Question

Altman and Bland report the survival times for patients with active hepatitis, half treated with prednisone and half receiving no treatment. \(^{12}\) The survival times (in months) (Exercise 1.25 and \(\mathrm{EX} 0125\) ) are adapted from their data for those treated with prednisone. 8 52 15757 \(\begin{array}{rr}142 & 162 \\ 144 & 165\end{array}\) a. Can you tell by looking at the data whether it is roughly symmetric? Or is it skewed? b. Calculate the mean and the median. Use these measures to decide whether or not the data are symmetric or skewed. c. Draw a box plot to describe the data. Explain why the box plot confirms your conclusions in part b.

Short answer

Answer: The data appears to be positively skewed, as the mean is greater than the median, and the box plot visually illustrates the skewness in the data distribution with more data points on the right side of the plot.

Step-by-step solution

a. Determining data symmetry

To visually examine the data, organize it in order: 8, 52, 142, 144, 157, 162, 165, 757 Now, observe the data and see if there's any apparent symmetry or skewness. Based on just looking at the data, it is difficult to definitively tell if the data is symmetric or skewed. We can observe that the values become more frequent as they increase, indicating a possible positive skew. However, a more rigorous approach is needed to determine the symmetry or skewness.

b. Calculating the mean and median

To find the mean, sum all the values and divide by the total number of values (n=8): Mean = (8+52+142+144+157+162+165+757) / 8 = 198.625 To find the median, arrange the values in order and find the middle value (the average of the two middle values if there is an even number of values): Median = (142 + 144) / 2 = 143 Since the mean is greater than the median, the data appears to be positively skewed.

c. Drawing a box plot

To create a box plot, first calculate the quartiles: Q1 (first quartile) = median of the lower half of data = (8 + 52) / 2 = 30 Q3 (third quartile) = median of the upper half of data = (162 + 165) / 2 = 163.5 Interquartile range (IQR) = Q3 - Q1 = 163.5 - 30 = 133.5 Next, draw a box with the lower boundary at Q1 (30), the upper boundary at Q3 (163.5), and the median (143) represented as a line inside the box. Draw whiskers extending from the box to the minimum and maximum values (8 and 757). The box plot shows that the data is positively skewed, as there are more data points on the right side of the box plot. This box plot confirms our conclusion from part b, which indicates the data is positively skewed. The mean is greater than the median, and the box plot visually illustrates the skewness in the data distribution.

Key concepts

Data Symmetry

Data symmetry refers to how evenly data points are distributed around the center in a dataset. When data is symmetric, the left and right sides of the data distribution look roughly the same when graphed. In the exercise, by ordering the data, students can start looking for symmetry.

Here are some key points to keep in mind about data symmetry:

• Symmetric data often resembles a bell shape, meaning data points gradually increase, peak in the middle, and decrease symmetrically.
• Lack of symmetry might indicate skewness, often making one side "heavier" than the other.
• Analyzing symmetry visually is a good start, but you must use calculations or plots (like a box plot) for confirmation.

While it might be tough to assess solely from numbers, tools like box plots give a clearer picture of symmetry versus skewness.

Mean and Median

The mean and median are central statistical measures that help us understand the distribution of a dataset. In the exercise, calculating these two values reveals a lot about data symmetry and potential skewness.

**Understanding Mean and Median**

• Mean: The average of all data points. It is calculated by summing all values and dividing by the number of values.
• Median: The middle value in a data set when arranged in order. It represents the midpoint where half of the data lies below and half above.

In a symmetric distribution, the mean and median will be close or equal. If they differ notably, as in the exercise where the mean (198.625) is greater than the median (143), it points towards skewness in the data.

Box Plot

A box plot is a valuable tool in statistical analysis, providing a visual summary of the distribution of data. It highlights central values, spreads, and potential skewness, all in one glance.

**Constructing a Box Plot**

• Identify quartiles: Q1 (25th percentile), Q2 (50th percentile or median), and Q3 (75th percentile).
• Draw a box from Q1 to Q3 with a line at the median (Q2).
• Extend whiskers from the box to the smallest and largest data points within a calculated range.

In the exercise, the box plot reveals a rightward skew with more data clustering on the lower end (left side), confirming the skewed nature deduced from the mean and median.

Skewness

Skewness describes the direction and extent of asymmetry in a distribution. It's a critical concept as it affects how data is interpreted and suggests potential outliers.

**Recognizing Skewness**

• A data set is positively skewed (or right-skewed) when mean > median. This means a longer or fatter tail on the right side, as seen in the exercise.
• Negatively skewed (or left-skewed) data has a longer left tail, occurring when mean < median.
• No skew means the dataset is symmetric, with mean equal to median.

Understanding skewness helps in making sense of the data's distribution, forecasts future trends, and informs on potential limits or areas needing adjustment in statistical analyses. Given that the mean is higher than the median in our exercise, it clearly suggests positive skewness.