Chapter 1, Problem 16SE

The article “Hydrogeochemical Characteristics of Groundwater in a Mid-Western Coastal Aquifer System” (S. Jeen. J. Kim, et al., Geosciences Journal, 2001:339–348) presents measurements of various properties of shallow groundwater in a certain aquifer system in Korea. Following are measurements of electrical conductivity (in microsiemens per centimeter) for 23 water samples.

2099    528    2030    1350    1018    384    1499

1265    375    424    789    810    522    513

488    200    215    486    257    557    260

461    500

1. a. Find the mean.
2. b. Find the standard deviation.
3. c. Find the median.
4. d. Construct a dotplot.
5. e. Find the 10% trimmed mean.
6. f. Find the first quartile.
7. g. Find the third quartile.
8. h. Find the interquartile range.
9. i. Construct a boxplot.
10. j. Which of the points, if any, are outliers?
11. k. If a histogram were constructed, would it be skewed to the left, skewed to the right, or approximately symmetric?

To determine

Find the mean.

Answer to Problem 16SE

The mean is 740.0.

Explanation of Solution

Given info:

The data shows that the measurements of the electrical conductivity.

Calculation:

Step by step procedure for finding the mean concentration using Minitab software is follows:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Measurements.
• Choose option statistics, and select Mean.
• Click OK.

Output using Minitab software is,

From the output the mean is 74.0.

To determine

Find the standard deviation.

Answer to Problem 16SE

The standard deviation is 550.

Explanation of Solution

Calculation:

Step by step procedure for finding the mean concentration using Minitab software is follows:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Measurements.
• Choose option statistics, and select Standard deviation.
• Click OK.

Output using Minitab software is,

From the output the standard deviation is 550.

To determine

Compute the median.

Answer to Problem 16SE

The median is 513.

Explanation of Solution

Calculation:

Step by step procedure for finding the median concentration is follows:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Measurements.
• Choose option statistics, and select Median.
• Click OK.

Output using Minitab software is,

From the output the median is 513.

To determine

Construct a dotplot.

Answer to Problem 16SE

The dotplot is,

Explanation of Solution

Calculation:

Step by step procedure to constructing dotplot using Minitab procedure is follows:

• Choose Graph > Dotplot.
• Choose One Y-Simple and then click OK.
• In Graph variables, enter Measurements.
• Click OK.

To determine

Compute the 10% trimmed mean.

Answer to Problem 16SE

The 10% trimmed mean is 657.16.

Explanation of Solution

Calculation:

The sample size n is 23.

Trimmed mean:

The trimmed mean is a measure of center that is designed to be unaffected by outliers.

The 10% of the value is,

$\begin{array}{c}10\%\left(n\right)=0.10\left(23\right)\\ =2.3\\ \approx 2\end{array}$

Therefore, trim the highest 2 and lowest 2 observations from the given data.

The observations after trimmed values are,

n	Measurements
1	257
2	260
3	375
4	384
5	424
6	461
7	486
8	488
9	500
10	513
11	522
12	528
13	557
14	789
15	810
16	1018
17	1265
18	1350
19	1499
Total	12,486
Mean	$\frac{12,486}{19}=657.16$

From the table, the trimmed mean is 657.16.

To determine

Compute the first quartile.

Answer to Problem 16SE

The first quartile of the concentrations is 384.

Explanation of Solution

Calculation:

Step by step procedure for finding the first quartile of the concentration is follows:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Measurements.
• Choose option statistics, and select First quartile.
• Click OK.

Output using Minitab software is,

From the output the first quartile is 384.

To determine

Compute the third quartile.

Answer to Problem 16SE

The third quartile is 1,018.

Explanation of Solution

Calculation:

Step by step procedure for finding the third quartile of the concentration is follows:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Measurements.
• Choose option statistics, and select Third quartile.
• Click OK.

Output using Minitab software is,

From the output the third quartile is 1,018.

To determine

Compute the interquartile range.

Answer to Problem 16SE

The interquartile range is 634.

Explanation of Solution

Calculation:

Step by step procedure for finding the third quartile of the concentration is follows:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Measurements.
• Choose option statistics, and select Interquartile range.
• Click OK.

Output using Minitab software is,

From the output the interquartile range is 634.

To determine

Construct a boxplot for the concentrations.

Answer to Problem 16SE

The boxplot for the given data is,

Explanation of Solution

Calculation:

Step by step procedure to constructing boxplot using Minitab procedure is follows:

• Choose Graph > Boxplot or Stat > EDA > Boxplot.
• Under One Y, choose Simple. Click OK.
• In Graph variables, enter Measurements.
• Click OK.

To determine

Identify the points in any outliers.

Answer to Problem 16SE

The points 2,030 and 2,099 are outliers.

Explanation of Solution

Calculation:

Outlier:

If the sample contain a few points that are much larger or smaller than the rest then the points are called outliers.

From the boxplot, it can be concluded that there are two outliers. They are 2,030 and 2,099.

To determine

Identify it is skewed to the left, skewed to the right or approximately symmetric if histogram were constructed.

Answer to Problem 16SE

The histogram is skewed to the right.

Explanation of Solution

Calculation:

Left and Right skewed:

If the sample is left skewed then the median is closer to the third quartile than to the first quartile and if the sample is right skewed then the median is closer to the first quartile than to the third quartile.

From the boxplot, it is observed that the median is closer to the first quartile than to the third quartile. That is, the data is skewed to the right.

Therefore, the histogram is skewed to the right.