As of 2011 , the following are the ages, in chronological order, at which U.S.
presidents were inaugurated:
$$
\begin{aligned}
&57,61,57,57,58,57,61,54,68,51,49,64,50,48,65,52,56,46, \\
&54,49,50,47,55,55,54,42,51,56,55,51,54,51,60,62,43,55, \\
&56,61,52,69,64,46,54,47
\end{aligned}
$$
Source: Time Almanac
Construct a grouped frequency distribution for the data. Use 41-45 for the
first class and use the same width for each subsequent class.
The Grouped Frequency Distribution is given by the following classes and their respective frequencies: 41-45: 0; 46-50: 7; 51-55: 10; 56-60: 7; 61-65: 5; 66-70: 3.
Step by step solution
01
Classifying the Ages
Start by categorizing each age into its corresponding class. The classes will be: 41-45, 46-50, 51-55, 56-60, 61-65, 66-70. If an age falls on the boundary of two classes, place it into the higher class.
02
Counting Each Age
Count how many ages fall into each class. This will be the frequency of the class. Remember that a frequency of 0 is valid and should also be included.
03
Creating the Frequency Distribution Table
Create a table with two columns. In the left column, list your classes in order. In the right column, list the corresponding frequencies from Step 2. Ensure each class has its own row.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Data Categorization
When creating a grouped frequency distribution, data categorization is a crucial first step. This involves organizing raw data into categories or groups. Here, the ages of U.S. presidents at their inaugurations are raw data. By placing each age into defined categories, we can analyze the data more efficiently.
For example: If you have a large dataset of numbers, categorizing it into groups can help detect patterns or trends. In this case, ages are categorized into intervals to form groups.
- Purpose: Simplifies data to make it more understandable.
- Process: Decide on categories that represent the original data accurately.
- Goal: Provide a clearer picture of the data distribution.
For our exercise, ages are categorized into intervals such as 41-45, 46-50, and so on. This helps us understand the age distribution of presidents when they assumed office.
Class Intervals
Class intervals are continuous ranges into which you group your data. They are fundamentally important when setting up a frequency distribution because they determine how the data is organized.
Selection: The choice of class interval affects how well the data is represented.
- Width of Intervals: Equal class width is chosen to maintain uniformity. E.g., our class intervals are 41-45, 46-50, etc., each with a width of 5.
- Boundaries: If a data point falls on the boundary between two classes, it is usually placed in the higher class (such as 50 falling into the 46-50 interval).
Class intervals aid in efficiently organizing data for a clearer visual presentation of its distribution. This method helps in detecting where the majority of data points concentrate within the data set.
Frequency Table
A frequency table is a compact way to summarize data, showcasing how often each class interval contains data points. Constructing this table involves two key steps: categorizing data into intervals and counting the occurrences.
Structure:
- Column 1 lists each class interval (like 41-45).
- Column 2 records the frequency of ages within that interval.
Making a frequency table can help reveal patterns. For example, the data might show that most presidents are inaugurated around a specific age group. This visual representation aids in comprehending and confirming how the data is distributed across different segments.
Mathematical Analysis
Mathematical analysis of a frequency distribution can uncover characteristics of a dataset that are not immediately apparent. This involves using statistical methods to discern insights from the data.
Common Analyses include:
- Central Tendency: Identifying the average (mean), median, or mode of the data.
- Spread of Data: Calculating measures like range or standard deviation to understand data variability.
For instance, in analyzing the frequency table of presidents' ages, you might calculate the average age. This helps in understanding common age tendencies. Such analyses give a pragmatic view by not only showing data but also explaining characteristics that can aid further assessments or decision-making.
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