JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Statistics

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and

\frac{205}{7}

respectively. Then the mean deviation about the mean of these 7 observations is :

Answer: (C)

Solution

The seven values are

170, 125, 230, 190, 210, a, b

with median

170

. Ordering the five known values:

125 < 170 < 190 < 210 < 230

; for

170

to remain the median (4th of 7 sorted values), both

a

and

b

must lie below

170

(giving three values below the median —

125, a, b

— and three above —

190, 210, 230

).
Sum of absolute deviations about the median:

∣125 - 170∣ + ∣190 - 170∣ + ∣210 - 170∣ + ∣230 - 170∣ + ∣ a - 170∣ + ∣ b - 170∣ = 165 + (170 - a) + (170 - b)

since

a, b < 170

. Given the mean deviation about the median is

\frac{205}{7}

, the sum of deviations is

205

165 + 340 - (a + b) = 205 ⟹ a + b = 300

Mean of all seven values:

\overset{x}{ˉ} = \frac{170 + 125 + 230 + 190 + 210 + a + b}{7} = \frac{925 + 300}{7} = \frac{1225}{7} = 175

Since

a, b < 170 < 175

, their deviations from the mean are

175 - a

and

175 - b

, summing to

350 - (a + b) = 50

.
Sum of absolute deviations about the mean:

∣125 - 175∣ + ∣190 - 175∣ + ∣210 - 175∣ + ∣230 - 175∣ + ∣170 - 175∣ + 50 = (50 + 15 + 35 + 55 + 5) + 50 = 210

Mean deviation about mean = \frac{210}{7} = 30

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Single correct option