Answered: Please find the range, sample standard…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Please find the range, sample standard deviation and inter-quartile range (IQR) of the following data set.

range = (Please enter an exact answer.) standard deviation (ss) = (Please show your answer to one decimal place.) IQR = (Please enter an exact answer.)

A new number, 254, is added to the data set above. Please find the new range, sample standard deviation and IQR of the new data set.

range = (Please enter an exact answer.) standard deviation = (Please show your answer to one decimal place.) IQR = (Please enter an exact answer.)

Which measure of spread is less affected by the addition of the extreme observation?

Range
standard deviation
IQR

Expert Solution

Step 1

From the data the minimum value is 28, and maximum value is 97.

The range of the data set is,

$\begin{array}{rcl} Range & = & Maximum value - Minimum value \\ = & 97 - 28 \\ = & 69 \end{array}$

Thus, the range is 69.

The mean of the data set is,

$\begin{array}{rcl} \bar{x} & = & \frac{\sum_{i} x}{n} \\ = & \frac{28 + 40 + 53 + 59 + 59 + 62 + 74 + 78 + 85 + 97}{10} \\ = & \frac{635}{10} \\ = & 63.5 \end{array}$

The mean is 63.5.

The standard deviation is,

$\begin{array}{rcl} s s & = & \sqrt{\frac{\sum_{i} {(x_{i} - \bar{x})}^{2}}{n - 1}} \\ = & \sqrt{\frac{{(28 - 63.5)}^{2} + {(40 - 63.5)}^{2} + \dots + {(97 - 63.5)}^{2}}{10 - 1}} \\ = & \sqrt{\frac{3, 870.5}{9}} \\ = & \sqrt{430.0556} \\ = & 20.7 \end{array}$

Thus, the standard deviation is 20.7.

The position of first quartile value for even sample size is,

$\begin{array}{rcl} First quartile & = & {[\frac{25}{100} (n + 1)]}^{t h} Data value \\ = & {[0.25 (10 + 1)]}^{th} Data value \\ = & 2 . 75^{th} Data value \end{array}$

The first quartile is,

$\begin{array}{rcl} First quartile & = & 2^{n d} Data value + 0.75 (3^{r d} Data value - 2^{n d} Data value) \\ = & 40 + 0.75 (53 - 40) \\ = & 40 + 9.75 \\ = & 49.75 \end{array}$

The first quartile $(Q_{1})$ is 49.75.

The position of third quartile value for even sample size is,

$\begin{array}{rcl} Third quartile & = & {[\frac{75}{100} (n + 1)]}^{t h} Data value \\ = & {[0.75 (10 + 1)]}^{th} Data value \\ = & 8 . 25^{th} Data value \end{array}$

The third quartile is,

$\begin{array}{rcl} Third quartile & = & 8^{t h} Data value + 0.25 (9^{t h} Data value - 8^{t h} Data value) \\ = & 78 + 0.25 (85 - 78) \\ = & 78 + 1.75 \\ = & 79.75 \end{array}$

The third quartile $(Q_{3})$ is 79.75.

The interquartile range (IQR) is,

$\begin{array}{rcl} I Q R & = & Q_{3} - Q_{1} \\ = & 49.75 - 79.75 \\ = & 30 \end{array}$

Thus, the interquartile range (IQR) is 30.

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