Statistics
Correcting mean and standard deviation
nta_pyq_2025_apr
Grade 12
Question:
The mean and standard deviation of 100 observations are 40 and 5.1 , respectively, By mistake one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are $\mu$ and $\sigma$ respectively, then 10($\mu$ + $\sigma$) is equal to
Step-by-Step Solution
Key Concept: Translate the given correcting mean and standard deviation into equations using$$\bar{x}$=$\tfrac{\sum x_i}${n}$and the variance formula.
$100(40) - 50 + 40$Actual means = $\mu$ = (4) 100 1 $\mu$ = 40 - = 39.9 10 Incorrect variance 2$\sumx$2 i 2 ¯ ¯¯ (5.1) = - (x) 100 2 2 2$\sumx$= 100 $\times$ (40$) + 100(5.1)$i 2 4 2$\sumx$= 16 $\times$$10 + (5.1)$$\times$$100 = 162601$i 2 2 2$\sumx$-$50 + 40$2 i 2 $\sigma$ = - ($\mu$) 100 2 2 $\sigma$ =$1617.01 - (39.9) = 25$$\sigma$ = 5 10($\mu$ + $\sigma$$) = 10(39.9 + 5) = 10$$\times$$44.9 = 449$
Correct Answer: 4