To explain why the other answer choices are wrong...
As Brent said, standard deviation (SD) is the variance (or more simply put, the spread) of a set of numbers. Adding the same value to each will effectively shift all of the values up or down the number line the same amount, but will not affect the spread.
1) |a|,|b|,|c|,|d|
If the set were all positive numbers, such as [1, 2, 3, 4], or all negative numbers, [-1, -2, -3, -4], then taking the absolute value would not affect the SD. If just one of those values was negative, though, as in Faraz's example: [-3, 1, 2, 4], then the SD of the absolute values would be different; [1, 2, 3, 4] - the spread is smaller.
2) a+1, b+1, c+1, d+1
Same SD, as discussed above.
3) 5a, 5b, 5c, 5d
Here, by multiplying each value by 5, we increase the range by a factor of 5 as well: [1, 2, 3, 4] --> [5, 10, 15, 20]. We can see that the set is more spread out, so the SD will be greater.
4) a^4,b^4,c^4,d^4
Taking terms to the 4th power will certainly affect the SD (unless every term is the same). If the terms are positive integers, they become more spread out: [1, 2, 3, 4] --> [1, 16, 81, 256]. If they're fractions, they become less spread out: [1, 1/2, 1/3, 1/4] --> [1, 1/16, 1/81, 1/256]. Any negative terms would become positive, etc.
5) 2a+1, 2b+1, 2c+1, 2d+1
Same issue as in 3). By multiplying each value by 2, the range is doubled. The +1 has no effect on SD, though.
For more on SD, see:
https://www.beatthegmat.com/call-for-hel ... tml#545420
