Problem 16 - Prealgebra

Question

Which measure of central tendency best summarizes the test scores shown below? Explain. \(97,99,95,89,99,100,87,85,89,92\) \(96,95,60,97,85\) (PICTURE NOT COPY)

Step-by-Step Solution

Answer

The median is the best measure as it is unaffected by the outlier (60) and provides a central value.

Step 1: Organize the Data

First, list the test scores in ascending order for easier analysis: \(60, 85, 85, 87, 89, 89, 92, 95, 95, 96, 97, 97, 99, 99, 100\).

Step 2: Calculate the Mean

To find the mean, add up all the scores and divide by the number of scores.Sum of scores: \(60 + 85 + 85 + 87 + 89 + 89 + 92 + 95 + 95 + 96 + 97 + 97 + 99 + 99 + 100 = 1265\)Number of scores: 15Mean = \(\frac{1265}{15} \approx 84.33\)

Step 3: Determine the Median

The median is the middle value when the data is in order. With 15 scores, the median is the 8th value: The 8th score is 95, so the median is 95.

Step 4: Find the Mode

The mode is the most frequently occurring score. The scores 85, 89, 95, 97, and 99 appear twice each, but 99 appears three times. Thus, the mode is 99.

Step 5: Analyze Which Measure is Best

Consider the distribution of the scores. The scores have a low outlier (60) which can skew the mean. The median and mode are more resistant to such an outlier. Comparatively, the median gives a central value relatively unaffected by extremes, providing a better representation of the overall performance of most students.

Key Concepts

Understanding Mean Calculation

The mean, often called the average, is the total sum of all numbers divided by how many numbers are in the dataset. In this exercise, we have the test scores: 60, 85, 85, 87, 89, 89, 92, 95, 95, 96, 97, 97, 99, 99, 100. First, you need to add these scores together, which gives us a total of 1265. Then, you divide this sum by the number of scores, which is 15 in this case. To represent it mathematically, the formula for the mean is:\[ \text{Mean} = \frac{\text{Sum of all scores}}{\text{Number of scores}} = \frac{1265}{15} \approx 84.33 \]Mean calculation provides a central value, but it's essential to consider factors like outliers, which can greatly influence the mean. For example, the score of 60 is significantly lower than the rest, which pulls the mean down and might not accurately reflect the central tendency of the dataset.

Determining the Median

The median is another measure of central tendency and represents the middle value in an ordered dataset. To find the median, it's crucial to first line up the numbers from smallest to largest. With our organized scores: 60, 85, 85, 87, 89, 89, 92, 95, 95, 96, 97, 97, 99, 99, 100, we have 15 numbers. The median is the eighth number because it is the middle point in a sequence of counting. Choosing the median involves these simple steps:

• Order the data from least to greatest.
• Select the middle number if the count is odd, pair the middle two numbers’ average if even.

In this list, the eighth score is 95. Thus, 95 is our median, providing a balanced picture of the dataset, especially in distributions with skewed data or outliers.

Analyzing the Mode

Mode is a measure that identifies the most frequently occurring number in a set of data. In our dataset of test scores, this can be very informative about the most common performance level among students. For our scores (60, 85, 85, 87, 89, 89, 92, 95, 95, 96, 97, 97, 99, 99, 100), several numbers like 85, 89, 95, 97, and 99 show up twice. However, 99 appears three times, making it the mode of this dataset. Mode Analysis:

• Helps to spot the most repeated scores, providing insights into the most common test performance.
• In our dataset, mode can indicate which score is typically seen more often than others.
• Doesn't get highly affected by outliers since it reviews frequency, not overall value.

In essence, the mode is valuable for understanding which score occurs most frequently in a dataset, providing a quick glance at the commonality within the data.