Question

A popular band on tour played a series of concerts in large venues. They always drew a large crowd, averaging 21,359 fans. While the band did not announce (and probably never calculated) the standard deviation, which of these values do you think is most likely to be correct: \(20,200,2000\), or 20,000 fans? Explain your choice.

Short answer

The most plausible standard deviation is 2,000 fans.

Step-by-step solution

Understanding the Problem

We need to determine which standard deviation is realistic given an average (mean) attendance of 21,359 fans. The standard deviation measures the spread of data around this average.

Considering the Magnitude

Reflect on the size of each standard deviation option relative to the mean. A standard deviation close to the mean (like 20,000) suggests an extreme variation, indicating significant deviations from the average at times, which is unrealistic for concert attendance data.

Evaluating Each Option

Examine each standard deviation value: - A value of 20 is too low given the context, suggesting almost no variation, which is unlikely for large events. - A value of 200 could indicate moderate and realistic variations in attendance numbers due to factors like venue size or geographic location. - A value of 2,000 indicates large variations, which could be plausible for international tours. - A value of 20,000 suggests extreme variation where it would be possible to have a concert with zero or possibly double the crowd size, which is not plausible.

Selecting the Most Reasonable Standard Deviation

Based on our evaluation, 2000 is the most plausible standard deviation. It reflects a reasonable range of variation given the average attendance, allowing for different venue sizes, locations, and potential local popularity fluctuations.

Key concepts

Standard Deviation

The concept of standard deviation is central in statistics for understanding the variance or spread in a set of data. Imagine you have attendance figures from several band concerts, each drawing slightly different numbers of fans. Instead of just knowing the average number of fans, the standard deviation gives insight into how much these figures fluctuate around that average.

A small standard deviation means the numbers are close to the average. In the context of a concert tour, this would suggest every show has nearly the same number of attendees. On the other hand, a large standard deviation means there is a wide variation in the numbers, so some concerts might be much bigger or smaller than others.

When you consider standard deviation values like 20, 2000, or 20,000 for a band with an average attendance of 21,359, you think about how those numbers represent differences from the typical crowd size. For instance, a value of 2000 suggests some concerts might be larger or smaller by up to a couple of thousand fans, which is reasonable given variances in venue capacity and ticket sales in different regions.

Mean

The mean, also commonly referred to as the average, is a fundamental statistical concept used to summarize a set of data with a single number. It is calculated by adding all the numbers in a dataset and then dividing by the number of entries in that dataset. In the context of our concert attendance example, the mean attendance of 21,359 fans represents the central point around which the number of attendees for each concert revolves.

Understanding the mean provides a snapshot of overall performance or trend. For the band, a mean attendance of 21,359 suggests that, on average, this is the crowd size they can expect, regardless of specific circumstances at each individual venue.

Given a mean of 21,359 fans, comparing potential standard deviations helps gauge whether fluctuations in attendance are typical or outliers. If one concert has considerably more or fewer fans, standard deviation helps indicate if such a difference is an anomaly or within a typical range of variation.

Data Variation

Data variation refers to how much the numbers in a dataset differ from one another and from the average (mean). This is important for any band or event organizer to understand because it affects planning and management.

Several factors can contribute to variation in concert attendance, including geographic location, day of the week, ticket prices, marketing efforts, and competing events. Analyzing data variation helps organizers understand and possibly predict performance for future events.

In terms of the original exercise, choosing a standard deviation of 2000 indicates acknowledging and responding to these variances. It suggests that while overall attendance might hover around the mean, certain concerts could see notable differences in crowd size, yet still remain within manageable expectations.