Question

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ______.

Options

• 0

• 1

• 2

• 3

Answer 1

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is 2.

Explanation:

Given n = 10, `barx` = 15

σ² = 15

`barx = (sumx_i)/10` = 15

σ² = `(sumx_i^2)/10 - (barx)^2` = 15

⇒ `sum^10x_i` = 150

⇒ `sum_(i = 1)^9x_i + 25` = 150

⇒ `sum_(i = 1)^9x_i` = 125

⇒ `sum_(i = 1)^9x_i + 15` = 140

Actual mean = `140/10` = 14 = `barx_"new"`

`sum_(i = 1)^9 (x_i^2 + 25^2 - 15^2)/10` = 15

⇒ `sum_(i = 1)^9x_i^2 + 625` = 2400 

⇒ `sum_(i = 1)^9x_i^2` = 1775

⇒ `sum_(i = 1)^9x_i^2 + 15^2` = 2000

= `(sumx_i^2)_"original"`

`σ_"original"^2 = ((sumx_i^2)_"original" - (barx_"new")^2)/10`

= `2000/10 - 14^2`

= 200 – 196

= 4

`("S.D")_"original"` = 6 = 2