Given:

Relation between X and Y: 3x + 4y = 20

Quartile deviation of X: 12


To find:

Quartile deviation of Y


Solution:

We know that quartile deviation is a measure of dispersion which is half the difference between the third and first quartiles. Mathematically, it can be represented as:


Quartile deviation = (Q3 - Q1) / 2

where Q3 and Q1 are the third and first quartiles respectively.


Now, we need to find the quartile deviation of Y, which can be done by finding the quartiles of Y. To do this, we can use the relation between X and Y given to us.


Finding Quartiles of Y:

Let's rearrange the given relation as:


4y = 20 - 3x

y = (20 - 3x) / 4

Now, we can use this equation to find the values of Y corresponding to the first and third quartiles of X.


First Quartile of X:

To find the first quartile of X, we need to find the value of X for which 25% of the data is less than or equal to X. This can be found using the formula:


Q1 = L + (n/4 - F) * i

where L is the lower limit of the first interval, n is the number of observations, F is the cumulative frequency of the interval preceding the quartile interval, and i is the width of the quartile interval.


Since we don't have the data, we can assume that the observations are evenly distributed and use the formula:


Q1 = X1 + (n/4 - 1) * (X2 - X1)

where X1 and X2 are the values of X corresponding to the two observations closest to the first quartile.


Let's assume that X1 = Q1 and X2 = Q2. Then, we have:


Q1 = Q1 + (n/4 - 1) * (Q2 - Q1)

Solving for Q1, we get:


Q1 = Q2 - (Q2 - Q1) / 4

Similarly, we can find the value of Y corresponding to Q1 of X using the equation:


y1 = (20 - 3Q1) / 4

Third Quartile of X:

To find the third quartile of X, we need to find the value of X for which 75% of the data is less than or equal to X. This can be found using the formula:


Q3 = L + (3n/4 - F) * i

where L is the lower limit of the first interval, n is the number of observations, F is the cumulative frequency of the interval preceding the quartile interval, and i is the width of the quartile interval.


Again, assuming that the observations are evenly distributed, we have:


Q3 = X3 + (3n/4 - 1) * (X4 - X3)

Let's assume that X

9 is the ans.
Since we know SD is affected by change of scale and not by change of origin .
S(y) = | b | × S ( x )
we can convert the given equation.
y = 20/4 - 3x/4
y = |3/4 | × 12
y = 9