future students become part of the
population, IMHO.

Do others agree with me here?

Ryan
On Fri, Jan 25, 2013 at 1:09 PM, Rich Ulrich <[hidden email]> wrote:

> If you are offering that as a suggestion for future development by SPSS,
> let me cast my vote as, "Please DON'T."   Or, SPSS, if you do, make it
> a special, harder-to-use option, like, "Available only through syntax";
> and then present it with a warning.
>
> It seems to me that the temptation to use it would be a smaller example
> of the temptation offered to statistical newbies by "Finite Population
> Correction"  for ANOVA, etc.   FPC is almost always inappropriate, but
> that
> conclusion is not obvious until you get a bit of sophisticated education
> in
> the theories of statistical inference.  Meanwhile, FPC seems to offer
> "better"
> results.
>
> --
> Rich Ulrich
>
>
> > Date: Thu, 24 Jan 2013 16:29:23 -0600
> > From: [hidden email]
>
> > Subject: Re: Calculation of Z Scores with Sample Standard Deviation
> > To: [hidden email]
>
> >
> > They are "Standardized scores" yet they use sample standard deviation to
> determine the value of "Z"
> >
> > But I want Z scores -- I'm assuming the group of scores that I have is
> actually a population. Thus, I was dismayed to see that SPSS lacks a choice
> to use population instead of sample values. Now I need to create my own
> transformations.
> >
> >
>  ...
>

[Rich Ulrich]

The topic seems to have moved to a consideration of the Finite
Sampling Correction.  Here are comments that I posted to the web
group sci.stat.math in November 22, 1999.

=== [after some other comments]
Two other things to note:
 - If the sampling of the population is not random, then the simple
formula does not apply.  This is typically the case when projecting
vote totals on election eve, which is the most frequent use of the
FPC.  (And it always a potential to be on guard against.)
 - If you have any intention of drawing implications about *other*
samples or populations anywhere, then the formula does not apply.  So
the FPC is not used in most research.


The derivation is simple.  Consider the counting of votes.  You know
the exact value of the what you have counted;  you simply want to sum
the Counted and the Uncounted (and you must know the total N), and
assume that the Uncounted will have the mean and variability that you
can project from Counted.


The practice is more complicated.  In the TV coverage of major
elections, the late-reporting precincts (precincts are the
vote-collection unit in the U.S.)  are not a random subset of the
total -- they tend to be rural ones, which are using slower methods
than the Voting Machines; so the estimate is (much) improved by
considering how those missing precincts voted in previous elections.  
  [Note: That was true in the US in 1999.  Today, the slow

  precincts are the urban ones.  This is the consequence of

  (at first, accidental; today, intentional)  "voter suppression"

  by Republicans, in the form or long lines -- too few machines.

  Voters who were in line on time can be voting after midnight.]

Thus, it does happen that a candidate may be the "projected winner"
while behind  in the most recent cumulative count.  The networks try
to "win" by  being earliest to  *correctly*  project a winner;  a
wrong projection counts as a serious discredit.
--
Rich Ulrich

> Date: Fri, 25 Jan 2013 14:11:41 -0800

> From: [hidden email]
> Subject: Re: Calculation of Z Scores with Sample Standard Deviation
> To: [hidden email]
>
> I agree that it is very rare to have an entire population in hand. Usually,
> one wishes to make an inference about some larger population, Ryan's future
> students being just one example.
>
> The only time I have operated /as if/ I've had the entire population is for
> teaching examples meant to illustrate the central limit theorem. (I.e.,
> I've created a small "population" of numbers, then drawn all possible
> samples of size n and computed the mean for each, plotted the sample means
> etc.) I gather that Evan is doing something similar to this.
>

[snip, Ryan's note about today's students versus all others, and my earlier note.]
...

[Ryan]

That was my point. I have, on a couple of occasions, consulted on projects where the clients had access to the "entire population," and after a short conversation, it was clear they planned on using these data to make inferences about individuals outside of their database without consciously knowing it. I think it is quite rare to truly have access to an "entire population of interest," especially when one factors in time.

Consequently, it makes perfect sense for SPSS and other software packages to assume one is dealing with a sample the vast majority of the time when calculating the standard deviation. Yes, it would be nice to have a built in option for such a scenario, but it should come with warnings, or as Rich suggested, make it difficult. (Similar things have been done with generalized linear mixed modeling procedures, for example, that most people probably don't realize.)

Ryan

On Jan 25, 2013, at 5:11 PM, Bruce Weaver <[hidden email]> wrote:

> I agree that it is very rare to have an entire population in hand.  Usually,
> one wishes to make an inference about some larger population, Ryan's future
> students being just one example.
>
> The only time I have operated /as if/ I've had the entire population is for
> teaching examples meant to illustrate the central limit theorem.  (I.e.,
> I've created a small "population" of numbers, then drawn all possible
> samples of size n and computed the mean for each, plotted the sample means
> etc.)  I gather that Evan is doing something similar to this.
>
> Cheers,
> Bruce
>
>
>
> Ryan Black wrote
>> Example. Suppose one collects data (e.g., GPA, engagement in various
>> activities) on all students currently enrolled at a University. If one
>> plans on making inferences about future students  (which often is the
>> case)
>> based on these data, then the students currently enrolled at a University
>> do NOT encompass the entire population; future students become part of the
>> population, IMHO.
>>
>> Do others agree with me here?
>>
>> Ryan
>> On Fri, Jan 25, 2013 at 1:09 PM, Rich Ulrich &lt;
>
>> rich-ulrich@
>
>> &gt; wrote:
>>
>>> If you are offering that as a suggestion for future development by SPSS,
>>> let me cast my vote as, "Please DON'T."   Or, SPSS, if you do, make it
>>> a special, harder-to-use option, like, "Available only through syntax";
>>> and then present it with a warning.
>>>
>>> It seems to me that the temptation to use it would be a smaller example
>>> of the temptation offered to statistical newbies by "Finite Population
>>> Correction"  for ANOVA, etc.   FPC is almost always inappropriate, but
>>> that
>>> conclusion is not obvious until you get a bit of sophisticated education
>>> in
>>> the theories of statistical inference.  Meanwhile, FPC seems to offer
>>> "better"
>>> results.
>>>
>>> --
>>> Rich Ulrich
>>>
>>>
>>>> Date: Thu, 24 Jan 2013 16:29:23 -0600
>>>> From:
>
>> EHarrington@
>
>>>
>>>> Subject: Re: Calculation of Z Scores with Sample Standard Deviation
>>>> To:
>
>> SPSSX-L@.UGA
>
>>>
>>>>
>>>> They are "Standardized scores" yet they use sample standard deviation
>>> to
>>> determine the value of "Z"
>>>>
>>>> But I want Z scores -- I'm assuming the group of scores that I have is
>>> actually a population. Thus, I was dismayed to see that SPSS lacks a
>>> choice
>>> to use population instead of sample values. Now I need to create my own
>>> transformations.
>>> ...
>
>
>
>
>
> -----
> --
> Bruce Weaver
> [hidden email]
> http://sites.google.com/a/lakeheadu.ca/bweaver/
>
> "When all else fails, RTFM."
>
> NOTE: My Hotmail account is not monitored regularly.
> To send me an e-mail, please use the address shown above.
>
> --
> View this message in context: http://spssx-discussion.1045642.n5.nabble.com/Calculation-of-Z-Scores-with-Sample-Standard-Deviation-tp5717675p5717698.html
> Sent from the SPSSX Discussion mailing list archive at Nabble.com.
>
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