Question

A company selling clothing on the Internet reports that the packages it ships have a median weight of 68 ounces and an IQR of 40 ounces. a) The company plans to include a sales flyer weighing 4 ounces in each package. What will the new median and IQR be? b) If the company recorded the shipping weights of these new packages in pounds instead of ounces, what would the median and IQR be? $(1\mathrm{lb}.=16\mathrm{oz}$.)

Short answer

New median: 72 ounces (4.5 pounds), new IQR: 40 ounces (2.5 pounds).

Step-by-step solution

Understand the effect of adding a constant weight

When a constant weight is added to all packages, the median weight will increase by that constant value, but the interquartile range (IQR) will remain unchanged. This is because the IQR is a measure of spread, and adding a constant does not affect how spread out the data is.

Calculate new median with flyer included

The original median weight of the packages was 68 ounces. With a sales flyer weighing 4 ounces added to each package, the new median weight is calculated by:$$68+4=72$$ ounces.

Observe the change in IQR

The original IQR of the package weights was 40 ounces. Since the IQR measures the middle spread of the data and is unaffected by adding a constant weight to each data point, the IQR remains:$$\text{IQR}=40$$ ounces.

Convert ounces to pounds for the new median

Since 1 pound equals 16 ounces, convert the new median weight of 72 ounces into pounds:$$72\text{ ounces}\times \frac{1\text{ pound}}{16\text{ ounces}}=4.5\text{ pounds}$$

Convert IQR from ounces to pounds

The IQR in ounces is 40 ounces. Converting this to pounds involves:$$40\text{ ounces}\times \frac{1\text{ pound}}{16\text{ ounces}}=2.5\text{ pounds}$$

Key concepts

Understanding Median

The median is the value that lies at the midpoint of a data set. This means half of the values are lower than the median, and half are higher. It's a useful measure of central tendency when the data includes outliers or is skewed, as it is less affected by extreme values than the mean.

• When you add a constant value to each data point in your dataset, the median itself will increase by that constant value.
• For example, if the median weight of packages is 68 ounces and a flyer weighing 4 ounces is added to every package, the new median becomes 72 ounces, reflecting the added weight.
• Importantly, the change in units, such as converting ounces to pounds, will also change the median proportionally according to the conversion factor. For instance, converting 72 ounces to pounds (since 1 pound = 16 ounces) yields a new median of 4.5 pounds.

Understanding the median helps provide a clearer picture of your data set's central value, particularly when planning and predicting future changes or requirements in data characteristics.

Interquartile Range Exploration

The interquartile range (IQR) measures the middle 50% of a dataset. It's the difference between the third quartile (Q3) and the first quartile (Q1), capturing the spread of the central half of the data. This statistic is particularly useful as it focuses on data dispersion and is robust against outliers.

• Adding a constant to each value in your data will leave the IQR unchanged. This is because the IQR expresses the difference between two quartiles, and a constant addition affects both quartiles equally, thereby not affecting their difference.
• For instance, if an IQR of package weights is 40 ounces, this value remains 40 ounces even after the addition of a 4-ounce flyer to each package.
• When converting units from ounces to pounds, the conversion factor similarly applies to the IQR. Thus, an IQR of 40 ounces translates to 2.5 pounds, reflecting the same relative spread of the data in different units.

Using the IQR helps you analyze the variability and spread of data effectively, especially when preparing for potential shifts or additions in the data.

The Process of Data Transformation

Data transformation involves changing the format, structure, or appearance of data. Common transformations include unit conversions or adding constants to dataset values. These processes can directly affect calculations or interpretations of statistical measures like the median and IQR.

• Adding a constant to each data point simply shifts the median by that constant without changing the dataset's spread, which means tools like the IQR remain static.
• Unit conversion, such as turning measurements from ounces to pounds, reforms each data point by the conversion factor used. This changes both the median and IQR numerically but preserves the data's relative distribution.
• Such transformations are essential in practical scenarios where data needs to be presented in the most relevant or understandable form, like converting weights from ounces to pounds for easier comprehension in contexts like shipping or inventory.

Understanding data transformation is crucial for accurately transforming and interpreting data in real-world applications, ensuring clarity and precision.