Multiple choice

In a set of $2n$ distinct observation, each of the observation below the median of all the observations is increased by $5$ and each of the remaining observations is decreased by $3$. Then the mean of the new set of observations :

  1. Increases by $1$.
  2. Decreases by $1$.
  3. Decreases by $2$.
  4. Increases by $2$.
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A Correct answer
Explanation

In 2n observations, n are below the median and n are above/at the median. Sum change = n*5 + n*(-3) = 2n. New mean change = 2n / 2n = 1.

AI explanation

Let the sum of the original 2n observations be S, so the original mean is S divided by 2n. The sum of the new set is calculated by adding 5 to each of the n lower observations and subtracting 3 from each of the n upper observations, which gives S + 5n - 3n, simplifying to S + 2n. The new mean is this new sum divided by 2n, which is S/2n + 1. The mean increases by 1.

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