3. Describing Data Numerically
Median
1:37 minutesProblem 3.1.13
Textbook Question
Critical Thinking. For Exercises 5–20, 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.
Caffeine in Soft Drinks 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?
0 0 34 34 34 45 41 51 55 36 47 41 0 0 53 54 38 0 41 47
Verified step by step guidance1
Step 1: Organize the data. Start by listing the caffeine values in ascending order: 0, 0, 0, 0, 34, 34, 34, 36, 38, 41, 41, 41, 45, 47, 47, 51, 53, 54, 55, 47.
Step 2: Calculate the mean. Add all the caffeine values together and divide by the total number of data points (n = 20). Use the formula: , where is the sum of all data points.
Step 3: Find the median. Since there are 20 data points (even number), the median is the average of the 10th and 11th values in the ordered list. Identify these values and compute their average.
Step 4: Determine the mode. The mode is the value(s) that appear most frequently in the dataset. Count the frequency of each value to identify the mode(s).
Step 5: Calculate the midrange. The midrange is the average of the smallest and largest values in the dataset. Use the formula: . Then, consider whether the statistics are representative of the population by discussing potential biases or limitations in the data collection process.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Descriptive Statistics
Descriptive statistics summarize and describe the main features of a dataset. This includes measures such as the mean (average), median (middle value), mode (most frequent value), and midrange (average of the highest and lowest values). These statistics provide a quick overview of the data's central tendency and variability, which is essential for understanding the distribution of caffeine levels in the given soft drinks.
Recommended video:
Guided course
05:53
Parameters vs. Statistics
Population vs. Sample
In statistics, a population refers to the entire group of individuals or items that we want to study, while a sample is a subset of that population. The question asks whether the statistics calculated from the 20 brands are representative of all cans consumed by Americans, highlighting the importance of understanding how well a sample reflects the broader population. This concept is crucial for making inferences about the population based on sample data.
Recommended video:
05:11Sampling Distribution of Sample Proportion
Statistical Inference
Statistical inference involves using data from a sample to make conclusions about a larger population. It includes techniques such as hypothesis testing and confidence intervals. In the context of the question, determining if the caffeine statistics are representative requires evaluating whether the sample of 20 brands can reliably inform us about the caffeine content in all cans of those brands consumed by Americans, which is a key aspect of inferential statistics.
Recommended video:
Guided course
05:53Parameters vs. Statistics
3:26mWatch next
Master Calculating the Median with a bite sized video explanation from Patrick
Start learning
