Question

In a study of milk production in sheep (for use in making cheese), a researcher measured the 3 -month milk yield for each of 11 ewes. The yields (liters) were as follows: \(^{28}\) $$ \begin{array}{llllrl} 56.5 & 89.8 & 110.1 & 65.6 & 63.7 & 82.6 \\ 75.1 & 91.5 & 102.9 & 44.4 & 108.1 & \end{array} $$ (a) Determine the median and the quartiles. (b) Determine the interquartile range. (c) Construct a boxplot of the data.

Short answer

Median (Q2) is 82.6 liters, Q1 is 65.6 liters, Q3 is 102.9 liters, and the interquartile range (IQR) is 37.3 liters.

Step-by-step solution

Sort the Data

First, arrange the milk yields in ascending order. Correctly ordering the data is necessary for finding medians and quartiles.

Find the Median (Q2)

With the data in order, locate the median, which is the middle number when there are an odd number of data points or the average of the two middle numbers when there are an even number of data points.

Find the First Quartile (Q1)

The first quartile, Q1, is the median of the lower half of the data (not including the median if there is an odd number of data points).

Find the Third Quartile (Q3)

Likewise, the third quartile, Q3, is the median of the upper half of the data (not including the median if there is an odd number of data points).

Determine the Interquartile Range

Subtract the value of the first quartile from the value of the third quartile to calculate the interquartile range (IQR).

Construct the Boxplot

Use the calculated medians and quartiles to draw a boxplot representing the spread of the data. The box spans from Q1 to Q3 with a line at the median. Whiskers extend from the box to the smallest and largest data points.

Key concepts

Median Calculation

The median serves as a measure of central tendency that identifies the midpoint of a given dataset. To calculate the median, follow these basic steps: Arrange the data in numerical order. If the number of observations is odd, the median is the middle value. For example, in a dataset of 11 values, the 6th value is the median after sorting. If the number of observations is even, calculate the median by averaging the two middle values.

Using the sheep milk yield data, after sorting the values, we find the 6th value in our 11-data point series to be the median. This value is a crucial part of a boxplot as it divides the box into two equal parts, reflecting the central tendency of the dataset.

Quartiles Calculation

Quartiles divide a data set into four equal parts. To calculate the quartiles:

• First Quartile (Q1): Also called the lower quartile, it is the median of the lower half of the dataset.
• Second Quartile (Q2): This is the median of the dataset.
• Third Quartile (Q3): The upper quartile, it is the median of the upper half of the dataset.

Step back to our example, since we have 11 observations, we exclude the median and find the medians of the lower 5 and upper 5 data points to determine Q1 and Q3, respectively. These values help in understanding the spread and skewness of the dataset.

Interquartile Range

The interquartile range (IQR) is a measure of statistical dispersion and is calculated by subtracting Q1 from Q3: \( IQR = Q3 - Q1 \). It essentially captures the middle 50% of the dataset. The IQR is also used in the construction of boxplots and is essential in identifying outliers. In the context of our milk yield data, once the quartiles are identified, the IQR is easily calculated and provides insight into the variability of the milk production among the ewes.

Data Visualization in Statistics

Data visualization in statistics involves the representation of data in a graphical format to make complex information easier to understand. A boxplot, also known as a box-and-whisker plot, is a visual tool for displaying the distribution of a dataset. It shows the median, quartiles, and potential outliers. The 'box' displays the central 50% of the data, bound by Q1 and Q3, and the 'whiskers' extend to show the range of the data, typically up to 1.5 times the IQR beyond the quartiles. Any data points outside of this range can be considered outliers and are plotted as individual points. In our milk yield study, the boxplot will visualize the spread and central tendency effectively.

Descriptive Statistics

Descriptive statistics summarize and describe the features of a dataset. They provide simple summaries about the sample and the measures. Key measures include:

• Measures of central tendency (mean, median, mode).
• Measures of variability (range, variance, standard deviation, IQR).
• Measures of position (percentiles, quartiles).

These statistics are foundational for the boxplot construction, as they shape our understanding of the data's central point and spread. In the sheep milk production study, using descriptive statistics allows for a precise and succinct description of the milk yield data.