I28-55

Question

During a science experiment, each measurement result was affected by the same calibration error and had to be adjusted. The corrected result  m  for each result was calculated as  m = -0.5n + 10 , where  n  was the original recorded value. The average (arithmetic mean) of the original measurements was 7.3 and the standard deviation of the original measurements was 2.9. 
 
Select for  Corrected Mean  the average (arithmetic mean) of the corrected results, and select for  Corrected Standard Deviation  the standard deviation of the corrected results. Make only two selections, one in each column.

Bunuel · Answer

Official Solution:

Since each original measurement  n  was transformed into a corrected result  m = -0.5n + 10 , the corrected mean is simply the average of all the  m  values. But since this transformation is applied identically to each individual value, we can apply the same transformation directly to the original mean.
 
That is, the corrected mean is:
 
Corrected mean = -0.5 * (original mean) + 10 
 
Corrected mean = -0.5 * 7.3 + 10 = -3.65 + 10 = 6.35
 
As for the standard deviation, two key properties are useful:
 
• If we multiply (or divide) all the numbers in a set by a non-zero constant, the standard deviation will be respectively multiplied or divided by the  absolute value  of that constant. For example, if the SD of  \{a, \ b, \ c\}  is  s , then the SD of  \{2a, \ 2b, \ 2c\}  will be  2s  and the SD of  \{-3a, \ -3b, \ -3c\}  will be  s*|-3|=3s .
 
• If we add (or subtract) a constant to all the numbers in a set, the standard deviation will NOT change. For example, if the SD of  \{a, \ b, \ c\}  is  s , then the SD of  \{a+4, \ b+4, \ c+4\}  or of  \{a-1, \ b-1, \ c-1\}  will also be  s .
 
Since the transformation is  m = -0.5n + 10 , the scaling factor is -0.5 and the shift is +10. The +10 has no effect on the standard deviation, and the multiplication by -0.5 scales the SD by  |-0.5| = 0.5 .

Correct answer:

Corrected Mean  "6.35"

Corrected Standard Deviation  "1.45"

Apoorv0194 · Answer

I did not quite understand the solution. Trying to understand correct Standard Deviation value

Bunuel · Answer

The formula given is m = -0.5n + 10.

The number -0.5 is a  scaling factor . It multiplies every original measurement by -0.5. 
 The number +10 is a  shift . It adds the same amount to every value.

Now, the rule:

Adding or subtracting a constant (like +10) does  not  change the standard deviation. 
 Multiplying every number by a constant changes the standard deviation by the  absolute value  of that constant.

So, since the original SD is 2.9, and the scaling factor is -0.5, the new SD becomes:
 
Corrected SD = |-0.5| * 2.9 = 0.5 * 2.9 = 1.45.

That’s how the corrected standard deviation is 1.45.

patnaik1988 · Answer

I like the solution - it’s helpful.

I28-55

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My Rewards

Corrected Mean	Corrected Standard Deviation
		17.3
		8.55
		6.35
		1.45
		-1.45
		-3.65