Statistics

C.O.V = σ/x? * 100

mean = Σx? f? /Σf? = (32+8α+9β)/ (8+α+β)=6
⇒2α+3β=16 . (i)
d? =x? −x? =−4,0,2,3
f? d? ²=64,0,4α,9β
Variance σ²=Σf? d? ²/Σf? =6.8
⇒ (64+4α+9β)/ (8+α+β)=6.8
⇒2.8α+ (−2.2β)=9.6
⇒28α−22β=96
14α−11β=48 . (ii)
Solving (i) and (ii),
⇒β=2, α=5
New mean=Σx? f? /Σf? =85/15=17/3

Var (1, 2, ., n) = (Σn²/n) - (Σn/n)² = 10.
(n (n+1) (2n+1)/6n) - (n (n+1)/2n)² = 10.
(n+1) (2n+1)/6 - (n+1)/2)² = 10.
(n+1)/12 * [2 (2n+1) - 3 (n+1)] = 10.
(n+1)/12 * (4n+2 - 3n-3) = 10.
(n+1) (n-1)/12 = 10.
n² - 1 = 120 ⇒ n² = 121 ⇒ n = 11.

Var (2, 4, ., 2m) = Var (2* (1, 2, ., m) = 2² * Var (1, 2, ., m) = 16.
4 * Var (1, 2, ., m) = 16.
Var (1, 2, ., m) = 4.
Using the formula from above: (m²-1)/12 = 4.
m² - 1 = 48 ⇒ m² = 49 ⇒ m = 7.
m + n = 7 + 11 = 18.

The data consists of n values of a and n values of -a.
Mean x? = (n*a + n* (-a) / 2n = 0 / 2n = 0.
Variance σ² = (Σx? ²)/N - x? ² = (n*a² + n* (-a)²) / 2n - 0² = 2na² / 2n = a².
If a value b is added to all observations, the new mean is x? ' = x? + b = 0 + b = b.
We are given the new mean is 5, so b=5.
Adding a constant does not change the variance. The new variance is still a².
We are given the new standard deviation is 20, so the new variance is 20² = 400.
Thus, a² = 400.
The required value is a² + b² = 400 + 5² = 400 + 25 = 425.

 x? = (2+4+10+12+14+x+y)/7 = 8
⇒ 42+x+y = 56 ⇒ x+y = 14
σ² = (Σx? ²/n) - (x? )²
16 = (4+16+100+144+196+x²+y²)/7 - (8)²
⇒ 16+64 = (460+x²+y²)/7
⇒ 560 = 460+x²+y² ⇒ x²+y² = 100
⇒ xy=48
(x-y)² = (x+y)² - 4xy = 4
|x-y| = 2

x? = Σf? x? / Σf? = (10 + 15x + 50) / (4+x)
= (60+15x)/ (4+x) = 15
σ² = 50 = Σf? x? ²/Σf? - (x? )²
50 = (50+225x+1250)/ (4+x) - (15)²
50 = (1300+225x)/ (4+x) - 225
⇒ 275 (4+x) = 1300 + 225x
⇒ 50x = 200 ⇒ x = 4

 x? = 10
⇒ x? = (63 + a + b)/8 = 10
⇒ a + b = 17
Since, variance is independent of origin.
So, we subtract 10 from each observation.
So, σ² = 13.5 = (79 + (a-10)² + (b-10)²)/8
⇒ a² + b² - 20 (a+b) = -171
⇒ a² + b² = 169
From (1) and (2) ; a = 12 and b = 5

The placement details of 2026 batch are yet to be released on the college page of St. John's National Academy of Health Sciences. However, the institute has released the NIRF report 2025 which includes the placement details for the batch 2024. The key highlights of St. John's National Academy of Health Sciences placements for the Class of 2023 and 2024 are tabulated below:

Let's take a look at the key highlights of St. John's National Academy of Health Sciences placements for the PG Class of 2024 in the table below: