Question

A botanist grew 15 pepper plants on the same greenhouse bench. After 21 days, she measured the total stem length \((\mathrm{cm})\) of each plant, and obtained the following values: \(^{53}\) $$ \begin{array}{lll} 12.4 & 12.2 & 13.4 \\ 10.9 & 12.2 & 12.1 \\ 11.8 & 13.5 & 12.0 \\ 14.1 & 12.7 & 13.2 \\ 12.6 & 11.9 & 13.1 \end{array} $$ (a) Calculate all three quartiles. (b) Compute the lower fence and the upper fence of the distribution. (c) How large would an observation in this data set have to be in order to be an outlier?

Short answer

Quartiles need to be calculated and the fences determined to find outliers. An observation would need to be less than the lower fence or greater than the upper fence to be classified as an outlier.

Step-by-step solution

- Order the data

First, arrange the stem lengths of the pepper plants in ascending order to prepare for quartile calculation.

- First Quartile (Q1) Calculation

Locate the median of the first half of the data set. For an odd number of data points, include the median in both halves of the data. For an even number, do not include the median. Since there are 15 data points, which is odd, the median will be included in both halves. Q1 will be the median of the first 8 data points.

- Third Quartile (Q3) Calculation

Locate the median of the second half of the data set. Q3 will be the median of the last 8 data points.

- Calculate the interquartile range (IQR)

Subtract Q1 from Q3 to find the IQR, which is the range within which the central 50% of the data lies.

- Lower Fence Calculation

Calculate the lower fence by subtracting 1.5 times the IQR from Q1. Any data point less than the lower fence is considered an outlier.

- Upper Fence Calculation

Calculate the upper fence by adding 1.5 times the IQR to Q3. Any data point greater than the upper fence is considered an outlier.

- Determine Outlier Threshold

The smallest observation larger than the upper fence will be considered an outlier. Similarly, the largest observation smaller than the lower fence will also be considered an outlier.

Key concepts

Data Set Organization

When working with statistical data, it's crucial to begin by organizing your data set. This helps to provide a clear overview of the data and lays the foundation for further analysis. In the exercise provided, the botanist has collected stem lengths of 15 pepper plants, which initially appear in no particular order.

To organize the data, we start by arranging the values in ascending order. This process not only gives us a better understanding of the range of measurements but is also a necessary step before we can calculate statistical measures such as the median and quartiles. In a typical data set organization, we would identify the smallest and largest values, calculate the range, and then move on to find the median, which is the middle value that divides the ordered data set into two equal halves. In the botanist's study, the median gives a sense of the central tendency of stem lengths across all plants.

Once organized, this data allows for a more transparent calculation of the quartiles, leading us to interpret the data's distribution accurately.

Interquartile Range Calculation

The interquartile range (IQR) is a measure of statistical dispersion and is essential in understanding the spread of the central half of the data points. To calculate the IQR, we need the first quartile (Q1) and the third quartile (Q3). Q1 is the median of the lower half of the data set, and Q3 is the median of the upper half.

In the given exercise, after lining up the data points in ascending order, the botanist divides the data set into two halves. Since the number of data points is odd, the median is included in both halves. Q1 is found from the median of the first 8 data points, while Q3 is the median of the last 8. The IQR is then found by subtracting Q1 from Q3 (\( \text{IQR} = Q3 - Q1 \)). It represents the range within which the middle 50% of the data lies, providing insight into the data's variability. This range is crucial for determining the normal variation in data, as well as identifying any outliers.

Outlier Determination

Identifying outliers is critical as they can indicate data points that differ significantly from the rest of the data set. These may be due to variability in the measurement or may indicate experimental error. In the context of our exercise, outliers can show which pepper plants behave abnormally in terms of growth.

To determine outliers, first, we calculate the 'fences' using the IQR. The lower fence is Q1 minus 1.5 times the IQR, and the upper fence is Q3 plus 1.5 times the IQR. Any data points that fall outside of these fences are considered outliers. This 1.5 IQR rule is a standard measure that signifies a data point is significantly higher or lower than the bulk of the data set.

After calculating the fences, the botanist examines the data points to find any that are below the lower fence or above the upper fence. These points would be classified as outliers. By identifying outliers, researchers like our botanist can make informed decisions about which data points to investigate further or possibly exclude from analysis to avoid skewing the results.