Chapter 4, Problem 49E

To determine

To Explain: the appropriate graphical displays and summary statistics.

Explanation of Solution

Given:

  

  

Calculation:

Median: First sort the data in ascending order. Following is the sorted data:

293.4, 358.7, 367.9, 381.9, 386.8, 408.3, 414.4, 418.4, 423.6, 426.7, 435.7, 435.7, 447.5,

449.8, 451.1, 454.6, 454.8, 458.5, 460.6, 461.7, 470.1, 482, 488.2, 491.7, 496, 505, 507.8,

508.8, 511.8, 512, 526.6, 526.9, 527.7, 529.4, 532.7, 537.1, 541.6, 544.4, 544.4, 545.5,

550.5, 552.5, 553.7, 555.1, 558.5, 564.4, 576.3, 578.6, 614.2, 615

These are 50 values so median is the average of 25^th (496) and 26^th (505) values. So median will be

  $\begin{array}{c}\text{Median=}\frac{496+505}{2}\\ =500.5\text{gallons}\end{array}$

Mean: sum of these 50 values is 24438.6 so mean will be

  $\begin{array}{c}\text{Mean=}\frac{24438.6}{50}\\ =488.8\text{gallons}\end{array}$

So standard deviation will be

  $\begin{array}{c}\text{SD}\left(s\right)=\sqrt{\frac{\left(x-\overline{x}\right)}{n-1}}\\ =\sqrt{\frac{231147.8}{49}}\\ =\sqrt{4717.30}\\ =68.7\text{gallons}\end{array}$

So standard deviation is 68.7 gallons

IQR:

To find IQR let us first find $Q_1$ and $Q_3$ . Lower quartile is middle value of lower half.

Lower half of data has 25 values. Upper half of data is

293.4, 358.7, 367.9, 381.9, 386.8, 408.3, 414.4, 418.4, 423.6, 426.7, 435.7, 435.7, 447.5,

449.8, 451.1, 454.6, 454.8, 458.5, 460.6, 461.7, 470.1, 482, 488.2, 491.7, 496

The lower quartile is 13^th value so

  $Q_1=447.5$

Upper half of data has 25 values. upper half of data is

505, 507.8, 508.8, 511.8, 512, 526.6, 526.9, 527.7, 529.4, 532.7, 541.6, 544.4,

544.4, 545.5, 550.5, 552.5, 553.7, 555.1, 558.5, 564.4, 576.3, 578.6, 614.2, 615

The lower quartile is 13^th value so

  $Q_3=544.4$

Therefore IQR is

  $\begin{array}{c}\text{IQR}=Q_3-Q_1\\ =544.4-447.5\\ =96.9\text{gallons}\end{array}$

Median and mean of the data is 500.5 gallons and 488.8 gallons respectively. Standard deviation of data is a 68.7 gallons means state by consumption per person varies about 68.7 gallons. Distribution is skewed to left slightly. So median works as a good measure of centre of distribution in comparison of mean.