Chapter 14.3, Problem 7CFU

To determine

To calculate: The arithmetic mean and the standard deviation of frequency distribution.

Answer to Problem 7CFU

The arithmetic mean is $30250$ and standard deviation is approximately $\mathrm{13226.39.}$

Explanation of Solution

Given information:

  

Formula used:

Standard Deviation of the Data in a Frequency Distribution − If $X_1,X_2,\mathrm{....},X_k$ are the class marks in a frequency distribution with k classes , and $f_1,f_2,\mathrm{.....},f_k$ are the corresponding frequencies, then the standard deviation $\sigma$ of the data in the frequency distribution is found as follows:

  $\sigma =\sqrt{\frac{{\displaystyle \sum _{j=1}^k{\left(X_j-\overline{X}\right)}^2.f_i}}{{\displaystyle \sum _{i=1}^k{f}_i}}}$

Formula of Arithmetic Mean

  $\overline{X}=\frac{{\displaystyle \sum f.X}}{f}$

Class Marks $=\frac{\text{Upper limit}+\text{Lower Limit}}{2}$

Calculation:

According to the question,

Class Limits	Class Marks $\left(X\right)$	$f$	$f.X$	$\left(X_i-\overline{X}\right)$	${\left(X_i-\overline{X}\right)}^2.f$
$0-10,000$	$5,000$	15	$75000$	$\begin{array}{l}=\left(5000-30250\right)\\ =-25,250\end{array}$	$9563437500$
$10,000-20,000$	$15,000$	30	$450000$	$\begin{array}{l}=\left(15,000-30250\right)\\ =-15,250\end{array}$	$6976875000$
$20,000-30,000$	$25,000$	50	$1250000$	$\begin{array}{l}=\left(25,000-30250\right)\\ =-5,250\end{array}$	$1378125000$
$30,000-40,000$	$35,000$	60	$2100000$	$\begin{array}{l}=\left(35,000-30250\right)\\ =4,750\end{array}$	$1353750000$
$40,000-50,000$	$45,000$	30	$1350000$	$\begin{array}{l}=\left(45,000-30250\right)\\ =14,750\end{array}$	$6526875000$
$50,000-60,000$	$55,000$	15	$825000$	$\begin{array}{l}=\left(55,000-30250\right)\\ =24,750\end{array}$	$9188437500$
		$200$	$6050000$		$34987500000$

Therefore, we have

  ${\displaystyle \sum f=200}$

  ${\displaystyle \sum f.X=}6050000$

The arithmetic mean −

  $\begin{array}{l}=\overline{X}=\frac{6050000}{200}\\ =\overline{X}=30250\end{array}$

Also,

${\displaystyle \sum {\left(X_i-\overline{X}\right)}^2.f}=34987500000$

The standard deviation −

$\begin{array}{l}\sigma =\sqrt{\frac{34987500000}{200}}\\ \sigma =\sqrt{174937500}\\ \sigma =13226.39\end{array}$

Hence , the arithmetic mean is $30250$ and standard deviation is approximately $\mathrm{13226.39.}$