Eleven shipping crates were measured before loading.

Question

Eleven shipping crates were measured before loading. The table shows each crate’s weight (kg) and volume (m^3). The last row lists the average across the main load. The standard deviations of both weight and volume were also calculated for this main load.

For an actual trip, one or more crates may be excluded from the load, and one substitute crate (BX-Sub) may be added. BX-Sub was measured with a weight of 260 kg and a volume of 4.2 m^3.

Crate 
 Weight (kg) 
 Volume (m^3) 
 
  BX-01 
 160 
 4.4 
 
  BX-02 
 180 
 3.8 
 
  BX-03 
 200 
 5.1 
 
  BX-04 
 220 
 4.0 
 
  BX-05 
 240 
 3.9 
 
  BX-06 
 260 
 4.2 
 
  BX-07 
 280 
 3.6 
 
  BX-08 
 300 
 4.6 
 
  BX-09 
 320 
 4.1 
 
  BX-10 
 340 
 3.5 
 
  BX-11 
 360 
 4.8 
 
   Average across main load  
  260  
  4.18

For each of the following statements, select  True  if the statement is true based on the information provided. Otherwise, select  False .

Bunuel · Accepted Answer

• Removing BX-06 will decrease the standard deviation of weights in the remaining load.

The standard deviation of a set shows how much the values deviate from the mean, that is, how widespread the set is. BX-06 is exactly at the mean weight (260), so it contributes zero deviation. Removing it leaves all nonzero deviations unchanged while reducing the count; here the mean of the remaining weights stays at 260, so those distances don’t shrink. With the same spread divided by fewer items, the average distance grows rather than falls. So, essentially, removing an item whose weight is equal to the mean weight makes the set more widespread, thus increasing the standard deviation.

Answer:  False

• Adding BX-Sub to the main load will reduce the standard deviation of weights.

BX-Sub’s weight is 260, the current mean. Adding a value equal to the mean does not add any new deviation and does not shift the mean; it simply increases the count. The same total spread spread over more items makes the average distance from the mean smaller, so the standard deviation goes down.

Answer:  True

• The standard deviation of volumes will be most decreased if BX-10 is removed.

The mean volume is about 4.18. BX-10, at 3.5, is 0.68 away from the mean. BX-03, however, at 5.1, is 0.92 away from the mean, which is farther than BX-10. Removing the single value with the largest absolute deviation reduces the standard deviation the most. Since BX-10 is not the farthest from the mean, removing it cannot produce the greatest decrease.

Answer:  False

I29-134

crimson_king · Answer

False, True, False

findingmyself · Answer

True, False, False

Removing BX-06 will decrease the standard deviation of weights in the remaining load.: Since BX-06 is equal to mean, Adding/Removing elements closer to mean will reduce SD

Adding BX-Sub to the main load will reduce the standard deviation of weights.BX sub =260, which is equal to mean. Adding an element equal to mean is going to increase SD and not reduce.

The standard deviation of volumes will be most decreased if BX-10 is removed.: Removing a value farther to mean brings in most reduction.

Study: Relation between mean and SD

crimson_king · Answer

Should be the opposite for the first two responses. Removing BX06 will increase std deviation since its equal to the mean. Any value that is near the mean will have the effect of reducing the standard deviation or the spread of the dataset

Dereno · Answer

Before going into the problem. Let’s first analyse a simple dataset to understand certain things in greater detail.

Let the data set be {1,2,3,4,5}

The sum = 15 and average (MEAN)= 15/5 =3.

The difference from mean = (-2,-1,0,1,2).

Summation of squares = 10

S.D = sqrt ( 10/5) = sqrt (2) = 1.414.

REMOVAL  : ( we are removing a number from the data set )

Case 1:  Removing a number which is  EQUAL  to mean

The new set is {1,2,4,5}

New sum = 12 and the new mean = 12/4 =3.

Difference from mean =( -2, -1, 1, 2)

Summation of squares = 10.

S.D = sqrt (10/4) = sqrt (2.5) = 1.5 . But, by reasoning we know sqrt 4 = 2, while the sqrt 9=3, a much greater value.

So, 1.5 is greater than 1.414. Thus, the  SD IS INCREASING.

case 2: removal of values which at the either ends of mean.

Which number causes more Expansion to S.D , a number which is closer to mean OR a number which is farther from the mean? The more a number is away from the mean, the more is helps in expanding S.D.

So, logically removing a number which causes more stress to standard deviation will result in  DECREASING S.D.

ADDITION :

Case 1:  Adding a number  EQUAL  to mean .

Remember the initial data set {1,2,3,4,5} , when we add 3 to the data set. Sum = 15+3=18. Number of elements increased to 6.

Mean = 18/6 = 3 ( the same mean)

Difference in mean = (-2,-1,0,0,1,2)

Summation of squares = 4+1+0+0+1+4 = 10

Till now, there is no difference is the outcomes.

S.D = sqrt ( 10/6). Now, If  the denominator value increases , the outcome decreases.

The actual value = 1.2 which is lesser than the initial value 1.414.

Add a value EQUAL to mean, S.D decreases.  ( which is the exact opposite of removal).

Case2:

Adding numbers to either ends away from mean.

When you add a number which has a greater difference from the mean, the impact on SD is higher ( increasing the standard deviation).

Adding a number closer to mean, has a lesser increase in standard deviation when compared to adding a number which is at a farther distance from the mean.

The difference from mean increases, eventually leading to increase in summation of squares.

Impact of standard deviation wrt adding or removal of numbers.

Left of Mean  
 At the Mean  
 Right of Mean  
 
  REMOVAL  
    Decreasing    
    Increasing    
   Decreasing    
 
  ADDING  
    Increasing   
    Decreasing    
    Increasing

Now, let’s look into the questions:

1)  Removing BX-06 will decrease the standard deviation of weights in the remaining load.

The value of BX-06 is 260, which is also equal to the mean weight 260. The standard deviation  INCREASES . Hence,  FALSE.

2)  Adding BX-Sub to the main load will reduce the standard deviation of weights.

BX-Sub has a weight of 260 kg, which is same as the mean weight of 260 kg.

The difference from mean is zero. The summation of squares will yield the same value as previously before addition. But, while calculating S.D the denominator value increases compared to previous set. So the  S.D DECREASES.

Hence,  TRUE.

3)  The standard deviation of volumes will be most decreased if BX-10 is removed.

The value of volume for BX-10 is 3.5, while the mean is 4.18. Thus, BX-10 is away from mean.

BX-03 has a value of 5.1. The difference from mean (5.1-4.18) > (4.18-3.5). Hence, removing BX-03 has a greater impact in standard deviation compared to BX-10 removal.

Hence,  FALSE .

mayankkamal999 · Answer

1. 
 BX-06 has weight of 260 which is equal to the mean of the main load 
 So, when a mean value is removed from the set the stdev of the sets will increase unless all the values in the set are equal (in that case all values=mean and there will be no change in stdev and stdev will be 0) 
 So when BX06 will be removed the stdev will increase 
 Hence  FALSE

2.  
 BX-Sub also has a weight of 260 i.e. equal to the mean of the main load 
 Adding a mean value to set will reduce the stdev unless all values are equal 
 Hence  TRUE

3. 
 Most change in stdev will happen if the value that you are adding or removing is most far from the mean as compared to other values 
 Value of BX-10 is 340 which has a difference of 80 from the mean while there are BX-11 or BX-01 which have values 360 and 160 respectively and both of them have a difference of 100 with the mean. 
 So BX-10 is not the one to induce the most change in stdev (Bx-11 or Bx01 could do it)  
 Hence  FALSE

Eleven shipping crates were measured before loading.

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Crate	Weight (kg)	Volume (m^3)
BX-01	160	4.4
BX-02	180	3.8
BX-03	200	5.1
BX-04	220	4.0
BX-05	240	3.9
BX-06	260	4.2
BX-07	280	3.6
BX-08	300	4.6
BX-09	320	4.1
BX-10	340	3.5
BX-11	360	4.8
Average across main load	260	4.18

True	False
		Removing BX-06 will decrease the standard deviation of weights in the remaining load.
		Adding BX-Sub to the main load will reduce the standard deviation of weights.
		The standard deviation of volumes will be most decreased if BX-10 is removed.

	Left of Mean	At the Mean	Right of Mean
REMOVAL	Decreasing	Increasing	Decreasing
ADDING	Increasing	Decreasing	Increasing

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Eleven shipping crates were measured before loading.

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