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Chapter 3: Problem 13

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, \((d)\) midrange, and then answer the given question. Listed below are measured amounts of caffeine (mg per 12 oz of drink) obtained in one can from each of 20 brands (7-UP, A\&W Root Beer, Cherry Coke, .. Tab). Are the statistics representative of the population of all cans of the same 20 brands consumed by Americans? $$\begin{array}{rrrrrrrrrr}0 & 0 & 34 & 34 & 34 & 45 & 41 & 51 & 55 & 36 & 47 & 41 & 0 & 0 & 53 & 54 & 38 & 0 & 41 & 47\end{array}$$

Short Answer

Expert verified
Mean: 32.55 mg, Median: 40.5 mg, Mode: 0 and 41 mg, Midrange: 27.5 mg. The statistics are somewhat representative but not fully conclusive for all cans consumed by Americans.

Step by step solution

01

- Arrange the Data in Ascending Order

First, arrange the given data in ascending order: \(0, 0, 0, 0, 34, 34, 34, 36, 38, 41, 41, 41, 45, 47, 47, 51, 53, 54, 55\).
02

- Calculate the Mean

The mean is found by adding all the values and then dividing by the number of values. Add all the values: \(0 + 0 + 0 + 0 + 34 + 34 + 34 + 36 + 38 + 41 + 41 + 41 + 45 + 47 + 47 + 51 + 53 + 54 + 55 = 651\). Then, divide by the number of values \(20\). Thus, the mean is \( \frac{651}{20} = 32.55 \mathbf{mg} \).
03

- Find the Median

The median is the middle value when the data is ordered. For an even number of values, the median is the average of the two middle numbers. In this case, the 10th and 11th values are both \(41\), so the median is the average of those two values: \( \frac{41 + 41}{2} = 40.5 \mathbf{mg} \).
04

- Determine the Mode

The mode is the value that appears most frequently. Here, the number \(0\) and \(41\) appear the most times (4 times each). So, the data set is bimodal with modes \(0\) and \(41 \mathbf{mg} \).
05

- Find the Midrange

The midrange is the average of the maximum and minimum values. The minimum value is \(0\) and the maximum value is \(55\). Therefore, the midrange is \(\frac{0 + 55}{2} = 27.5 \mathbf{mg} \).
06

- Answer the Given Question

The statistics provided (mean, median, mode, midrange) can offer a representative snapshot of caffeine content per can for the sampled brands, but they may not be fully indicative of all cans consumed by Americans due to variability in consumption habits and potential differences in brand popularity. Therefore, it is likely that the statistics are somewhat representative, but not completely conclusive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

mean
The mean, often known as the average, is calculated by adding all the values in a data set and then dividing the sum by the number of values. For example, in our caffeine data set, we add up all the given values to get a total of 651 mg. We then divide this total by the number of observations, which is 20. Hence, the mean caffeine content is \(\frac{651}{20} = 32.55\) mg. The mean provides a single value that represents the central point of the data set. It is very useful when you want to understand the overall trend or typical value in the data.
median
The median is the middle value in a data set when the values are arranged in ascending order. Unlike the mean, it is not influenced by extreme values or outliers. In our example, we first order the caffeine amounts: 0, 0, 0, 0, 34, 34, 34, 36, 38, 41, 41, 41, 45, 47, 47, 51, 53, 54, 55. Since we have 20 values, the median is the average of the 10th and 11th values. Here, both the 10th and 11th values are 41, so the median is \(\frac{41 + 41}{2} = 40.5\) mg. The median helps to identify the central tendency of a data set and is especially useful when the data involves outliers.
mode
The mode is the value that appears most frequently in a data set. For the caffeine example, the values 0 and 41 both appear four times. Thus, our data set is bimodal, with modes 0 and 41 mg. The mode is useful for identifying the most common or frequent occurrences within the data set. It can help reveal patterns or trends, especially in qualitative data.
midrange
The midrange is calculated by averaging the maximum and minimum values of the data set. This value gives an indication of the middle point of the data's range. In our example, the minimum value is 0 mg and the maximum value is 55 mg. Therefore, the midrange is \(\frac{0 + 55}{2} = 27.5\) mg. While it is straightforward to calculate, the midrange can be heavily influenced by outliers, similar to the mean.
data analysis
Data analysis is the process of inspecting, cleaning, transforming, and modeling data to discover useful information, make conclusions, and support decision-making. In this exercise, we calculated descriptive statistics like mean, median, mode, and midrange to understand the caffeine content in various drinks. Each measure provides different insights:
  • Mean: Gives us the average caffeine content.
  • Median: Shows the middle value, useful when there are outliers.
  • Mode: Reveals the most common caffeine levels.
  • Midrange: Indicates the central point of the data's range.
Combining these descriptive statistics provides a more comprehensive picture of the data set. However, it's crucial to note that these statistics might not fully represent the entire population of caffeine content in drinks consumed by Americans, especially due to sample variability and brand popularity differences.

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Most popular questions from this chapter

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