Question 3:

The largest of 50 measurements is 3.84 kg. If the range  is 0.46 kg ,find the smallest measurement.

Solution:

L = 3.84     R = 0.46

R = L – S

0.46 = 3.84 – S

S = 3.84 – 0.46

S = 3.38 kg

Question 4:

The standard deviation of 20 observations is 5. If each observation is multiplied by 2,find the standard deviation and variance of the resulting observations.

Solution:

Here 5 is the standard deviation of 20 observations. Now we need to find the standard deviation and variance if each observation is multiplied by 2.

Standard deviation (σ) = 2 5

Question 5:

Calculate the standard deviation of the first 13 natural numbers.

Solution:

Formula:

= √(n² -1)/12

= √(13² -1)/12

= √(169 -1)/12

= √168/12

= √14

= 3.7416

= 3.74

Therefore the standard deviation for first 13 numbers is 3.74

Question 6 :

Calculate the standard deviation of the following data

(i) 10,20,15,8,3,4

Solution:

First we have to write the given data in the ascending order.

3,4,8,10,15,20

Since the given numbers are small we can use direct method to find standard deviation.

 x x² 3 9 4 16 8 64 10 100 15 225 20 400 Σ x = 60 Σ x² = 814 σ  = √(Σ x²/n) - (Σ x/n)²        =  √(814/6) - (60/6)²        = √[135.67 - (10)²]        = √(135.67 - 100)        = √35.67        = 5.97

(ii) 38,40,34,31,28,26,34

Solution:

First we have to write the given data in the ascending order.

26,28,31,34,34,38,40

Since the given numbers are little large we can use actual mean method to find standard deviation.

x̄ = Σ x/n

= (26 + 28 + 31 + 34  + 34 + 38 + 40)/7

= 231/7

= 33

 x d = x -x̄d = x - 33 d² 26 26 - 33 = -7 49 28 28 - 33 = -5 25 31 31 - 33 = -2 4 34 34 - 33 = 1 1 34 34 - 33 = 1 1 38 38 - 33 = 5 25 40 40 - 33 = 7 49 Σ d² = 154 σ  = √(Σ d²/n)        =  √154/7  = √22    = 4.690     = 4.69