Question

There are 10 employees in a particular division of a company. Their salaries have a mean of $70,000, a median of $55,000, and a standard deviation of $20,000. The largest number on the list is $100,000. By accident, this number is changed to $1,000,000.
a. What is the value of the mean after the change?
b. What is the value of the median after the change?
c. What is the value of the standard deviation after the change?

Answer 1

Solution:

a. original $\bar{x} =70,000 =\frac{\sum x}{n}$

$\therefore \sum x = 70,000 \times 10 =700,000$

$modified \sum x = 700,000 - 100,000 + 1,000,000 = 1,600,000$

$modified \bar{x} = \frac{1,600,000}{10} =160,000$

b. Median remains the same.

c. $original s^{2}=(20,000)^{2}=400000000=\frac{1}{n-1}[\sum x^{2}- n\bar{x}^{2}]$

$\therefore orig. \sum x^{2} = (10-1)(20000)^{2} + 10(70000)^{2}$

                                                  $=52600000000$

$mod. \sum x^{2} = 52600000000-100000^{2} +1000000^{2}$

                                  $=1.0426E+12$

$mod.s^{2} = \frac{1}{n-1} [\sum x^{2} - n\bar{x}^{2}]$

                       $=\frac{1}{9} [1.0426E+12 - 10 \times 160000^{2}]$

                       $=87400000000$

$modified s= \sqrt{s^{2}} =\sqrt{87400000000} =295634.91$