Question

A veterinary anatomist investigated the spatial arrangement of the nerve cells in the intestine of a pony. He removed a block of tissue from the intestinal wall, cut the block into many equal sections, and counted the number of nerve cells in each of 23 randomly selected sections. The counts were as follows. $$ \begin{array}{llllllll} 35 & 19 & 33 & 34 & 17 & 26 & 16 & 40 \\ 28 & 30 & 23 & 12 & 27 & 33 & 22 & 31 \\ 28 & 28 & 35 & 23 & 23 & 19 & 29 & \end{array} $$ (a) Determine the median, the quartiles, and the interquartile range. (b) Construct a boxplot of the data.

Short answer

The median is the middle value once sorted, Q1 is the median of the first half of the data, Q3 is the median of the second half, and IQR is Q3 minus Q1. A boxplot is constructed using these statistics.

Step-by-step solution

Organize the Data

First, sort the data in ascending order for ease of calculation.

Find the Median

Determine the middle value of the dataset, which is the median.

Find the First Quartile (Q1)

Identify the median of the first half of the data to find the first quartile (Q1).

Find the Third Quartile (Q3)

Identify the median of the second half of the data to find the third quartile (Q3).

Calculate the Interquartile Range (IQR)

Subtract the first quartile (Q1) from the third quartile (Q3) to get the interquartile range (IQR).

Construct the Boxplot

Use the obtained values (minimum, Q1, median, Q3, and maximum) to draw a boxplot.

Key concepts

Median

The median is a form of average, more specifically, the middle value in a dataset after it has been ordered from lowest to highest. If the data set has an odd number of observations, the median is the number that sits right in the middle. However, if there is an even number of observations, the median is found by calculating the average of the two middle numbers.

In our example, the median helps the veterinary anatomist understand the typical count of nerve cells in the sections of pony intestine by identifying the central tendency of his observations. To find this, the data is organized in ascending order and is then split into two equal parts. The anatomist would find the value that falls in the middle of this ordered set, or if the count was even, the average of the two central values, to determine the median nerve cell count.

Quartiles

Quartiles are values that divide a dataset into four equal parts, providing a deeper understanding of the distribution of data. The three quartiles are commonly referred to as:

• First Quartile (Q1) - This is the median of the lower half of the dataset, excluding the median if there's an odd number of data points.
• Second Quartile (Q2) - This is essentially the median of the entire dataset.
• Third Quartile (Q3) - This is the median of the upper half of the dataset, also excluding the median for odd-sized datasets.

The veterinary anatomist in our exercise calculates Q1 and Q3 to assess the variability in the count of nerve cells. These values indicate the lower and upper quarters of his data, helping to understand where most of the data points lie with respect to the entire range.

Interquartile Range

The interquartile range (IQR) is a measure of statistical dispersion that is represented by the difference between the third and first quartiles (IQR = Q3 - Q1). It describes the middle 50% of the data and is useful in showing how spread out the data points are around the median.

For the anatomist's research, the IQR helps to identify the spread of nerve cell counts around the central value, excluding the outliers that can skew the understanding of what a 'typical' section might look like. A small IQR suggests that the nerve cells are fairly consistently distributed among the sections, while a large IQR would indicate more variability.

Boxplot

A boxplot, also known as a box and whisker plot, graphically displays a dataset's distribution through five-summary statistics: the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and the maximum. It consists of a box from Q1 to Q3, with a line inside the box indicating the median, and 'whiskers' that extend from the box to the minimum and maximum values.

The boxplot created by the veterinary anatomist allows for immediate visual analysis of the data. It enables the identification of the central spread and any potential outliers, which are points that lie beyond the whiskers – often cases worth individual consideration. The clear graphical representation makes it easy to compare different datasets and can reveal symmetry, skewness, and the presence of outliers at a glance.