Question

Construct a relative frequency histogram for these 50 measurements using classes starting at 1.6 with a class width of .5. Then answer the questions. $$\begin{array}{llllllllll}3.1 & 4.9 & 2.8 & 3.6 & 2.5 & 4.5 & 3.5 & 3.7 & 4.1 & 4.9 \\ 2.9 & 2.1 & 3.5 & 4.0 & 3.7 & 2.7 & 4.0 & 4.4 & 3.7 & 4.2 \\ 3.8 & 6.2 & 2.5 & 2.9 & 2.8 & 5.1 & 1.8 & 5.6 & 2.2 & 3.4 \\ 2.5 & 3.6 & 5.1 & 4.8 & 1.6 & 3.6 & 6.1 & 4.7 & 3.9 & 3.9 \\ 4.3 & 5.7 & 3.7 & 4.6 & 4.0 & 5.6 & 4.9 & 4.2 & 3.1 & 3.9\end{array}$$ What fraction of the measurements are less than \(5.1 ?\)

Short answer

Answer: The fraction of measurements less than 5.1 is 0.86.

Step-by-step solution

Identify the Classes

Divide the data into classes, starting at 1.6 with class width of 0.5. Note that the last measurement is 6.2 which will be the last boundary. Thus, the classes will be: - Class 1: [1.6, 2.1) - Class 2: [2.1, 2.6) - Class 3: [2.6, 3.1) - ... - Class 10: [5.6, 6.1) - Class 11: [6.1, 6.6)

Tally the Data in Each Class

Count the number of measurements that fall within each class: - Class 1: 1 - Class 2: 6 - Class 3: 6 - Class 4: 10 - Class 5: 11 - Class 6: 9 - Class 7: 4 - Class 8: 2 - Class 9: 0 - Class 10: 1 - Class 11: 0

Calculate the Relative Frequency

To calculate the relative frequency of each class, divide the number of measurements in each class by the total number of measurements (50): 1. \( \frac{1}{50} = 0.02 \) 2. \( \frac{6}{50} = 0.12 \) 3. \( \frac{6}{50} = 0.12 \) 4. \( \frac{10}{50} = 0.20 \) 5. \( \frac{11}{50} = 0.22 \) 6. \( \frac{9}{50} = 0.18 \) 7. \( \frac{4}{50} = 0.08 \) 8. \( \frac{2}{50} = 0.04 \) 9. \( \frac{0}{50} = 0.00 \) 10. \( \frac{1}{50} = 0.02 \) 11. \( \frac{0}{50} = 0.00 \) Using the calculated relative frequencies, construct the histogram.

Answer the Question

We are asked to find the fraction of the measurements that are less than 5.1. Observing the histogram, we add the relative frequencies of classes 1 through 6: \(0.02+0.12+0.12+0.20+0.22+0.18=0.86\) Therefore, the fraction of measurements less than 5.1 is \(\boxed{0.86}\).

Key concepts

Class Width

The concept of class width is critical in the organization and presentation of data for analysis. Class width refers to the range of values included within each interval or 'bin' when data is divided into classes or categories for a frequency distribution. It's essentially the difference between the upper and lower boundaries of a class.
For example, if we're constructing a frequency distribution with classes that have a lower limit of 1.6 and an upper limit of 2.1, the class width would be the difference between these two values, which in this case is 0.5. Uniform class width across all classes ensures each data point is categorized without overlap or gaps, providing clarity and consistency in the data's graphical representation, such as in histograms.
Class width is calculated by taking the range of all data points (largest minus smallest value) and dividing by the desired number of classes. However, in the given exercise, we are provided with a class width, simplifying the process. The right selection of class width is vital as too narrow may result in an over-complicated model, while too wide could oversimplify the data, potentially missing important patterns.

Data Classification

Data classification involves organizing raw data into meaningful and manageable groups or classes. This is a foundational step in statistical analysis allowing us to interpret and extract insights. In our exercise, data classification entails grouping 50 measurements into classes, each covering a range of values determined by the class width.
The process starts by determining a starting point, often the smallest value in the data set or a convenient number slightly lower. Subsequent classes are formed by adding the class width to the starting value, creating non-overlapping intervals that encompass all possible data values. Proper classification ensures that every data point is included in the analysis and allows for subsequent steps such as frequency and relative frequency calculation.

Relative Frequency Calculation

Relative frequency represents the proportion of the total number of data points that are within a particular class. It is calculated by dividing the number of data points in a class by the total number of data points. To express as a percentage, this proportion can then be multiplied by 100.
In the exercise provided, to calculate the relative frequency for each class, you divide the count of data points in the class by 50 (the total number of measurements), as demonstrated in the step by step solution. These calculations provide insights into the distribution of data relative to the whole, enabling comparisons between classes and highlighting trends within the dataset.
Understanding relative frequency is important for interpreting data distributions since it normalizes the frequencies, allowing us to compare data sets of different sizes on an equal footing.

Histogram Construction

A histogram is a visual representation of data that allows us to see frequencies of data points within defined ranges or classes. Constructing a histogram involves plotting the classes along the horizontal axis (X-axis) and the frequencies or relative frequencies along the vertical axis (Y-axis).
Each class is represented by a bar whose height corresponds to the quantity it represents—either frequency or relative frequency. In the context of the exercise, a relative frequency histogram is constructed, with each bar's height being the relative frequency of measurements recorded in each class. This graphical representation gives a clear and immediate picture of the data distribution and is instrumental for quickly identifying patterns such as central tendency, spread, and skewness within the data.
When we construct the histogram for the exercise, we graph the classes starting at 1.6 and the relative frequencies calculated in the previous step, resulting in a histogram that accurately reflects the distribution of the 50 measurements.