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 from 2.6 up to but not including \(4.6 ?\)

Short answer

Answer: \(\frac{33}{50}\)

Step-by-step solution

Arrange data into classes

We are given that the classes start at 1.6 with a class width of 0.5. So, we create the following classes: [1.6 - 2.1), [2.1 - 2.6), [2.6 - 3.1), [3.1 - 3.6), [3.6 - 4.1), [4.1 - 4.6), [4.6 - 5.1), [5.1 - 5.6), [5.6 - 6.1), and [6.1 - 6.6) Now we will count the number of measurements that fall into each class.

Count measurements in each class

We go through the data and count the number of measurements that fall into each class: [1.6 - 2.1): 1 (1.8) [2.1 - 2.6): 6 (2.5, 2.5, 2.2, 2.5, 2.8, 2.9) [2.6 - 3.1): 8 (3.1, 2.7, 2.8, 3.4, 2.9, 3.1, 2.9, 3.0) [3.1 - 3.6): 8 (3.5, 3.5, 3.6, 3.6, 3.7, 3.7, 3.9, 3.5) [3.6 - 4.1): 8 (3.6, 4.0, 3.7, 4.0, 3.9, 3.7, 4.2, 3.9) [4.1 - 4.6): 9 (4.5, 4.1, 4.4, 4.3, 4.2, 4.6, 4.9, 4.0, 4.9) [4.6 - 5.1): 7 (4.8, 4.7, 5.1, 5.0, 4.9, 5.1, 5.6) [5.1 - 5.6): 4 (5.1, 5.7, 5.6, 5.6) [5.6 - 6.1): 1 (6.1) [6.1 - 6.6): 1 (6.2)

Calculate relative frequencies

Now we will calculate the relative frequency of each class by dividing the count of each class by the total number of measurements (50): 1/50, 6/50, 8/50, 8/50, 8/50, 9/50, 7/50, 4/50, 1/50, 1/50 These relative frequencies will be the basis for the histogram.

Construct a histogram

Using the relative frequencies from Step 3, create a histogram with the classes on the x-axis and the relative frequencies on the y-axis. Each class will be represented by a bar with the height equal to its relative frequency.

Find the fraction of measurements between 2.6 and 4.6

We are asked to find the fraction of measurements that are from 2.6 up to but not including 4.6. We will look at the relative frequencies of the classes that fall in this range: [2.6 - 3.1): 8/50 [3.1 - 3.6): 8/50 [3.6 - 4.1): 8/50 [4.1 - 4.6): 9/50 Now, add up the relative frequencies of these classes: Fraction of measurements = (8 + 8 + 8 + 9) / 50 = 33/50 So, the fraction of measurements between 2.6 and 4.6 is \(\frac{33}{50}\).

Key concepts

Data Classification

Data classification is the process of organizing raw data into categories or classes that have similar attributes. It's a critical first step in data analysis, as it simplifies and clarifies the data, making it easier to understand and interpret. For instance, when constructing a histogram, we classify continuous numerical data into 'bins' or 'classes', each covering a specific range of values.

Data classification helps in creating a clear visual representation of the distribution of data points, which can reveal patterns that might not be evident in an unorganized list of numbers. The classes should be mutually exclusive, meaning each data point belongs to one class only, and collectively exhaustive, meaning all possible data points are included within the classes. In the exercise provided, data was classified into classes with a width of 0.5, starting at 1.6.

Statistical Measurements

Statistical measurements are quantities that summarize or describe characteristics of a set of data. Some common statistical measurements include mean (average), median (middle value), mode (most frequent value), range (difference between the highest and lowest values), and standard deviation (measure of how spread out the values are).

In the context of constructing histograms like in our exercise, frequency and relative frequency are key statistical measurements. Frequency counts how often each value or range of values (class) occurs, while relative frequency is the proportion of total data points that fall within each class, often presented as a fraction or percentage. These measurements give us insights into the 'shape' of the data, indicating the concentration and spread of values across different intervals.

Histogram Construction

Histogram construction is a method for visualizing the distribution of data values. A histogram is a type of bar chart where each bar represents a class categorized by the range of data it includes. The height of each bar corresponds to the frequency or relative frequency of the values within that class.

To construct a histogram from a given set of data, we first determine a suitable class interval or width, which should be consistent throughout the chart. Next, we tally the data into these classes, and then we draw the bars. The exercise given involves plotting a relative frequency histogram specifically, where the height of the bars reflects the proportion of data in each class relative to the total dataset. The step-by-step solution provides a practical application of histogram construction, where organizing the data into intervals allows for a clear, graphical representation of frequency within specific ranges.

Probability and Statistics

Probability and statistics are interrelated fields of mathematics that involve analyzing and interpreting numerical data. Probability refers to the likelihood of a particular event occurring and is a vital concept in statistics, which is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data.

In statistics, relative frequency is tied to probability, since it can be viewed as an estimate of the likelihood of an event occurring based on empirical data. For example, in the exercise we analyzed, the relative frequency histogram provides a visual approximation of probabilities for different ranges of data points. By calculating the sum of relative frequencies for certain classes, we indirectly estimate the probability of a measurement falling between any two points. Hence, the fraction of measurements between 2.6 and 4.6 from the dataset gives us a probability-like understanding obtained via statistical methods.