Question

A specialty foods company sells "gourmet hams" by mail order. The hams vary in size from \(4.15\) to \(7.45\) pounds, with a mean weight of 6 pounds and standard deviation of \(0.65\) pounds. The quartiles and median weights are \(5.6,6.2\), and \(6.55\) pounds. a) Find the range and the IQR of the weights. b) Do you think the distribution of the weights is symmetric or skewed? If skewed, which way? Why?

Short answer

a) Range: 3.3 pounds, IQR: 0.95 pounds. b) Distribution is skewed left (median > mean).

Step-by-step solution

Calculate the Range

The range of a data set is the difference between the maximum and minimum values. Given the weights range from 4.15 to 7.45 pounds, the formula for range is: \[ \text{Range} = \text{Maximum} - \text{Minimum} = 7.45 - 4.15 = 3.3 \text{ pounds} \].

Calculate the Interquartile Range (IQR)

The IQR is the difference between the third quartile (Q3) and the first quartile (Q1). From the problem, these quartiles are given as 6.55 pounds (Q3) and 5.6 pounds (Q1). The formula is: \[ \text{IQR} = Q3 - Q1 = 6.55 - 5.6 = 0.95 \text{ pounds} \].

Determine the Symmetry of the Distribution

To determine symmetry, we compare the median to the mean. The median is 6.2 pounds, and the mean is 6 pounds. Since the median (6.2) is greater than the mean (6), the distribution is likely skewed left. Additionally, the placement of the quartiles Q1 and Q3 supports this, as the lower quartile Q1 is closer to the median than Q3 is.

Key concepts

Understanding Data Distribution

In statistical analysis, understanding data distribution is vital as it gives us insights into how data points are spread across different values. When we talk about data distribution, we're referring to the way in which data is arranged.
The distribution could be symmetric, skewed, or take another shape.
With symmetric distributions, data points are evenly spread around the center (mean or median). On the other hand, a skewed distribution means the data is not evenly spread. Among the characteristics of skewness, data might have a tail longer on one side than on the other.
- **Uniform distribution**: All data points have equal frequency.
- **Normal distribution (bell curve)**: Data is symmetrically distributed about the mean.
- **Skewed distribution**: More data points lie on one side of the mean or median.
In our "gourmet hams" example, although the mean weight of 6 pounds and median of 6.2 pounds are close, the slight difference can indicate skewness. The placement of quartiles suggests the distribution might be skewed left, highlighting the importance of data distribution in analysis.

Exploring Range and Interquartile Range

When analyzing data, especially weights or measures, quantifying variability with the range and interquartile range (IQR) is key.
**Range** gives us a basic picture of spread by providing the difference between the largest and smallest data points. It's useful for a quick, high-level view, but can be sensitive to outliers. The calculation for the range in our example is simple:

• Maximum weight: 7.45 pounds
• Minimum weight: 4.15 pounds
• Range = 7.45 - 4.15 = 3.3 pounds

**Interquartile Range (IQR)** offers a more robust measure of variability by focusing on the middle 50% of data. It helps to understand the spread without being affected by outliers. For the given ham weights, the first quartile (Q1) and third quartile (Q3) are key:

• First Quartile (Q1): 5.6 pounds
• Third Quartile (Q3): 6.55 pounds
• IQR = Q3 - Q1 = 6.55 - 5.6 = 0.95 pounds

By using both measurements, we're better equipped to understand how data points are distributed around the central value.

Identifying Skewness in Data

Skewness is a critical concept in statistics that helps us understand the asymmetry in data distribution. When data is skewed, it means the bulk of data values lie on one side of the mean more than the other.
There are two types of skewness:

• **Positive skew (right skew)**: The tail on the right side is longer or fatter than the left side. This means most data points cluster at the lower end.
• **Negative skew (left skew)**: The tail on the left side is longer or fatter than the right side, indicating that most data points cluster at the higher end.

To determine skewness in a dataset, one method is to compare the mean and median. If the mean is greater than the median, expect a right skew. If the mean is less than the median, expect a left skew.
In the gourmet hams example, the mean is 6 pounds, and the median is 6.2 pounds. Because the median exceeds the mean, we suspect a slight left skew. Observing the quartiles further supports this, where the lower quartile is closer to the median, implying a concentration of higher values skewing leftward.
Overall, identifying skewness can provide deeper insights into data characteristics, beyond central tendencies, helping in more effective data analysis and decision-making.