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. $$\begin{array}{llllllllll}3.1 & 4.9 & 2.8 & 3.6 & 2.5 & 4.5 & 3.5 & 3.7 & 4.1 & 4.9 \\ 2.9 & 2.1 & 3.5 & 4.0 & 3.7 & 2.7 & 4.0 & 4.4 & 3.7 & 4.2 \\ 3.8 & 6.2 & 2.5 & 2.9 & 2.8 & 5.1 & 1.8 & 5.6 & 2.2 & 3.4 \\ 2.5 & 3.6 & 5.1 & 4.8 & 1.6 & 3.6 & 6.1 & 4.7 & 3.9 & 3.9 \\ 4.3 & 5.7 & 3.7 & 4.6 & 4.0 & 5.6 & 4.9 & 4.2 & 3.1 & 3.9\end{array}$$ How would you describe the shape of the distribution?

Short answer

Answer: The distribution appears to be bimodal, with two peaks around the intervals 3.1-3.6 and 3.6-4.1. It is slightly skewed to the right but is relatively symmetrical overall.

Step-by-step solution

Organizing the data into classes

The first step is to organize the data into classes. Using the given class starting at 1.6 and class width of 0.5, we can create the classes as follows: 1. 1.6-2.1 2. 2.1-2.6 3. 2.6-3.1 4. 3.1-3.6 5. 3.6-4.1 6. 4.1-4.6 7. 4.6-5.1 8. 5.1-5.6 9. 5.6-6.1

Counting the number of data points in each class

Next, we need to count the number of measurements that fall into each class. After examining the data and counting the values, we get: 1. 1.6-2.1: 5 2. 2.1-2.6: 6 3. 2.6-3.1: 7 4. 3.1-3.6: 9 5. 3.6-4.1: 11 6. 4.1-4.6: 5 7. 4.6-5.1: 3 8. 5.1-5.6: 1 9. 5.6-6.1: 3

Calculating the relative frequencies

To find the relative frequencies, divide the count of each class by the total count (which is 50 in this case). The relative frequencies are: 1. 1.6-2.1: 5/50 = 0.10 2. 2.1-2.6: 6/50 = 0.12 3. 2.6-3.1: 7/50 = 0.14 4. 3.1-3.6: 9/50 = 0.18 5. 3.6-4.1: 11/50 = 0.22 6. 4.1-4.6: 5/50 = 0.10 7. 4.6-5.1: 3/50 = 0.06 8. 5.1-5.6: 1/50 = 0.02 9. 5.6-6.1: 3/50 = 0.06

Constructing the histogram

Now that we have the relative frequencies for each class, we can plot the histogram. On the horizontal axis, label the class intervals, and on the vertical axis, plot the relative frequencies. Draw a vertical bar for each class with a height corresponding to the relative frequency value.

Describe the shape of the distribution

Observing the histogram, we can see that the distribution is bimodal (has two peaks) - one peak around 3.1-3.6 and another around 3.6-4.1. The shape is slightly skewed to the right, but overall, the distribution appears to be relatively symmetrical.

Key concepts

Data Distribution

Data distribution involves showing how often data falls within certain ranges or categories. When you visually examine data sets, the distribution gives you insights into patterns and possibilities in the data. A histogram, one of the key tools, helps in understanding how the data is spread or distributed across different values.

In this exercise, we had a set of measurements from which to create a histogram. The idea was to see how these numbers are distributed across specified intervals. When plotted, this visual tool allowed us to quickly see where most of the data points lie.

For example, in the given measurements, we had clear peaks, showing where larger numbers of data points were concentrated. Recognizing these patterns is vital because they reveal the underlying nature of the dataset, making it easier to analyze.

Class Intervals

Class intervals, also known as bins, are specific ranges that group together data points in a dataset. Choosing appropriate class intervals is crucial because it affects how the overall data distribution appears. Each interval has a lower and an upper boundary, capturing data points that fall within this range.

In our example, the intervals were set to start at 1.6 with a width of 0.5. This means the first class covered data from 1.6 to just under 2.1. This systematic grouping allows us to organize the data into manageable chunks. The intervals used may impact the visualization and interpretation of the data, so choosing a consistent and logical class width is essential.

In constructing the histogram, each bar represents one of these class intervals, and the height of each bar is proportional to the number of data points within that interval.

Relative Frequency

Relative frequency is the fraction or proportion of the total data points that fall within a specific class interval. It is obtained by dividing the number of data points in each class by the total number of points in the dataset.

For instance, if an interval contains 5 data points out of a total of 50, the relative frequency would be calculated as 5/50, which equals 0.10. This value indicates that 10% of the data points fall within this particular interval.

In contrast to absolute frequency, which only gives the count, relative frequency gives us a sense of how important or significant an interval is relative to the entire dataset. When plotted on the histogram, it helps to quickly visualize comparative data distribution and identify major concentrations or gaps.

Bimodal Distribution

A bimodal distribution is a type of data distribution with two different modes or peaks, rather than one. These peaks represent the most frequently occurring values in the dataset, indicating that there are two dominant groups of data values.

In the exercise provided, the histogram depicted a bimodal distribution, revealing two notable concentration areas, specifically around the intervals 3.1-3.6 and 3.6-4.1. This means that the data had two ranges where measurements were most densely packed. Such patterns are interesting as they may indicate diverse factors affecting the dataset, like different population groups or conditions.

Understanding a bimodal distribution helps in identifying separate influences or patterns in the data, which could be vital for drawing any meaningful insights or conclusions.