Question

The length of time (in months) between the onset of a particular illness and its recurrence was recorded for \(n=50\) patients: $$ \begin{array}{rrrrrrrrrr} 2.1 & 4.4 & 2.7 & 32.3 & 9.9 & 9.0 & 2.0 & 6.6 & 3.9 & 1.6 \\ 14.7 & 9.6 & 16.7 & 7.4 & 8.2 & 19.2 & 6.9 & 4.3 & 3.3 & 1.2 \\ 4.1 & 18.4 & .2 & 6.1 & 13.5 & 7.4 & .2 & 8.3 & .3 & 1.3 \\ 14.1 & 1.0 & 2.4 & 2.4 & 18.0 & 8.7 & 24.0 & 1.4 & 8.2 & 5.8 \\ 1.6 & 3.5 & 11.4 & 18.0 & 26.7 & 3.7 & 12.6 & 23.1 & 5.6 & .4 \end{array} $$ a. Construct a relative frequency histogram for the data. b. Would you describe the shape as roughly symmetric. skewed right, or skewed left? c. Find the fraction of recurrence times less than or equal to 10 months.

Short answer

Answer: The shape of the histogram is skewed right, and the fraction of recurrence times less than or equal to 10 months is 9/10.

Step-by-step solution

Organize the data

First, sort the data in ascending order:$$ \begin{array}{rrrrrrrrrr} .2 & .2 & .3 & .4 & 1.0 & 1.2 & 1.3 & 1.4 & 1.6 & 1.6 \\ 2.0 & 2.1 & 2.4 & 2.4 & 2.7 & 3.3 & 3.5 & 3.7 & 3.9 & 4.1 \\ 4.3 & 4.4 & 5.6 & 5.8 & 6.1 & 6.6 & 6.9 & 7.4 & 7.4 & 8.2 \\ 8.2 & 8.3 & 8.7 & 9.0 & 9.6 & 9.9 & 11.4 & 12.6 & 13.5 & 14.1 \\ 14.7 & 16.7 & 18.0 & 18.0 & 18.4 & 19.2 & 23.1 & 24.0 & 26.7 & 32.3 \end{array} $$ Decide on an appropriate number of intervals (bins) and their widths. Six intervals with a width of 5 should work well in this case.

Find frequencies and relative frequencies

Divide the data into six intervals of width 5: - \(0 \le x < 5\) - \(5 \le x < 10\) - \(10 \le x < 15\) - \(15 \le x < 20\) - \(20 \le x < 25\) - \(25 \le x < 30\) Find the frequencies and relative frequencies for each interval: Interval | Frequency (f) | Relative Frequency ---------------|---------------|------------------- \(0 \le x < 5\) | 30 | 30/50 \(5 \le x < 10\) | 15 | 15/50 \(10 \le x < 15\)| 4 | 4/50 \(15 \le x < 20\)| 5 | 5/50 \(20 \le x < 25\)| 3 | 3/50 \(25 \le x < 30\)| 1 | 1/50

Construct a relative frequency histogram

Using the relative frequencies obtained in Step 2, draw a histogram. On the horizontal axis, label the intervals, and on the vertical axis, represent the relative frequencies as bars.

Determine the shape of the histogram

Look at the histogram created in Step 3 and observe the overall shape. If the histogram looks relatively symmetric, with a higher frequency of data in the middle and lower frequencies on either side, it is symmetric. If the histogram has a long tail on the right side (higher values), it is skewed right. If the histogram has a long tail on the left side (lower values), it is skewed left.

Find the fraction of recurrence times less than or equal to 10 months

Count the number of recurrence times less than or equal to 10 months: 45 (from the sorted data). Divide this number by the total number of patients (50) to find the fraction: $$\frac{45}{50} = \frac{9}{10}$$Therefore, the fraction of recurrence times less than or equal to 10 months is \(\frac{9}{10}\).

Key concepts

Relative Frequency

In statistics, relative frequency is the proportion of times a particular outcome occurs relative to the total number of occurrences. It gives us a clearer understanding of the data set by showing how often each outcome happens compared to others.
For example, in the context of the original exercise, the frequency of recurrence times falling within specific intervals was calculated, and the relative frequency was determined by dividing each interval's frequency by the total number of observations, which was 50.
This calculation helps us understand the distribution of data across different intervals.

• For instance, in the interval \(0 \leq x < 5\), there were 30 occurrences. The relative frequency is \(\frac{30}{50} = 0.6\), meaning 60% of the data lies in this interval.
• Similarly, for the interval \(5 \leq x < 10\), the relative frequency is \(\frac{15}{50} = 0.3\), representing 30% of the data.

Relative frequencies are useful because they provide context regarding the occurrence of outcomes within a dataset.

Histogram

A histogram is a graphical representation of data distribution, where data is divided into intervals, and the frequency of data points within each interval is depicted as bars.
In the exercise, a relative frequency histogram was created using intervals of width 5 months. This type of histogram helps in visualizing the frequency of data relative to the total dataset.

• The horizontal axis of the histogram represents the intervals of possible outcome values. For this data, the intervals were \(0 \leq x < 5\), \(5 \leq x < 10\), etc.
• The vertical axis represents the relative frequencies of occurrence within each interval.
• Bars of different heights indicate how frequent specific ranges of data values are in comparison to others.

This representation is beneficial for quickly understanding the spread and concentration of the data at a glance.

Data Distribution

Data distribution refers to how data points are spread or distributed across different values. Understanding the data distribution helps in interpreting the underlying patterns and tendencies in the dataset.
The original exercise involved organizing the data according to recurrence times and visualized this distribution through a histogram. Such distribution can inform researchers about common or rare events in a dataset.

• A uniform data distribution suggests that each outcome is equally likely.
• A normal distribution, also known as a bell curve, indicates data symmetrically distributed with most occurrences around a central peak.
• Analyzing data distribution provides insights into the variance and standard deviation, which quantifies the amount of variation or dispersion in a set of values.

Understanding data distribution allows us to derive meaningful insights and inform decision-making processes.

Skewness

Skewness is a measure of the asymmetry of the distribution of data points around the mean. In simple terms, it informs us about the tilt or imbalance in the data.
In the context of the histogram from this exercise, skewness helps determine the shape of the data distribution.

• If the data is symmetric, it has a skewness close to zero, meaning there is no notable tilt, and the histogram peaks around the center equally spreading on both sides.
• An observation of a long tail on the right indicated a right or positive skew, where most data points appear on the left side of the graph.
• Conversely, a left or negative skew results when the long tail lays on the left side, with most data points clustered on the right.

Understanding skewness in your data is crucial as it impacts statistical methods and interpretation results, providing insights into potential biases or tendencies.