Question

A company's customer service hotline handles many calls relating to orders, refunds, and other issues. The company's records indicate that the median length of calls to the hotline is \(4.4\) minutes with an IQR of \(2.3\) minutes. a) If the company were to describe the duration of these calls in seconds instead of minutes, what would the median and IQR be? b) In an effort to speed up the customer service process, the company decides to streamline the series of pushbutton menus customers must navigate, cutting the time by 24 seconds. What will the median and IQR of the length of hotline calls become?

Short answer

The new median is 240 seconds, and the IQR remains 138 seconds.

Step-by-step solution

Convert Minutes to Seconds

First, we need to convert the median and IQR from minutes to seconds. Since there are 60 seconds in a minute, we will multiply both the median and the IQR by 60.- Median: \( 4.4 \text{ minutes} \times 60 = 264 \text{ seconds} \)- IQR: \( 2.3 \text{ minutes} \times 60 = 138 \text{ seconds} \)

Determine Median Time Change

The company plans to reduce the call duration by 24 seconds due to an improved menu. We need to subtract 24 seconds from the median duration in seconds.- New Median: \( 264 \text{ seconds} - 24 \text{ seconds} = 240 \text{ seconds} \)

Confirm IQR is Unchanged

The interquartile range (IQR) measures the spread of the middle 50% of data and is unaffected by changes to the central location like shifts in the median. Thus, the IQR remains the same.- IQR: Still \( 138 \text{ seconds} \)

Summary of Results

After converting the values to seconds and adjusting for the procedural change, the new median and IQR are: - New Median: 240 seconds - IQR: 138 seconds

Key concepts

Understanding the Median

The median is a type of average that gives us the middle value in a list of numbers. It is a very important statistical measure, especially in customer service data like call durations. In our exercise, the median call length was initially 4.4 minutes. This means that half the calls are shorter than 4.4 minutes, and the other half are longer. To find the median, you must first order all the data from smallest to largest. If there's an odd number of data points, the median is the middle one. If there's an even number, it's the average of the two middle numbers. This measure is particularly useful because it is not affected by extremely large or small values, known as outliers. These can skew data significantly when using the mean, but the median gives a better sense of a typical call length here. The exercise asks us to convert this median from minutes to seconds, emphasizing its flexibility and practical application. It easily adjusts with the data's measurement unit, staying applicable whether in minutes or seconds.

Conversion Units: From Minutes to Seconds

Conversion units are crucial when comparing or analyzing data in different forms. The ability to convert appropriately ensures you can understand and interpret data consistently across various contexts. In our problem, the median call duration is 4.4 minutes, which needs to be expressed in seconds. Since there are 60 seconds in a minute, we can multiply the number of minutes by 60 to convert it to seconds. Here, the median of 4.4 minutes becomes 264 seconds because \[ 4.4 \times 60 = 264 \]Similarly, the interquartile range (IQR), initially 2.3 minutes, becomes 138 seconds after conversion, calculated as:\[ 2.3 \times 60 = 138 \] This type of conversion ensures you're accurately working with data in contexts where time might be tracked differently, and it maintains the same relationships and comparisons across different units. Understanding this concept is essential for accurate data representation and helps in making dependable conclusions.

Data Spread: Understanding IQR

The interquartile range (IQR) is a measure of data spread, providing insight into the variability within a dataset. Specifically, the IQR focuses on the middle 50% of the data, which involves the difference between the third quartile (Q3) and the first quartile (Q1). In our problem, the initial IQR in minutes is 2.3 minutes, indicating a moderately narrow spread during customer service calls. Calculating the spread in a dataset is crucial because it helps organizations understand how consistent or varied their data is. A small IQR suggests that data points are close to the median, signifying consistency. Conversely, a large IQR implies more variability. In this exercise, after converting the IQR to seconds, it remains unchanged by the additional seconds reduced from the process. This consistency highlights an important trait of IQR — it measures the spread irrespective of specific changes to the central location, such as adjustments to the median. Therefore, even after a procedural change aiming to speed up customer calls, the IQR in seconds stays at 138, showing that the call durations' relative spread remains stable.