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25 Dec 2025, 08:17
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Records = {24, 38, 53, 23, 56, 34}
In ascending {23, 24, 34, 38, 53, 56}
Now one is erroneous.
Mean for the initial six = 228/6 = 38
Spread = {15, 14, 4, 0, 15, 18}
Now Most decrease will happen when the farthest outlier from mean is removed.
The farthest from the mean is 56, as its difference is 18 from the mean
Most increase happens when the value closest to the mean is removed hence it would be 38 itself as the difference with mean is 0
Hence
Most Decrease = 56
Most Increase = 38
In ascending {23, 24, 34, 38, 53, 56}
Now one is erroneous.
Mean for the initial six = 228/6 = 38
Spread = {15, 14, 4, 0, 15, 18}
Now Most decrease will happen when the farthest outlier from mean is removed.
The farthest from the mean is 56, as its difference is 18 from the mean
Most increase happens when the value closest to the mean is removed hence it would be 38 itself as the difference with mean is 0
Hence
Most Decrease = 56
Most Increase = 38
Bunuel
25 Dec 2025, 08:20
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Standard deviation SD is calculated by taking all distances between set elements and the mean. In the current set, the average \(m=228/6=38.\)
Therefore, original SD is assessing how far every element is from 38, and we can also show that: \([23+15, 24+14, 34+4, 38+0, 53-15, 56-18].\)
So, to change SD, we can do two things:
Therefore, the answer is 56 for max decrease and 38 for max increase.
Therefore, original SD is assessing how far every element is from 38, and we can also show that: \([23+15, 24+14, 34+4, 38+0, 53-15, 56-18].\)
So, to change SD, we can do two things:
- remove the most outlying value (in this case 56), which will trigger the largest decrease; or
- remove the least outlying value (basically 38 equal to the mean), and this will give us the largest spike in SD.
Therefore, the answer is 56 for max decrease and 38 for max increase.
msignatius
25 Dec 2025, 08:28
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First, we arrange the volumes in ascending order: 23, 24, 34, 38, 53, 56. As we're looking at Standard Deviation, we must realize first that SD is always calculated from the mean. Now, the mean of the 6 numbers here - 228 / 6 is 38.
Now, we know that the if we remove one of these volumes, it will impact not just the SD, but also the mean. Let's ensure we're calculating the mean with every change.
Currently, 38 is the mean, and 23, the farthest from it at the lower end, is 15 down, and 56, the farthest from it at the higher end, is 18 up.
Now, to increase or decrease the SDs, let's remember, if there are a couple of numbers bunched up away from the mean, removing one of them will reduce the SD, as you can almost imagine the pull of the farther term is reducing. Similarly, to further spread it out, let's ensure this distance - 15 down and 18 up - is as imbalanced as possible. That can be helped a lot with readjusting the mean.
Now, if you want to reduce the SD the most - we can see that the two values in the 50s are, together, further away from 38 than the two values in the 20s - 18 and 15 more VS 14 and 15 less.
However, that's not enough to see. The mean will adjust too, afterall.
We can see the change when we remove each value, starting with 56.
We'll have 23 + 24 + 34 + 38 + 53 = 172 / 5 = mean of 34.4. We now see that the lower end, 23, is 11.4 down, and the higher end, 53, is 18.6 up.
If we, similarly, remove 53, we'll have a 175 / 5 = mean of 35. At the lower end, that's 12 down; at the higher end, that's 21 up.
Removing 38, will keep the mean at 38, - 15 down, and 18 up. No change.
Removing 34, will take the mean up to 38.8. We now see that the lower end is now 15.8 down, and the higher end, 56, is 17.2 up.
Removing 23 will take the mean up to 41, from which the lower end is now 18 down, and the higher end is now 15 up.
Removing 24, will take the mean up to 40.8, from which the lower end is now 17.8 down, and the higher end is now 15.2 up.
If you look at the deviation when you remove 56, then it's 11.4 + 10.4 + 0.4 + 3.4 + 18.6 or 44.2
If you look at the deviation when you remove 53, then it's 12 + 11 + 1 + 3 + 18 or 45;
Upon removing 38, the deviation is now 15 + 14 + 4 + 15 + 18 or 66 - a substantial increase.
Upon removing 34, the deviation is now, 15.8, 14.8, 0.8, 14.2, 17.2 or 62.8
Removing 23 gives us, 17 + 7 + 3 + 12 + 15 or 54, and we can tell removing 24 will only have a similar effect.
Clearly, the most reduction in SD happens when you remove 56, and the highest addition happens when you remove 38.
Now, we know that the if we remove one of these volumes, it will impact not just the SD, but also the mean. Let's ensure we're calculating the mean with every change.
Currently, 38 is the mean, and 23, the farthest from it at the lower end, is 15 down, and 56, the farthest from it at the higher end, is 18 up.
Now, to increase or decrease the SDs, let's remember, if there are a couple of numbers bunched up away from the mean, removing one of them will reduce the SD, as you can almost imagine the pull of the farther term is reducing. Similarly, to further spread it out, let's ensure this distance - 15 down and 18 up - is as imbalanced as possible. That can be helped a lot with readjusting the mean.
Now, if you want to reduce the SD the most - we can see that the two values in the 50s are, together, further away from 38 than the two values in the 20s - 18 and 15 more VS 14 and 15 less.
However, that's not enough to see. The mean will adjust too, afterall.
We can see the change when we remove each value, starting with 56.
We'll have 23 + 24 + 34 + 38 + 53 = 172 / 5 = mean of 34.4. We now see that the lower end, 23, is 11.4 down, and the higher end, 53, is 18.6 up.
If we, similarly, remove 53, we'll have a 175 / 5 = mean of 35. At the lower end, that's 12 down; at the higher end, that's 21 up.
Removing 38, will keep the mean at 38, - 15 down, and 18 up. No change.
Removing 34, will take the mean up to 38.8. We now see that the lower end is now 15.8 down, and the higher end, 56, is 17.2 up.
Removing 23 will take the mean up to 41, from which the lower end is now 18 down, and the higher end is now 15 up.
Removing 24, will take the mean up to 40.8, from which the lower end is now 17.8 down, and the higher end is now 15.2 up.
If you look at the deviation when you remove 56, then it's 11.4 + 10.4 + 0.4 + 3.4 + 18.6 or 44.2
If you look at the deviation when you remove 53, then it's 12 + 11 + 1 + 3 + 18 or 45;
Upon removing 38, the deviation is now 15 + 14 + 4 + 15 + 18 or 66 - a substantial increase.
Upon removing 34, the deviation is now, 15.8, 14.8, 0.8, 14.2, 17.2 or 62.8
Removing 23 gives us, 17 + 7 + 3 + 12 + 15 or 54, and we can tell removing 24 will only have a similar effect.
Clearly, the most reduction in SD happens when you remove 56, and the highest addition happens when you remove 38.
Bunuel
