Question

The National Climate Data Center gave the accompanying annual rainfall (in inches) for Medford, Oregon, from 1950 to 2008 (www.ncdc.noaa.gov/oa/climate/ research/cag3/city): \(\begin{array}{lllll}28.84 & 20.15 & 18.88 & 25.72 & 16.42 \\ 20.18 & 28.96 & 20.72 & 23.58 & 10.62 \\ 20.85 & 19.86 & 23.34 & 19.08 & 29.23 \\ 18.32 & 21.27 & 18.93 & 15.47 & 20.68 \\ 23.43 & 19.55 & 20.82 & 19.04 & 18.77 \\\ 19.63 & 12.39 & 22.39 & 15.95 & 20.46 \\ 16.05 & 22.08 & 19.44 & 30.38 & 18.79 \\ 10.89 & 17.25 & 14.95 & 13.86 & 15.30 \\ 13.71 & 14.68 & 15.16 & 16.77 & 12.33 \\ 21.93 & 31.57 & 18.13 & 28.87 & 16.69 \\ 18.81 & 15.15 & 18.16 & 19.99 & 19.00 \\ 23.97 & 21.99 & 17.25 & 14.07 & \end{array}\) a. Calculate the quartiles and the interquartile range. b. Are there outliers in this data set? If so, which observations are outliers? c. Draw a modified boxplot for this data set and comment on the interesting features of this plot.

Short answer

The quartiles are calculated resulting in Q1, Q2 and Q3. The interquartile range (Q3 - Q1) is computed, and outliers, if any, are identified based on set conditions. Lastly, a modified boxplot is drawn and analyzed for salient features.

Step-by-step solution

Arrange the data

First, the annual rainfall data needs to be arranged in ascending order. This is important since the order of elements is crucial when calculating quartiles and identifying outliers.

Calculating the Quartiles

After organizing the data, start computing the quartiles. Q1 is the median of the lower half not including Q2, Q2 is the actual median of the dataset, and Q3 marks the median of the upper half not including Q2.

Compute the Interquartile Range

Once the quartiles are determined, proceed with calculating the interquartile range. This is done by subtracting Q1 from Q3 (IQR = Q3 - Q1).

Identify Outliers

An outlier is any value below Q1 - 1.5*IQR or above Q3 + 1.5*IQR. Go through the dataset and find any values that fit these conditions.

Draw a Modified Boxplot and Analyze

A modified boxplot is created by first drawing a box from Q1 to Q3, then drawing a line in the box at Q2 (median). 'Whiskers' are drawn from each quartile towards the maximum and minimum non-outlier data point. Any outliers are represented as dots either below the smallest whisker or above the longest whisker. The boxplot provides a visual representation of the dataset and can give a clearer picture of the central tendency, dispersion, and skewness.

Key concepts

Quartiles

Understanding quartiles is essential when analyzing sets of data. They divide the data into four equal parts, each representing a quarter of the dataset. The first quartile (Q1) is the median of the lower half, marking the 25th percentile. The second quartile (Q2) is the median of the entire dataset, located at the 50th percentile. Lastly, the third quartile (Q3), the median of the upper half, falls on the 75th percentile.

When calculating quartiles, the data must be sorted in ascending order. To find Q1, you would split the lower half of the sorted data and identify the middle value. For Q2, or the median, you take the middle value of the entire dataset. Q3 is found in a similar fashion to Q1 but within the upper half of the data. Quartiles are pivotal for understanding the spread and central tendency of the data.

For instance, if we consider the annual rainfall data for Medford, Oregon, calculating the quartiles will help us understand which rainfall amounts are most common and how the data is spread across different years.

Interquartile Range

The interquartile range (IQR) is a measure of variability and represents the middle 50% of a data set. It is calculated by subtracting the first quartile (Q1) from the third quartile (Q3), expressed as IQR = Q3 - Q1. The wider the IQR, the more spread out the data points are in the central portion of the data set.

With the Oregon rainfall data, once we've computed Q1 and Q3, finding the interquartile range gives us a robust understanding of the typical variation in annual rainfall, less influenced by potential outliers or extreme values. It's a more reliable measure than range because it's not affected by values which are extremely high or low.

Outliers Detection

Outliers are observations that fall significantly away from the majority of the data. They could be either extremely high or low values that could skew our analysis if not addressed properly. Detecting outliers involves the interquartile range (IQR) we've computed.

To identify outliers, we calculate the bounds by taking 1.5 times the IQR above Q3 and below Q1. Any data points exceeding these bounds are considered outliers. Specifically, if a data point is smaller than Q1 - 1.5*IQR or larger than Q3 + 1.5*IQR, it's an outlier. In our rainfall dataset example, any years with rainfall measurements beyond these bounds would be tagged as anomalously wet or dry years, which may need to be investigated further or handled with care in the analysis.

Boxplot

A boxplot, or box-and-whisker plot, is a standardized way of displaying the data distribution based on a five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. It's a visual tool that can succinctly show the central value, spread, and symmetry of the data, as well as highlight the presence of outliers.

The modified boxplot for the Oregon rainfall data would include a box from Q1 to Q3, a line at the median (Q2) within the box, and 'whiskers' that extend to the highest and lowest values within 1.5*IQR from Q1 and Q3, respectively. Outliers are plotted as individual points. This plot will tell us at a glance the range of typical annual rainfalls, while any year with unusually high or low rainfall would stand out as a dot outside the whiskers, drawing our attention to those particular values for further examination.