Problem 3 - AP Psychology Premium Prep 2021

Question

In the following set of test scores, what is the mode and what is the range? Test Scores: \(71,67,75,65,66,32,69,70,72,82,\) 73,68,75,68,75,78 (A) Mode: 68; Range: 75 (B) Mode: 69; Range: 50 (C) Mode: 75; Range: 70.5 (D) Mode: 75; Range: 50

Step-by-Step Solution

Answer

The short answer is: (D) Mode: 75; Range: 50.

Step 1: Arrange the scores in ascending order

First, we should arrange the given test scores in ascending order to make it easier to determine the mode and range. The ordered test scores are: \(32, 65, 66,67, 68, 68, 69, 70, 71, 72, 73, 75, 75, 75, 78, 82\).

Step 2: Determine the mode

Next, we should find the most frequent score in the ordered test scores. The frequency of scores in the list is as follows: - 68 appears 2 times - 75 appears 3 times No other score appears more than once. So, the mode is 75 since it has the highest frequency.

Step 3: Calculate the range

Finally, we should find the range by subtracting the lowest score from the highest score in the ordered test scores: Range = Highest Score - Lowest Score = 82 - 32 = 50

Step 4: Match the result with the given options

The calculated mode is 75, and the calculated range is 50. Matching these results with the given options, we find that the correct answer is: (D) Mode: 75; Range: 50

Key Concepts

Understanding Mode in Statistics

The mode is a key concept in statistics that refers to the number which appears most frequently in a data set. In simple terms, it is the "most common" number. If you're analyzing test scores, identifying the mode can tell you what the most common score is among students. This can give insights into how often a particular score was achieved and indicates the score level that many students clustered around. When dealing with a set of numbers, make sure to first list them in order. Then, count how often each number appears. The number that shows up the most is your mode. Sometimes, you might encounter a data set with no mode or multiple modes, if no number or several numbers appear the most frequently. In the example problem, after ordering the scores, we found that the number 75 appears most frequently, three times, making it the mode. If you're preparing for an AP Psychology test, it's helpful to understand what the mode tells us about common trends or popular outcomes within your data.

Calculating Range in Statistics

The range is a simple concept that helps you understand the spread of a data set. It is the difference between the highest and the lowest numbers. Think of it as the "distance" between the largest score and the smallest score in your list. This measure gives you an idea of how varied or concentrated the data points are.To calculate the range, subtract the smallest number from the largest number in your data set. This gives you a single number, which helps you grasp the overall span of the scores.In the exercise example, the highest score is 82, and the lowest is 32. So, the range is calculated as follows:\[ \text{Range} = 82 - 32 = 50 \]A larger range means a more diverse data set, while a smaller range indicates that the scores are more similar to each other.

Analyzing Test Scores

Analyzing test scores involves looking at various statistical measures like the mode and range to gain insights into student performance. By analyzing these scores, you can uncover trends about how students are doing overall and where most scores tend to cluster. You begin by arranging the scores and identifying key figures: the mode shows the most common score, and the range tells you how spread out the scores are. This analysis can be critical in understanding the performance level of different groups or identifying outliers whose results deviate from the norm. In the given exercise, we identified that the mode was 75, indicating that a significant number of students scored similar to this number. The range was 50, suggesting there is a considerable disparity between the lowest and highest scores. When preparing for an AP Psychology test, analyzing test scores in this manner can help you identify areas of focus, showing both strength and room for improvement. Recognizing patterns in test scores can aid in developing strategies and study plans, guiding you towards more effective preparation.