SOLUTION: A certain set of data has a mean of 20 and a Standard deviation of 4.5. It is found that two pieces of data have been omitted, one of value 30 and the other of value 25. Find the m

The other tutor's answer for the correct standard deviation is wrong.
Also he used the formula for the standard deviation of a sample, and
this student has informed me that this is to be considered as an 
entire population, not as a sample from a population.

He gave:

4.5^2=20.25
20.25*22=445.5 sum of (each data point minus mean)^2

But that is the sum of each data point minus the mean of the 23)^2
not (the sum of each data point minus the mean of the 25)^2
He incorrectly assumed this was the latter.

It's a bit more complicated than what he did.

He used the definition of the standard deviation
of a sample



but you must use the "shortcut" formula for the
standard deviation of a population 



for the standard deviation instead.

Here is the entire correct problem:

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Let  = the unknown sum of the 23 numbers
Let  = the unknown sum of the 25 numbers

Then 

(1)   

Let  be the given mean of the 23 numbers

Let  be the unknown mean of the 25 numbers. 

Since , then
 

Multiplying both side by 23 gives



From equation (1) above



Therefore


 
That's the mean of the 25 numbers, which the other tutor 
calculated correctly.
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Now we use the short-cut formula for the standard deviation
of a population:



We will use the above formula with both sides squared,
for convenience:



Let  be the given standard deviation of the 23 numbers

Let  be the unknown standard deviation of the 25 numbers

Let  be the unknown sum of the squares of the 25 numbers.

Therefore,

(2)  

Using the formula for the 23 numbers:


 
Substituting given values,







Multiply both sides by 23



From equation (2) above,



Using the formula for the 25 numbers,



Substituting known values











Edwin