Question

3.36 Construct a histogram corresponding to each of the five frequency distributions, \(\mathrm{I}-\mathrm{V},\) given in the following table, and state whether each histogram is symmetric bimodal, positively skewed, or negatively skewed: \begin{tabular}{cccccc} & \multicolumn{5}{c} { Frequency } \\ \cline { 2 - 6 } Class Interval & I & II & III & IV & V \\ 0 to \(<10\) & 5 & 40 & 30 & 15 & 6 \\ 10 to \(<20\) & 10 & 25 & 10 & 25 & 5 \\ 20 to \(<30\) & 20 & 10 & 8 & 8 & 6 \\ 30 to \(<40\) & 30 & 8 & 7 & 7 & 9 \\ 40 to \(<50\) & 20 & 7 & 7 & 20 & 9 \\ 50 to \(<60\) & 10 & 5 & 8 & 25 & 23 \\ 60 to \(<70\) & 5 & 5 & 30 & 10 & 42 \\ \hline \end{tabular}

Short answer

The student would need to provide five histograms, each based upon the frequency distributions provided, and an explanation based on their observations of the graphs illuminating whether each histogram is symmetric, bimodal, positively skewed, or negatively skewed.

Step-by-step solution

Construct Histogram for Frequency Distribution I

For this task, the student would plot the given class intervals on the X-axis and the corresponding frequencies from column I on the Y-axis. Each class interval would be represented by a bar, the height of which corresponds to the given frequency. The shape of the histogram should be examined - is it symmetric, bimodal or skewed?

Construct Histogram for Frequency Distribution II

Repeat Step 1, but this time plot the frequencies from column II. Examine the resulting histogram's shape - is it symmetric, bimodal or skewed?

Construct Histogram for Frequency Distribution III

Repeat Step 1, but this time plot the frequencies from column III. Once again, the student should examine the shape of the resulting histogram.

Construct Histogram for Frequency Distribution IV

Repeat Step 1, but now plot the frequencies from column IV. The student should pay attention to the shape of the resulting histogram.

Construct Histogram for Frequency Distribution V

Finally, the student should repeat the process from Step 1 one last time, but with the frequencies from column V. The student again should examine the shape of the resulting histogram.

Determine Histogram Shapes

Using the constructed histograms, the student should determine whether each histogram is symmetric (appears uniformly shaped around its middle), bimodal (has two noticeable peaks), positively skewed (most data gathered towards lower values with few outliers at high values), or negatively skewed (most data gathered towards high values with few outliers at low values).

Key concepts

Frequency Distribution

A frequency distribution is a way to summarize data by showing the number of observations within each interval. In the context of histograms, these intervals are the bins represented on the X-axis. Each bin corresponds to a range of values called a class interval. The frequency, represented on the Y-axis, reflects how many data points fall within each class interval.

Creating a frequency distribution involves a few simple steps:

• Determine the range of the data by subtracting the smallest value from the largest.
• Decide on the number of bins, which will influence how detailed your distribution looks.
• Calculate the bin width by dividing the data range by the number of bins.
• Count the number of data points falling within each bin.

Once this setup is complete, you can translate this information into a histogram which provides a visual impression of the data's spread and essential trends.

Shape Analysis

When analyzing shapes in histograms, the goal is to understand the distribution of data visually. The shape can reveal important features of your data and may indicate underlying population parameters. There are a few common types of shapes to look for when analyzing histograms:

• Symmetric: If the left and right side of the histogram are mirror images, then it indicates that the data is uniformly distributed around the center.
• Skewed: If one tail is longer than the other, we say that the histogram is skewed either positively or negatively.
• Bimodal: This occurs when there are two distinct peaks in the histogram, suggesting two different modes or groups within the dataset.
• Uniform: Indicating that each interval has roughly the same frequency.

Shape analysis provides insights into data trends and potential anomalies, helping in further statistical analyses.

Symmetric Distribution

A symmetric distribution is a type of data distribution where the left half of the graph is a mirror image of the right half. This balance means the central tendency, typically mean and median, tend to coincide. When looking at a histogram, a symmetric distribution appears bell-shaped, and it often signifies a normal distribution pattern.

Understanding a symmetric distribution is key because:

• It suggests there are as many values lower than the mean as there are higher.
• This balance in data can simplify further statistical inference, like calculating probabilities.
• Many statistical tests assume the data follows a symmetric or normal distribution.

In practical terms, a symmetric histogram simplifies the analysis, as it often points to a normal, Gaussian process underlying the data.

Skewness Analysis

Skewness refers to the asymmetry in the distribution of data, and it can be either positive or negative.

• Positive skew (right-skewness): The tail on the right side is longer or fatter than the left side. Most values accumulate towards the lower end with a few high-value outliers.
• Negative skew (left-skewness): The tail on the left side is longer or fatter than the right side. Most data values gather at higher numbers, with a few low-value outliers.

Understanding skewness is crucial because:

• It can affect mean and median, positioning the mean towards the longer tail.
• It helps identify potential outliers and anomalies within data.
• Knowing the skewness helps in choosing the right statistical tests and interpretations.

Being aware of skewness is vital for accurate data analysis, as it can heavily influence the results of analytical models.