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Chapter 15: Problem 7

In \(7-9,\) find the mean, median, range, and interquartile range for each set of data to the nearest tenth. $$ \begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 50 & {3} \\ {45} & {8} \\\ {40} & {12} \\ {35} & {15} \\ {30} & {11} \\ {25} & {7} \\ {20} & {4} \\\ \hline\end{array} $$

Short Answer

Expert verified
Mean: 35, Median: 35, Range: 30, IQR: 5.

Step by step solution

01

Calculate the Mean

To find the mean, first, calculate the total of all data points by multiplying each value \(x_i\) by its frequency \(f_i\), then sum these products. Divide the total sum by the total frequency (i.e., the sum of all \(f_i\)). \[\text{Total} = (50 \times 3) + (45 \times 8) + (40 \times 12) + (35 \times 15) + (30 \times 11) + (25 \times 7) + (20 \times 4) = 35 \times 60\]\[\text{Mean} = \frac{35 \times 60}{60} = 35\]
02

Calculate the Median

To find the median, determine the central point in the ordered data set. First, list all data points in order (considering their frequencies). The data set based on cumulative frequency is: 20(4), 25(7), 30(11), 35(15), 40(12), 45(8), 50(3). Calculate the cumulative frequencies:\[0, 4, 11, 22, 37, 49, 57, 60\]The middle value (or values) in this series is the median position: \[ \frac{60+1}{2} = 30.5 \]The median falls in the frequency of 35.
03

Calculate the Range

The range is the difference between the highest and lowest values in the data set.\[\text{Range} = 50 - 20 = 30\]
04

Calculate the Interquartile Range (IQR)

The interquartile range is the difference between the third quartile (\(Q_3\)) and the first quartile (\(Q_1\)). Find the positions of \(Q_1\) and \(Q_3\):- \(Q_1\) position: \[ \frac{60}{4} = 15 \]th value falls at 35.- \(Q_3\) position: \[ \frac{3 \times 60}{4} = 45 \]th falls at 40\[\text{IQR} = Q_3 - Q_1 = 40 - 35 = 5\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean
The mean, often referred to as the average, provides us with an insight into the central tendency of a data set. Calculating the mean involves finding the sum of all the data points, each multiplied by their respective frequency, and then dividing this sum by the total number of data points (total frequency). This gives us an idea of where the center of the data is located.

To understand this better, let's look at an example:
  • Multiply each data point by its frequency to get a series of products.
  • Add all these products together to get the total sum.
  • Divide this total sum by the total frequency (i.e., the number of data points).
In our case, after calculating, the mean turned out to be 35. This means that the average experience or tendency of the entire group is close to 35.
Median
The median is another measure of central tendency, but it is different from the mean. Instead of considering arithmetic averages, the median is the middle value of a data set when the numbers are arranged in order. If the data set contains an odd number of observations, the median is the middle number. If it's even, the median is the average of the two middle numbers.

In our example:
  • First, arrange the data in order while respecting their frequencies.
  • Next, identify the position of the median using the formula \( \frac{n+1}{2} \) for finding the central position.
  • For the given data, the median position is 30.5, which hints us at the value just past the 30th position.
Fortunately, this lies within the 35 range, making 35 the median in this context.
Range
The range is a measure that provides a basic snapshot of the spread or extent of a data set. Calculating this is simple: subtract the smallest value from the largest value in the data set. It gives a rough idea of how spread out the values in a dataset are.

For example, in our data:
  • The highest value is 50, and the smallest is 20.
  • By subtracting the smallest from the largest (50 - 20), we obtain a range of 30.
This calculation shows that the values within the data vary within a span of 30 units. This is useful, but limited as it only considers two points and not the entire dataset's distribution.
Interquartile Range
The interquartile range (IQR) is a more refined measure of spread than the range as it considers the middle 50% of the data. The IQR is found by subtracting the first quartile (\(Q_1\)) from the third quartile (\(Q_3\)). It focuses on the data around the central median portion and is less influenced by outliers or extreme values.

To calculate the IQR for our example:
  • Identify the position of \(Q_1\) by calculating \( \frac{n}{4} = 15\). This value falls at 35.
  • Locate \(Q_3\) by calculating \( \frac{3n}{4} = 45\), which falls at 40.
  • Subtract \(Q_1\) from \(Q_3\) to get the IQR: 40 - 35 = 5.
This 5-unit span indicates the spread of the middle half of our dataset. The IQR is crucial in providing insights into the variation within the dataset while minimizing the impact of extreme values.

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Most popular questions from this chapter

In \(11-17 :\) a. Draw a scatter plot. b. Does the data set show strong positive linear correlation, moderate positive linear correlation, no linear correlation, moderate negative linear correlation, or strong negative linear correlation? c. If there is strong or moderate correlation, write the equation of the regression line that approximates the data. Jack Sheehan looked through some of his favorite recipes to compare the number of calories per serving to the number of grams of fat. The table below shows the results. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline \text { Calories } & {310} & {210} & {260} & {290} & {320} & {245} & {293} & {220} & {260} & {350} \\ \hline F a t & {5} & {11} & {12} & {14} & {16} & {7} & {10} & {8} & {8} & {15} \\\ \hline\end{array} $$

In \(9-13 :\) a. Create a scatter plot for the data. b. Determine which regression model is the most appropriate for the data. Justify your answer. c. Find the regression equation. Round the coefficient of the regression equation to three decimal places. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ \hline y & {2.7} & {2.3} & {2.0} & {1.7} & {1.5} & {1.2} & {1.1} & {0.9} & {0.8} & {0.7} \\ \hline\end{array} $$

The grades on a math test of 25 students are listed below. $$\begin{array}{llllllllllll}{86} & {92} & {77} & {84} & {75} & {95} & {66} & {88} & {84} & {53} & {98} & {87} & {83} \\ {74} & {61} & {82} & {93} & {98} & {87} & {77} & {86} & {58} & {72} & {76} & {89}\end{array}$$ a. Organize the data in a stem-and-leaf diagram. b. Organize the data in a frequency distribution table. c. How many students scored 70 or above on the test? d. How many students scored 60 or below on the test?

In \(3-8,\) find the mean, the median, and the mode for each set of data. $$ \begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 50 & {8} \\ {40} & {12} \\\ {30} & {17} \\ {20} & {10} \\ {10} & {3} \\ \hline\end{array} $$

In \(3-8,\) find the mean, the median, and the mode for each set of data. $$ \begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline \$ 1.10 & {1} \\ {\$ 1.20} & {5} \\ {\$ 1.30} & {8} \\ {\$ 1.40} & {6} \\ {\$ 1.50} & {6} \\\ \hline\end{array} $$
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