Let x₁, x₂, ... x_n be n observations. Let w_i = lx_i + k for i = 1, 2, ...n, where l and k are constants. If the mean of x_i’s is 48 and their standard deviation is 12, the mean of w_i’s is 55 and standard deviation of wi’s is 15, the values of l and k should be

Given x₁, x₂, ... x_n be n observations

And Mean of these n observations,

And their standard deviation, SD_x=12.

Another series of n observations is given such that

w_i = lx_i + k for i = 1, 2, ...n, where l and k are constants

And mean of these n observations,

And their standard deviation, SD_w=15

Applying the given condition for mean we get

w_i = lx_i + k

Substituting the corresponding given values of means, we get

55 = l(48) + k…….(i)

Now we know

If standard deviation of x series is s, then standard deviation of kx series is ks,

So standard deviation of x₁, x₂, ... x_n is SD_x,

And hence the standard deviation of lx₁, lx₂, ... lx_n is lSD_x.

Similarly,

If standard deviation of x series is s, then standard deviation of k+x series is s,

So standard deviation of lx₁, lx₂, ... lx_n is lSD_x,

And hence the standard deviation of lx₁+k, lx₂+k, ... lx_n+k is lSD_x.

So applying the given condition for standard deviation, we get

SD_w=lSD_x

Substituting the given values, we get

15=l(12)

Now substituting the value of l in equation (i), we get

55 = (1.25)(48) + k

55=60+k

⇒ k=55-60=-5

Hence the values of k and l are -5 and 1.25 respectively