Problem 79 - Prealgebra

Question

A student has quiz scores of \(65,72,70,88,70,\) and 73 Find each of the following: a. mean score b. median score c. mode of the scores d. range of scores

Step-by-Step Solution

Answer

Mean: 73; Median: 71; Mode: 70; Range: 23.

Step 1: Calculate the Mean Score

To find the mean score, sum up all the quiz scores and then divide by the number of scores. The scores are 65, 72, 70, 88, 70, and 73.Sum of scores: \[65 + 72 + 70 + 88 + 70 + 73 = 438\]Number of scores: 6Mean = \(\frac{438}{6} = 73\)

Step 2: Find the Median Score

The median is the middle number when the scores are arranged in numerical order. First, sort the scores: 65, 70, 70, 72, 73, 88.Since there are an even number of scores, the median is the average of the two middle numbers (70 and 72):\[\text{Median} = \frac{70 + 72}{2} = 71\]

Step 3: Determine the Mode of the Scores

The mode is the score that appears most frequently. In the set of scores, 65, 70, 70, 72, 73, and 88, the number 70 appears twice, while all others appear only once. Thus, the mode is 70.

Step 4: Calculate the Range of the Scores

The range is calculated as the difference between the highest and lowest scores. Highest score: 88 Lowest score: 65 Range = 88 - 65 = 23

Key Concepts

Mean Calculation

The mean is a fundamental statistical concept that represents the average of a set of numbers. It provides a sense of the central tendency of the numbers. To calculate the mean, you'll need to add up all the values and then divide by the number of values in the dataset.

In the given exercise, the student has quiz scores of 65, 72, 70, 88, 70, and 73. Here's a step-by-step breakdown for calculating the mean:

• Step 1: Sum all the scores: \[ 65 + 72 + 70 + 88 + 70 + 73 = 438 \]
• Step 2: Count the number of scores, which is 6 in this case.
• Step 3: Divide the total by the number of scores: \[ \text{Mean} = \frac{438}{6} = 73 \]

So, the mean score is 73, which tells us the average score of the quizzes.

Median Calculation

The median is another way to determine the central tendency of a dataset, representing the middle value when all numbers are arranged in order. The median gives a better sense of the center if there are outliers that skew the mean.

Here's how to find the median for the student's quiz scores:

• Step 1: Organize the scores in ascending order: 65, 70, 70, 72, 73, 88.
• Step 2: As there are 6 scores, we have an even number of values.
• Step 3: The median will be the average of the two middle numbers (70 and 72):\[ \text{Median} = \frac{70 + 72}{2} = 71 \]

Thus, the median score is 71, which indicates the midpoint of the quiz scores.

Mode Calculation

The mode of a dataset is the number that appears most frequently within it. Unlike the mean and median, the mode doesn't necessarily provide a sense of the middle, but it helps us identify which score occurred most often.

To find the mode of the quiz scores 65, 72, 70, 88, 70, and 73:

• List the scores and count the frequency of each score.
• In this data, the scores: 65 appears once, 72 appears once, 70 appears twice, 88 appears once, and 73 appears once.
• 70 is the number that occurs most frequently, with a frequency of 2.

Therefore, the mode of the scores is 70, highlighting the score that appears most often in the dataset.

Range Calculation

The range of a dataset is a simple measure of spread that represents the difference between the highest and lowest values. It gives you a sense of the interval in which all values lie.

To calculate the range for the student's quiz scores, follow these steps:

• Step 1: Identify the highest score, which is 88, and the lowest score, which is 65, among the values 65, 72, 70, 88, 70, and 73.
• Step 2: Subtract the lowest score from the highest score:\[ \text{Range} = 88 - 65 = 23 \]

Thus, the range is 23, illustrating the span between the lowest and highest quiz scores.