Commerce (English Medium) Class 11 - CBSE Question Bank Solutions

Show that the two formulae for the standard deviation of ungrouped data 

\[\sigma = \sqrt{\frac{1}{n} \sum \left( x_i - X \right)^2_{}}\] and 

\[\sigma' = \sqrt{\frac{1}{n} \sum x_i^2 - X^2_{}}\]  are equivalent, where \[X = \frac{1}{n}\sum_{} x_i\]

Find the standard deviation for the following distribution:

\[\lim_{x \to 0} \frac{5^x - 1}{\sqrt{4 + x} - 2}\]

\[\lim_{x \to 0} \frac{\log \left( 1 + x \right)}{3^x - 1}\]

\[\lim_{x \to 0} \frac{a^x + a^{- x} - 2}{x^2}\]

\[\lim_{x \to 0} \frac{a^{mx} - 1}{b^{nx} - 1}, n \neq 0\]

\[\lim_{x \to 0} \frac{a^x + b^x - 2}{x}\]

\[\lim_{x \to 0} \frac{9^x - 2 . 6^x + 4^x}{x^2}\] 

\[\lim_{x \to 0} \frac{8^x - 4^x - 2^x + 1}{x^2}\]

\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{x}\] 

\[\lim_{x \to 0} \frac{a^x + b^x + c^x - 3}{x}\] 

\[\lim_{x \to 2} \frac{x - 2}{\log_a \left( x - 1 \right)}\]

\[\lim_{x \to 0} \frac{5^x + 3^x + 2^x - 3}{x}\]

\[\lim_{x \to \infty} \left( a^{1/x} - 1 \right)x\]

\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{\sin kx}\]

\[\lim_{x \to 0} \frac{a^x + b^ x - c^x - d^x}{x}\]

\[\lim_{x \to 0} \frac{e^x - 1 + \sin x}{x}\]

\[\lim_{x \to 0} \frac{\sin 2x}{e^x - 1}\] 

\[\lim_{x \to 0} \frac{e\sin x - 1}{x}\] 

Find the standard deviation for the following data: