Question

how do we derive the formula: lower limit = mid value - class width/2​

Answer 1

→ Lower limit of a given class is the least member of the class. E.g.: lower limit of the class 15 - 20 is 15.

→ Upper limit of a given class is the greatest member of the class. E.g.: upper limit of the class 15 - 20 is 20.

→ Class width is the difference between upper limit and lower limit. E.g.:- Class width of 15 - 20 is 20 - 15 = 5.

→ The middle most member in the class is the mid value of the class. It's given by the half of the sum of upper and lower limits. E.g.:- mid value of the class 15 - 20 is (15 + 20)/2 = 17.5.

We know that,

$\displaystyle\longrightarrow\sf{Mid\ Value=\dfrac {Lower\ Limit+Upper\ Limit}{2}}$

Or,

$\displaystyle\longrightarrow\sf{Lower\ Limit=2\times Mid\ Value-Upper\ Limit\quad\quad\dots (1)}$

And,

$\displaystyle\longrightarrow\sf{Class\ Width=Upper\ Limit-Lower\ Limit}$

Or,

$\displaystyle\longrightarrow\sf{Upper\ Limit=Lower\ Limit+Class\ Width\quad\quad\dots (2)}$

Substituting (2) in (1),

$\displaystyle\longrightarrow\sf{Lower\ Limit=2\times Mid\ Value-(Lower\ Limit+Class\ Width)}$

$\displaystyle\longrightarrow\sf{Lower\ Limit=2\times Mid\ Value-Lower\ Limit-Class\ Width}$

$\displaystyle\longrightarrow\sf{2\times Lower\ Limit=2\times Mid\ Value-Class\ Width}$

$\displaystyle\longrightarrow\sf{Lower\ Limit=\dfrac {2\times Mid\ Value-Class\ Width}{2}}$

$\displaystyle\longrightarrow\sf{\underline {\underline {Lower\ Limit=Mid\ Value-\dfrac {Class\ Width}{2}}}}$

Hence Derived!

Answer 2

Step-by-step explanation: