Question

Use the information given to find a convenient class width. Then list the class boundaries that can be used to create a relative frequency histogram. 7 classes for \(n=50\) measurements; minimum value \(=10 ;\) maximum value \(=110\)

Short answer

Answer: The class boundaries for the relative frequency histogram are 10, 25, 40, 55, 70, 85, 100, and 115.

Step-by-step solution

Calculate the range of the data

We need to find the range of the data given the minimum and maximum values. The range is the difference between the maximum and the minimum value. Range = Maximum Value - Minimum Value Range = 110 -10 Range = 100

Find the class width

Now, we need to divide the range by the number of classes (7) to find the class width. Class Width = Range / Number of Classes Class Width = 100 / 7 Class Width ≈ 14.3 We can round the class width to 15 for convenience.

Determine the class boundaries

To find the class boundaries, we will begin with the minimum value and add the class width successively until we have covered all the classes. 1st class boundary: 10 (minimum value) 2nd class boundary: 10 + 15 = 25 3rd class boundary: 25 + 15 = 40 4th class boundary: 40 + 15 = 55 5th class boundary: 55 + 15 = 70 6th class boundary: 70 + 15 = 85 7th class boundary: 85 + 15 = 100 8th class boundary: 100 + 15 = 115 (one step beyond our maximum value for the sake of histograms) The class boundaries for the relative frequency histogram are: 10, 25, 40, 55, 70, 85, 100, and 115.

Key concepts

Class Width

When constructing histograms, determining the class width is a crucial step. The class width tells us how wide each bin in the histogram should be to cover our data. For example, in our problem, we have a range of data from 10 to 110 and we need to divide this into 7 classes.
To find the class width, you take the difference between the highest and lowest values (the range) and divide it by the number of classes. This formula looks like this:
time Class Width = \( \frac{\text{Range}}{\text{Number of Classes}} \).After calculating, if your result isn't a whole number, it's often best to round up to a convenient whole number to simplify the construction of your histogram. In our case, \( \frac{100}{7} \approx 14.3 \), which we rounded to 15 for easy use.
This process ensures each class or bin in your histogram will cover an equal portion of your data range.

Range Calculation

Range indicates how spread out the numbers in your data set are. Calculating the range gives you a quick sense of the overall spread. It's a simple, yet powerful step to define the extent of your data spread. To find the range, subtract the smallest number in the dataset from the largest number.
time Range = Maximum Value - Minimum Value. For example, if the smallest measurement in your data is 10, and the largest is 110, then the range is calculated as: 110 - 10 = 100.
Knowing your data's range is foundational before you move forward to create a histogram, as it helps you determine the class width and overall configuration of your data depiction.

Relative Frequency

Relative frequency represents how often something occurs relative to the total number of observations. It's a great way to understand the proportion of your data within each class or bin in a histogram.
To calculate relative frequencies, take the frequency count of individual class data and divide by the total number of observations. time Relative Frequency = \( \frac{\text{Class Frequency}}{\text{Total Number of Observations}} \).Relative frequencies can be helpful for comparing different data sets or for emphasizing specific trends within your dataset.
In the context of a histogram, calculating relative frequencies and plotting them gives a normalized view, allowing easier comparison between different datasets or samples. It shows the shape and spread of the data distribution effectively.