Question

C-sections A study in Switzerland examined the number of cesarean sections (surgical deliveries of babies) performed in a year by samples of male and female doctors. Here are summary statistics for the two distributions:

(a) Based on the computer output, which distribution would you guess has a more symmetrical shape? Explain your answer.

(b) Explain how the IQRs of these two distributions can be so similar even though the standard deviations are quite different.

(c) Does either distribution have any outliers? Justify your answer.

Short answer

Part (a) Female doctors distribution.

Part (b) The standard deviation is not resistant, however, the interquartile range is.

Part (c) In male doctors.

Step-by-step solution

Part (a) Step 1. Given information.

The given information is:

Part (a) Step 2. Guess the shape of the distributions.

The values of mean and median for the given data about male doctors are 41.333 and 34 respectively.

We can see that the mean value is higher than the median, from it we can conclude that the skewness of the distribution is towards the right.

Its reason is that the mean may be influenced by the unusual data values if we compare it with the median making the mean value larger than the median.

The values of mean and median for the given data about female doctors are 19.1 and 18.5 respectively.

The mean and median for this data are very close, indicating that the distribution is very symmetric.

Its reason is that the mean may be influenced by the unusual data values if we compare it with the median we will find no unusual large or small data values.

Therefore, female doctors have a distribution with more symmetry.

Part (b) Step 1. Explain the similarity between the IQRs of these two distributions.

The standard deviation is not resistant to change, although the IQR is.

This means that outliers have a substantial influence on the standard deviation, but not on the interquartile range (IQR), which is nevertheless influenced by outliers.

When the distributions (or one of the distributions) contain outliers, the standard deviations can be substantially different yet the interquartile ranges IQRs are similar.

Part (c) Step 1. Identify any outliers in the distributions.

Observations that are more than 1.5 times the IQR above the third quarter or below the first quarter are considered outliers.

Male Doctors:

$$\begin{gathered} IQR=50-27=23 \\ {Q}_3+1.5IQR=50+1.5\left(23\right) \\ =84.5 \\ {Q}_1-1.5IQR=27-1.5\left(23\right) \\ =-7.5 \end{gathered}$$

We know that the maximum value is 86 which is larger than 84.5 making 86 an outlier.

Therefore, we can say that outliers are present in this given data.

Female Doctors:

$$\begin{gathered} IQR=29-10=19 \\ {Q}_3+1.5IQR=29+1.5\left(19\right) \\ =57.5 \\ {Q}_1-1.5IQR=10-1.5\left(19\right) \\ =-18.5 \end{gathered}$$

We can say that there are no outliers are present in this given data.