12 Days of Christmas GMAT Competition 2025 - Day 8: A quality

A quality inspector recorded the fill volumes, in milliliters, of six bottles from a production run, with the following results:

{24, 38, 53, 23, 56, 34}

The inspector believes exactly one of the six measurements is erroneous and will discard that measurement. To evaluate the impact of the discard, he will calculate the standard deviation of the original six measurements and the standard deviation of the remaining five measurements.

Select for Most decrease the measurement which, if discarded, would produce the greatest decrease in standard deviation, and select for Most increase the measurement which, if discarded, would produce the greatest increase in standard deviation. Make only two selections, one in each column.

Given list {24, 38, 53, 23, 56, 34}
For SD
Most increase comes when u discard a value closer to mean.
Most decrease comes when u discard a value far away from mean.

Mean = (24+38+53+23+56+34)/6 = 228/6 = 38

Most decrease in sd remove 56 (18 mL away from mean, next best 23, 53 each 15 units away)
Most increase in sd remove 38 (exactly mean)

linnet

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25 Dec 2025, 01:47

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Data set: 23,24,34,38,53,56
Sorted Data: 23,24,34,38,53,56
Lowest extreme= 23
Highest extreme :56
Middle: 34-38
most decrease i SD= btw 23-56
56 s further from center than 23
Most decrease is 56
Most increase in SD= Btw the middle ; 34 ad 38
34 is slightly closer to the center of all values
Most increase is 34

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25 Dec 2025, 02:10

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Bunuel

Let's understand this concept first :-

for increasing standard deviation : - We need to discard number which is nearest to mean, because of this the standard deviation of the whole data will increase.

For decreasing standard deviation : - We need to discard number which is farthest from mean.

so we have number { 23, 24, 34, 38, 53, 56 } ---- > average ( mean ) here is 38

Most decrease : -- >

Number which is farthest from mean is 56 ( 56 - 38 = 18 )

so removing this will decrease S.D most

Most Increase : -- >

Number which is nearest to mean is 38 (as 38 is the mean)

so removing this will increase S.D most

Our answers are 56 & 38

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25 Dec 2025, 02:15

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Let's calculate the average of the given values.
The average is 38.

For maximum increase, we can remove a value with the least distance from the mean. We have 38, which is the same as the mean, and the distance is 0, so this will give the maximum increase.

For maximum decrease, we remove a value with the greatest distance from the mean. 56 is 18 units away from 38, which is the maximum(23 is just 15 units away); this will give the maximum decrease.

So 56 for maximum decrease and 38 for maximum increase.

Bunuel

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25 Dec 2025, 02:35

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Bunuel

arrnaging in noncreasing order
23 24 34 38 53 56
mean is 38

greatest decrease is when we remove farthest element from mean = 56

so greatest increase would be if we remove close to mean = 38

56,38

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25 Dec 2025, 03:18

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Bunuel

The six bottle reading volumes in milli litres are {23, 24, 34, 38, 53, 56}.

The sum = 228

Mean = 228/6 = 38

Standard Deviation (S.D) is the measure of how spread the data is, with respect to mean.

Case 1:

Now we need remove a data, which makes the SD increase , SD decrease also.

So, if we remove a value, which is closest or equal to mean. The spread increases.

Then, removing 38, which is the mean, will result in the mean =38.

But, the values around the mean is diminished, there by pushing the spread (S.D).

S.D INCREASE most happens at value 38.

Case 2:

When we remove the data which is at the farthest distance from the mean (38) = 56.

This pushes the values towards the mean. New mean = 34.4.

The values are pulled towards the mean, with the left side having more spread than the right side. The skewness is more towards the left hand side.

Thus, the S.D DECREASES most at 56.

Most S.D increase = 38

Most S.D decrease = 56

For general understanding :

Imagine a normal bell shaped curve, or a tent in bell shape, which has a long pole at the centre (mean).

Values closer or equal to mean:

When you add more data at the mean position or closer to the mean, the clustering of data happens at that point. This increases the height of the curve, thereby pulling the outlier data towards the mean. So, the spread decreases.

When you remove values which are equal to mean or closer to mean, assume u remove all structures which is holding the curve as bell shaped near the mean. This will result in data falling, and the spread moving across in both directions, thus spread increases.

Values away from mean:

When you add more data away from mean, your stretching the curve beyond its actual shape, so the spread increases.

When you remove data away from mean, your releasing the tension held by the values, so that it shrinks to it original shape. So, spread decreases.

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25 Dec 2025, 03:40

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We have the set {24,38,53,23,56,34} we need to find which number if removed will result in most decrease and increase in standard deviation. Ideally if we arrange the numbers in increasing order of set , it will become {23 ,24,34,38,53,56} . Now seeing this set if we remove 56 then the numbers will become more compact and standard deviation will be less since the mean of this set is 38 and distance between 38 and 56 is highest. And as we move close to mean , standard deviation will start decreasing . If we remove 38 , the mean still remains same and standard deviation will most increase because variance will be larger.

So answer is Remove 56 to get most decrease in SD. Remove 38 to get most increase in SD.

Bunuel

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25 Dec 2025, 04:25

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Mean of the series = 24+38+53+23+56+34/6 = 228/6 = 38

Most decrease is when we remove the value farthest from the mean i.e. 56
Most increase is when we remove the value closer to the mean i.e. 38

Answer=> Most decrease => 56, most increase => 38

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25 Dec 2025, 04:43

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{24, 38, 53, 23, 56, 34}
{23, 24, 34, 38, 53, 56}

Standard deviation is a quantity expressing by how much the members of a group differ from the mean value for the group.

mean = (23+24+34+38+53+56)/6 = 38

Most decrease
Standard deviation is most reduced by removing the value farthest from the mean, since that value contributes most to the spread.
56-38=18 > 38-23=15
So, remove 56.

Most increase
Removing a value close to the mean makes the remaining data more spread out.
The mean, 38, is one of the values.
So, remove 38.

Most decrease=56
Most increase=38

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25 Dec 2025, 04:55

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A quality inspector recorded the fill volumes, in milliliters, of six bottles from a production run, with the following results:

{24, 38, 53, 23, 56, 34}

The inspector believes exactly one of the six measurements is erroneous and will discard that measurement. To evaluate the impact of the discard, he will calculate the standard deviation of the original six measurements and the standard deviation of the remaining five measurements.

Results: (23,24,34,38,53,56)
Mean= (23+24+34+38+53+56)/6= 228/6= 38

We know that removing a value far from mean, descreases SD the most.
And removing a value close to mean, increases SD the most.

Distance of all points from mean,
38-23= 15
38-24= 14
38-34= 4
38-38= 0 (nearest to mean)
38-53= 15
38-56= 18 (farthest from mean)

Most decrease in SD= 56
Most increase in SD= 38

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25 Dec 2025, 05:02

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arithmetic mean of the values is 38

Standard deviation measures how much the individual values in a dataset spread out from the average. A low standard deviation means most numbers are close to the mean and a high one means they are more scattered.

Removing the value that is most distant from the average shrinks the spread the most.
The largest deviation is 56 (56-38=18), so removing it produces the greatest decrease in standard deviation.

If you remove a value close to the mean, the leftover data becomes more dispersed.
Discarding 38 removes the central value and leaves the remaining data more dispersed, producing the greatest increase in standard deviation.

Most decrease=56 and Most increase=38

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25 Dec 2025, 05:22

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average of the six measurements = 38

Standard deviation quantifies how widely individual values deviate from the average.

The value furthest from the mean is the biggest contributor to spread, so discarding it decreases standard deviation the most.
The furthest value can be 56 (56-38=18) or 23 (38-23=15).
The answer is 56

Deleting a number that is near the mean causes the rest of the data to be more spread apart.
We can remove the mean, 38.

Most decrease=56
Most increase=38

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25 Dec 2025, 05:36

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Mean = (24+38+53+23+56+34)/6 = 38

Standard deviation tells you the typical distance between each data point and the mean of a dataset.

Standard deviation drops the most when you take out the number farthest from the mean.
The number farthest from 38 is 56.

Taking out a value near the average often makes the remaining set less clustered, increasing variability.
38 is not only near the average, it's the average.

Most decrease=56 and Most increase=38

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25 Dec 2025, 05:55

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Mean of given 6 terms = 38

Removing an extreme value have greatest decrease on SD. Here it is 56.

And removing 38 (the mean) is going to have the greatest increase on SD compared to rest of the numbers.

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25 Dec 2025, 06:34

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given six readings {23,24,34,38,53,56} of which any one value is to be discarded.
mean of the above values is 38.
since SD=square root of[ {summation of the squares of differences of each values from the mean}/total no of values]
hence discarding the value which is farthest from the mean will decrease the SD most and vice versa.
hence most decrease=56 which is 18 units (maximum) away from 38
and most increase=38 which is same as the mean.

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25 Dec 2025, 06:42

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To solve this without heavy calculations, we can use Standard Deviation (SD).
Standard deviation will measure how far the data points are from the mean.

Sum = 23 + 24 + 34 + 38 + 53 + 56 = 228
Mean = 228/6 = 38

Most decrease in standard deviation will reduce SD the most, remove a value farthest from the mean (an extreme).
Distances from mean: 23 → 56 → farthest
Removing 56 shrinks the spread the most. Most increase in standard deviation To increase SD, remove a value near the center, leaving extremes farther apart.

Closest to mean: 38
Removing 38 increases spread the most.

Final Answer Most decrease: 56
Most increase: 38

Bunuel

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25 Dec 2025, 06:58

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Bunuel