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Magazine Chapter 15: Problem 13
In \(13-18,\) find the mean and the median for each set of data to the nearest tenth. $$ \begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 2 |-25 & {2} \\ {16-20} & {3} \\ {11-15} & {12} \\ {6-10} & {6} \\ {1-5} & {1} \\\ \hline\end{array} $$
Short Answer
Expert verified
Mean: 9.9; Median: 13.
Step by step solution
01
Identify the Midpoint of Each Interval
For each interval, find the midpoint, which we will call \( x_i \). To find the midpoint of an interval, calculate the average of its upper and lower bounds. For instance, for the interval \( 2|-25 \), the midpoint is \((2 + (-25))/2 = -11.5\). Do this for each interval.
02
Create a Frequency Distribution Table with Midpoints
Create a new column in the table for midpoints \( x_i \):1. \( 2|-25 \): midpoint = \(-11.5\)2. \( 16-20 \): midpoint = \(18\)3. \( 11-15 \): midpoint = \(13\)4. \( 6-10 \): midpoint = \(8\)5. \( 1-5 \): midpoint = \(3\)
03
Calculate the Mean
The mean is calculated using the formula \( \mu = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \).1. Calculate \( x_i \cdot f_i \) for each interval: - \(-11.5 \cdot 2 = -23\) - \(18 \cdot 3 = 54\) - \(13 \cdot 12 = 156\) - \(8 \cdot 6 = 48\) - \(3 \cdot 1 = 3\)2. Sum the products: \(-23 + 54 + 156 + 48 + 3 = 238\)3. Sum the frequencies: \(2 + 3 + 12 + 6 + 1 = 24\)4. Calculate the mean: \( \mu = \frac{238}{24} \approx 9.9\).
04
Determine the Cumulative Frequencies
Determine the cumulative frequencies to help find the median.1. \(2\) (for interval \(2|-25 \))2. \(2 + 3 = 5\) (for interval \(16-20 \))3. \(5 + 12 = 17\) (for interval \(11-15 \))4. \(17 + 6 = 23\) (for interval \(6-10 \))5. \(23 + 1 = 24\) (for interval \(1-5 \))Check that the last cumulative frequency matches the total number of observations \(24\).
05
Calculate the Median
The median is the value at the \( \frac{n+1}{2} \)-th position in the ordered distribution, where \( n \) is the total frequency.1. \( n = 24 \), so \( \frac{24 + 1}{2} = 12.5 \).2. Identify the interval containing the 12.5th value using cumulative frequencies: - The 12.5th position falls in the \(11-15\) interval (since cumulative frequency is \(17\) here).3. Therefore, the midpoint \(13\) of the \(11-15\) interval is the median.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Mean Calculation
The mean is a central measure of a data set, representing an average value. It helps in understanding the overall behavior of the data. To calculate the mean, especially in a grouped frequency distribution like our exercise, follow these simplified steps:
- First, determine the midpoint of each class interval. The midpoint is calculated by taking the average of the upper and lower boundaries of each interval.
- Next, multiply each midpoint by the frequency of its corresponding class interval. This gives you a weighted value that accounts for the frequency of each interval.
- Add together all the products from the step above. This total is the numerator in our mean calculation formula.
- Now, sum up all the frequencies; this is the denominator of our formula.
- Finally, calculate the mean by dividing the total sum of products by the total sum of frequencies, \[\mu = \frac{\sum (x_i \cdot f_i)}{\sum f_i}\]where \(x_i\) represents the midpoint and \(f_i\) the frequency.
Exploring Median Calculation
The median is another measure of central tendency, indicating the middle value of a dataset. It's particularly useful when the data is skewed or contains outliers. To identify the median in a frequency distribution:
- First, calculate cumulative frequencies, which are the running totals of frequencies up to each class interval.
- Total the overall frequencies and use the formula \(\frac{n+1}{2}\) to find the median position, where \(n\) is the total frequency.
- Next, locate which class interval contains this median position by comparing it to your list of cumulative frequencies.
- The median is identified by finding the class interval's midpoint associated with where the median position falls.
Demystifying Frequency Distribution
A frequency distribution is a way of organizing data into intervals, displaying the count or frequency of observations within each interval. It allows for a quick view of how data is distributed across different ranges:
- Each interval or class represents a range of values within which data points fall, and its corresponding frequency denotes how many data points lie within that interval.
- This distribution helps in visualizing the shape, spread, and central tendencies like the mean, median, and mode.
- The frequency distribution can often be shown as a table or graph (like histograms), making it easier to identify patterns or trends within the data.
- When dealing with grouped data, the midpoint of each interval is often used for calculating central tendencies as it simplifies the representation of a potentially large dataset.
