Question

The Cornell Lab of Ornithology holds an annual Christmas Bird Count (www.birdsource.org), in which bird watchers at various locations around the country see how many different species of birds they can spot. Here are some of the counts reported from sites in Texas during the 1999 event: \(\begin{array}{lllllll} 228 & 178 & 186 & 162 & 206 & 166 & 163 \\ 183 & 181 & 206 & 177 & 175 & 167 & 162 \\ 160 & 160 & 157 & 156 & 153 & 153 & 152 \end{array}\) a) Create a stem-and-leaf display of these data. b) Write a brief description of the distribution. Be sure to discuss the overall shape as well as any unusual features.

Short answer

Stem-and-leaf plot shows a slightly left-skewed distribution with most counts around 160s and 170s.

Step-by-step solution

Organize the Data

The first step is to organize the data into a list sorted in ascending order. This will help in creating the stem-and-leaf plot. The sorted data are: 152, 153, 153, 156, 157, 160, 160, 162, 162, 163, 166, 167, 175, 177, 178, 181, 183, 186, 206, 206, 228.

Determine Stems and Leaves

In a stem-and-leaf plot, each number is split into a stem (the leading digit(s)) and a leaf (the last digit). Here, the tens digit forms the stem, and the units digit is the leaf. For example, in the number 152, the stem will be 15, and the leaf will be 2.

Construct the Stem-and-Leaf Plot

List the stems in a vertical column in ascending order and attach each leaf to the appropriate stem. The stem-and-leaf plot is as follows: 15 | 2 3 3 6 7 16 | 0 0 2 2 3 6 7 17 | 5 7 8 18 | 1 3 6 20 | 6 6 22 | 8

Analyze the Distribution

The distribution of the data is slightly skewed left, with the majority of counts centered around the 160s and 170s. There is a cluster of data points in the 160s and 170s, indicating that bird counts were most frequently in this range during the 1999 event. The presence of a couple of higher counts, like 206 and 228, however, adds some skewness, but not enough to classify it as strongly skewed.

Key concepts

Data Analysis

Data analysis is a vital process that helps us make sense of numerical information, such as the bird count data from the Christmas Bird Count event. By organizing and representing this data visually, we can draw insights and better understand its features. One effective method used in data analysis is the stem-and-leaf plot. This plot helps us maintain the original data while also providing a visual display that is easier to read and interpret than raw numbers.
To create a stem-and-leaf plot, start by listing the data in ascending order. This makes it simple to see patterns and trends. In our dataset, we have numbers ranging from 152 to 228. Each number is divided into a 'stem' and a 'leaf.' Typically, the stem consists of all but the last digit (usually the tens place), while the leaf is the final digit (the ones place).
Using this method, we can easily categorize the data. For example, numbers in the 150s have a stem of 15, and their unit digits become the leaves. This organization allows for quick identification of the number of counts that fall within specific ranges, making it easier to analyze and derive conclusions about the data set.

Distribution Shape

The shape of a distribution gives us insights into how data points are spread across different values. In our bird count data, the shape of the distribution is shown through the arrangement of numbers in the stem-and-leaf plot. Observing the distribution shape is key to interpreting the data's overall pattern and any exceptions present.
In this exercise, most of the bird counts fall within the 160s and 170s. This indicates that a majority of participating locations reported bird counts in this range. The distribution of these counts forms a central cluster, suggesting a concentration or peak in this interval. This form of distribution can be described as having a central tendency, where the bulk of data points are gathered around the median value.
By closely observing the distribution shape, we can get a sense of consistency and any variations among the datasets. Identifying if the distribution is uniform, unimodal, bimodal, or multimodal helps in making informed interpretations. A unimodal distribution, like in our example, indicates a single peak or most common value, which is crucial for effective data analysis.

Skewness

Skewness in a dataset refers to the asymmetry or tilt of the data points around its center, which influences how data is distributed. In this context, skewness helps us assess whether the distribution leans more to one side, impacting conclusions and interpretations.
In our bird count example, the data is slightly skewed to the left. This means that there are some higher value outliers in the 200s and 220s (like 206 and 228) that stretch the distribution towards the lower end. However, the left skewness here is mild, indicating that while there are extreme values, they don't heavily distort the overall distribution's shape.
Understanding skewness helps identify any anomalies in our data. A skewed distribution may affect statistical measures, such as the mean, median, and mode, making it critical to consider skewness during data analysis. Recognizing whether a distribution is left-skewed, right-skewed, or symmetrical allows us to use appropriate statistical tools to assess and predict patterns accurately.