Just for completeness, here's the code in R.  I followed Bruce's code.

n = c(40, 60, 50)

mean = c(10, 15, 20)

sd = c( 1, 2, 2)

grand.mean = function(N, M) {weighted.mean(M, N)}

grand.sd = function(N, M, S) {

Mbar = weighted.mean(M, N)

SSw = (N-1)*S^2

SSb = N*(M-Mbar)^2

SSbtwn = sum(SSb)

SSwithin = sum(SSw)

SStot = SSbtwn + SSwithin

SD =sqrt(SStot/(sum(N)-1))

return(SD)

}

sum(n)

grand.mean(n, mean)

grand.sd(n, mean, sd)

Hello Mohammed.  This is a basic ANOVA problem.  From the group data, you can compute SS_within and SS_between, and then sum them to get SS_total.  Here is an example using Stata.  If you don't use Stata, you should be able to translate the code to another package fairly easily. 

* Try to duplicate results from calcualtor at

* http://atozmath.com/CONM/Ch2_CombinedSD.aspx

clear

input n m s

40 10 1

60 15 2

50 20 2

end

* n = sample size

* m = mean

* s = SD

egen N = total(n) // N = column total for n

egen SUM = total(n*m) // SUM = column total for n*m

generate MEAN = SUM/N

egen SSw = total((n-1)*s^2) // SS within rows

egen SSb = total(n*(m-MEAN)^2) // SS between rows

generate SStot = SSb+SSw // Total SS

generate SD = sqrt(SStot/(N-1))

list N MEAN SD

* Note: The website divides by N, not N-1

* when computing the grand SD.

Results:

N = 150

MEAN = 15.33333

SD = 4.260269

This approach can take as many rows of data as you like.

HTH.

Hi Bruce, thanks a whole lot. Should I be worried that the website divides by N, not N-1? Cos I have ran a few numbers on it.

Plus I am not very good with syntax and command, however, I am downloading R now. I self-learned those a longtime ago, before I ditched them for a simpler SPSS.

If you have access to SPSS, try this syntax, replacing my lines of data with your own.

* Try to duplicate results from calcualtor at

* http://atozmath.com/CONM/Ch2_CombinedSD.aspx .

NEW FILE.

DATASET CLOSE ALL.

**************************************************.

* NOTE: Replace the data lines between BEGIN DATA

* and END DATA with your own data.

**************************************************.

DATA LIST list / n (F5.0) m s (2F8.2).

BEGIN DATA

40 10 1

60 15 2

50 20 2

END DATA.

* n = sample size

* m = mean

* s = SD .

COMPUTE Sum = n*m. /* Sum of scores within each group.

COMPUTE SSw = (n-1)*s**2. /* SS within each group.

AGGREGATE

/OUTFILE=* MODE=ADDVARIABLES

/BREAK=

/Ntot=SUM(n)

/SumTot=SUM(Sum).

COMPUTE GrandMean = SumTot/Ntot.

COMPUTE SSb = n*(m-GrandMean)**2.

AGGREGATE

/OUTFILE=* MODE=ADDVARIABLES

/BREAK=

/SSbtwn=SUM(SSb)

/SSwithin=SUM(SSw).

COMPUTE SStot = SSbtwn+SSwithin.

COMPUTE GrandSD = SQRT(SStot/(Ntot-1)).

FORMATS Ntot(F5.0) / GrandMean GrandSD (F10.5).

LIST Ntot GrandMean GrandSD /CASES FROM 1 to 1.

OUTPUT using my data:

Ntot GrandMean GrandSD

150 15.33333 4.26027

HTH.

p.s. - Unless you have population data, which seems very unlikely, you should divide by n-1, not N, when computing the variance & SD.

Just for completeness, here's the code in R.  I followed Bruce's code.

n = c(40, 60, 50)

mean = c(10, 15, 20)

sd = c( 1, 2, 2)

grand.mean = function(N, M) {weighted.mean(M, N)}

grand.sd = function(N, M, S) {

Mbar = weighted.mean(M, N)

SSw = (N-1)*S^2

SSb = N*(M-Mbar)^2

SSbtwn = sum(SSb)

SSwithin = sum(SSw)

SStot = SSbtwn + SSwithin

SD =sqrt(SStot/(sum(N)-1))

return(SD)

}

sum(n)

grand.mean(n, mean)

grand.sd(n, mean, sd)

Dear Bruce & Salvatore,

Thank you very much.

It just occurred to me that any program that performs one-way ANOVA from summary data can be used to do this too, and more easily.  E.g., in Stata, one can use the user-written program -aovsum-.  E.g.,

* Type -findit aovsum- to find and install

* user-written program -aovsum-, which performs

* one-way ANOVA from summary data.

aovsum, n(40 60 50) m(10 15 20) sd(1 2 2)

Here is the first part of the output (copy and view in fixed font to make it clearer):

Groups | Summary of Response variable

(cells) | Mean Std. Dev. Obs.

------------+------------------------------------

1 | 10 1 40

2 | 15 2 60

3 | 20 2 50

------------+------------------------------------

Total | 15.333333 4.2602686 150

The Total row at the bottom shows the grand mean, grand SD and total N.

In SPSS, the ONEWAY command can be used to perform one-way ANOVA on summary data.  See the example at the link given below.  To get the table of descriptive stats showing the grand mean & SD, you'll need to add a STATISTICS sub-command that includes option DESCRIPTIVES.  I don't have SPSS on this machine, so cannot test, but something like this should work:

NEW FILE.

MATRIX DATA

variables=groupvar rowtype_ depvar /factors groupvar.

BEGIN DATA

1 N 40

1 MEAN 10

1 SD 1

2 N 60

2 MEAN 15

2 SD 2

3 N 50

3 MEAN 20

3 SD 2

END DATA.

ONEWAY depvar by groupvar

 /MATRIX=in(*)

  /STATISTICS=DESCRIPTIVES.

HTH.

http://www-01.ibm.com/support/docview.wss?uid=swg21476127

Dear Mohammed,

Despite you have your problem solved by using the Bruce and Salvatore SPSS and R approaches you (or other users) may also find useful to use the attached Excel spreadsheet.

Regards, Luis

 Thank you very much, Dr. Luis

The pooled mean is straightforwardly obtained as MEAN = (N1* MEAN1+N2* MEAN2+...+N13*MEAN13)/(N1+N2+...+N13) and the pooled variance is obtained as SD2 = [(N1-1)*SD12+(N2-1)*SD22+...+(N13-1)*SD132]/(N1+N2+...+N13-13). Therefore the pooled standard deviation is SD=SQRT(SD2). In the above, MEANi, SDi and Ni are respectively the mean, standard deviation and the size of the ith group. You can use any software to calculate these.

@Ette:  What you are calling SD2 is really SSwithin-groups/dfwithin-groups = MSwithin-groups.  I believe what the OP is asking for is SQRT(SSTotal / dfTotal).

As noted above, a convenient way to get SSTotal is via software that can do one-way ANOVA from summary data (e.g., aovsum in Stata, ONEWAY in SPSS). 

HTH.