Question

QD: this is related to the way we measure errors in statistics and machine learning.

i) What x value minimizes the sum of the absolute distances from the values 1, 3, and 8?

ii) Repeat but for sum of squared distances rather than absolute distances.

iii) How do your answers in (i) and (ii) relate to the mean and the median of 1, 3, and 8?

iv) What x value minimizes the sum of the absolute distances from the values 1, 3, 7, 8?

v) What x value minimizes the sum of the squared distances from the values 1, 3, 7, 8?

vi) is your observation in (iii) still true?

Answer 1

1.

let the value be a. s=(1=a)^2+(3-a)^2+(8-a)^2. we have to minimise S . so we will differentiate S wrt a and equate it 0. by this we will get a=4.

2.

s=|1-a|+|3-a|+|8-a| this will be minimised if |1-a| and |8-a| will be minimised. which implies 1<a<8 and |3-a| will be minimised if a=3. hence a=3 will minimise s

3.

mean=4,median=3. so sum of squares is minimised wrt mean and absolute values are minimised  wrt median.

4. by same method we wll find that 4.75 will minimise squared distance and 5 will minimise the absolute distance.

5. yes the observation is still true.