Statistics

To calculate the variance of the first 10 natural numbers that are multiples of 3 , we use the formula for variance:
σ2=‌

1

n

‌

n

∑

i=1

(xi)2−(x)2
Here, let xi=3k where k∈[1,10]. So the numbers are 3×1,3×2,...,3×10.
First, we find the sum of squares of these numbers:
σ2=‌

1

n

×9‌

10

∑

k=1

k2−9(k)2
Calculating step-by-step:
The formula for the sum of squares of the first n natural numbers is:

n

∑

k=1

k2=‌

n(n+1)(2n+1)

6

Substituting n=10 :

10

∑

k=1

k2=‌

10×11×21

6

The mean k of the first 10 natural numbers is:
k=‌

1

10

‌

10

∑

k=1

k=‌

10×11

2×10

Plugging these into the variance formula:
σ2=‌

9

10

×‌

10×11×21

6

−9(‌

10×11

20

)2
This simplifies to:
σ2=‌

9

10

×385−9×30.25
Finally, calculating the result:
σ2=‌

9

10

×385−272.25=74.25
Thus, the variance of the first 10 multiples of 3 is 74.25 .