Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the mean, the median, and the mode for each set of data. $$ \begin{array}{|r|r|}\hline x_{i} & {f_{i}} \\ \hline 10 & {1} \\ {9} & {1} \\\ {8} & {3} \\ {7} & {7} \\ {6} & {6} \\ {5} & {2} \\ {4} & {2} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

The mean is approximately 6.36, the median is 7, and the mode is 7.

Step 1: Understand the Data

The table provided indicates a frequency distribution where \(x_i\) represents the data values and \(f_i\) represents their respective frequencies. We will use this table to find the mean, median, and mode.

Step 2: Calculate the Mean

The mean can be found using the formula: \( \bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \). First, we calculate \(\sum (x_i \cdot f_i) = (10 \times 1) + (9 \times 1) + (8 \times 3) + (7 \times 7) + (6 \times 6) + (5 \times 2) + (4 \times 2) = 140\), then \(\sum f_i = 22\). So, the mean is \( \frac{140}{22} \approx 6.36 \).

Step 3: Calculate the Median

To find the median, first determine the cumulative frequencies. The cumulative frequencies are: 1, 2, 5, 12, 18, 20, 22. The total frequency is 22, so the median position is \(\frac{22+1}{2} = 11.5\), which means the median is the 12th value in the ordered dataset. Since the 12th value falls in the data value 7 (cumulative frequency 12), the median is 7.

Step 4: Calculate the Mode

The mode is the data value with the highest frequency. From the table, the value 7 has the highest frequency of 7. Thus, the mode is 7.

Key Concepts

Frequency Distribution

A frequency distribution is a way to organize data by showing how often each value occurs. This is useful in understanding the structure of the data set.

In a frequency distribution table:

• Each unique data value is listed, often in rows.
• Alongside, the frequency of each value, or how many times it appears, is noted.

In the example from the exercise, you can see that data values like 10, 9, 8, etc., are paired with their respective frequencies, such as 1, 1, 3, respectively.

This tabular format makes it much easier to see at a glance which values occur most frequently and aids in identifying patterns or trends quickly.

Cumulative Frequency

Cumulative frequency is a progressive total of the frequencies through the classes in a distribution. It tells you how many data points are below or equal to a particular value. To calculate cumulative frequency, you add the frequency of each value to the sum of the frequencies of all previous values.

In the exercise, to find the cumulative frequencies:

• Start at the top of the frequency list. Begin with the first frequency, which in this case is 1.
• For the next number, you add the current frequency to the prior cumulative frequency. For example, 1 (from 10) and 1 (from 9) gives 2.
• Continue this process down the list: 5, 12, 18, 20, and finally 22.

This process creates a running total and demonstrates the accumulation of frequencies, helping pinpoint the median or quantiles.

Statistical Measures

Statistical measures like mean, median, and mode give a quick snapshot of the data's characteristics and distribution.

• Mean is the average of the data values. It's found by multiplying each data value by its frequency, summing all these products, and then dividing by the total number of data points. In this exercise, the mean is calculated to be approximately 6.36.


• Median is the middle value when the data is sorted in order. By looking at cumulative frequency, we can determine that the median in this example is the 12th data point, which corresponds to the data value 7.


• Mode is the data value with the highest frequency. It represents the most common data point in the set. Here, the mode is 7, since it has the highest frequency of 7.

Understanding these statistical measures helps illustrate central tendencies and variability in a dataset, providing insights into its overall structure.