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Range and mean absolute deviation

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5387566
Eight ocean buoys reported these wave heights: \(1.8\,\text{m}, 2.4\,\text{m}, 2.1\,\text{m}, 3.0\,\text{m}, 2.7\,\text{m}, 1.9\,\text{m}, 2.6\,\text{m}, 2.3\,\text{m}\). Find the range.

Hints

- Find only the least and greatest values. - The order of the other values does not affect the range. - Include the unit in your answer.

Solution

1. The least value is \(1.8\,\text{m}\). 2. The greatest value is \(3.0\,\text{m}\). 3. The range is \(3.0 - 1.8 = 1.2\,\text{m}\).

Answer

The range is \(1.2\,\text{m}\).
5387746
A data set has a minimum of \(7.5\) and a range of \(12.8\). Find the maximum.

Hints

- Relate the minimum, maximum, and range. - Work backward from the given difference. - Check your result by subtraction.

Solution

1. The range equals the maximum minus the minimum. 2. Therefore, the maximum is \(7.5 + 12.8 = 20.3\).

Answer

The maximum is \(20.3\).
5387656
The ordered data set is \(11, 12, 13, 14, 15, 16, 17, 18, 42\). The value \(42\) is identified as an outlier and is removed for a second analysis. Find the range with and without the outlier.

Hints

- Identify the greatest value in each version of the data set. - The least value does not change. - Compare the two ranges.

Solution

1. With the outlier, the range is \(42 - 11 = 31\). 2. Without the outlier, the greatest value is \(18\). 3. The range without the outlier is \(18 - 11 = 7\).

Answer

The range is \(31\) with the outlier and \(7\) without the outlier.
5316926
A student-run business recorded its monthly profits and losses in the bar graph. Positive values represent profits, and negative values represent losses. a) In which months did the business make a profit? In which months did it have a loss? b) Find the range of the monthly results. c) Find the business's net result for January through June.
Figure for problem 531692

Hints

- Read each bar's value and pay attention to its sign. - The range is the greatest value minus the least value. - Add all six positive and negative values to find the net result.

Solution

1. The positive values occur in January, April, and May. The negative values occur in February, March, and June. 2. The greatest value is \(\$500\), and the least value is \(-\$300\). The range is \(\$500 - (-\$300) = \$800\). 3. The net result is \(\$400 + (-\$200) + (-\$300) + \$100 + \$500 + (-\$100) = \$400\). 4. The business made a net profit of \(\$400\).

Answer

a) Profit: January, April, and May; loss: February, March, and June b) \(\$800\) c) A net profit of \(\$400\)
5350446
The bar graph shows how far a reservoir's water level was above or below its target level during five months. The differences are in meters. Find the range of the measurements. During which month was the water level farthest below the target?
Figure for problem 535044

Hints

- Identify the greatest and least values on the graph. - Subtracting a negative value increases the difference. - Find the bar that extends farthest below zero.

Solution

1. The greatest difference is \(3\,\text{m}\) in July. 2. The least difference is \(-2.5\,\text{m}\) in May. 3. The range is \(3 - (-2.5) = 5.5\,\text{m}\). 4. The water level was farthest below the target in May.

Answer

The range is \(5.5\,\text{m}\). The water level was farthest below the target in May.
5351226
The bar graph shows the daily high temperatures for one week in degrees Celsius. a) Find the range of the temperatures. b) Suppose Sunday had been much colder, making the range exactly twice its original value. What temperature would Sunday have needed to be?
Figure for problem 535122

Hints

- Find the original maximum and minimum. - Double the original range. - The colder Sunday would become the new minimum.

Solution

1. The greatest temperature is \(23\,^\circ\text{C}\), and the least is \(18\,^\circ\text{C}\). 2. The original range is \(23 - 18 = 5\,^\circ\text{C}\). 3. Twice the original range is \(2 \times 5 = 10\,^\circ\text{C}\). 4. A colder Sunday would become the new minimum while \(23\,^\circ\text{C}\) remains the maximum. Let the new Sunday temperature be \(T\). Then \(23 - T = 10\), so \(T = 13\). 5. Sunday would need to be \(13\,^\circ\text{C}\).

Answer

a) \(5\,^\circ\text{C}\) b) \(13\,^\circ\text{C}\)
5388056
Data set A is \(9, 10, 10, 11\), and data set B is \(0, 10, 10, 20\). Both have a mean of \(10\). Is the equal mean enough to describe the data sets as similar? Explain.

Hints

- Compare the distances between the values, not only their centers. - Find a simple measure of spread for each data set. - Equal sums can be distributed very differently.

Solution

1. Each data set has a sum of \(40\), so each mean is \(40 \div 4 = 10\). 2. Data set A has a range of \(11 - 9 = 2\). 3. Data set B has a range of \(20 - 0 = 20\). 4. Data set A is tightly clustered around the mean, while data set B has values much farther from the mean. Therefore, the equal means are not enough to describe the data sets as similar.

Answer

No. Data set A has a range of \(2\), while data set B has a range of \(20\). Their spreads are very different despite the equal means.

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