Question

An exam is given in a certain class. The average (arithmetic mean) of the highest score and the lowest score on the exam is equal to $x$. If the average score for the entire class is equal to $y$ and there are $z$ students in the class, where $z>5,$ then in terms of $x,y,$ and $z,$ what is the average score for the class, excluding the highest and lowest scores? A. $$\frac{zy-2x}{z}$$ B. $$\frac{zy-2}{z}$$ C. $$\frac{zx-y}{z-2}$$ D. $$\frac{zy-2x}{z-2}$$ E. $$\frac{zy-x}{z+2}$$

Short answer

D. $\frac{zy-2x}{z-2}$

Step-by-step solution

Define Variables

Let the highest score be denoted as $H$ and the lowest score be denoted as $L$. Given that the average of the highest and lowest score is equal to $x$, we can write this relationship as: $\frac{H+L}{2}=x$.

Express H + L in terms of x

From the equation $\frac{H+L}{2}=x$, solve for $H+L$: $H+L=2x$.

Express Total Sum of Scores

Let the sum of all scores of the class be denoted as $S$. Given that the average score of all $z$ students is $y$, we can write the sum $S$ as: $S=zy$.

Express Sum of Remaining Scores

To find the sum of the scores excluding the highest and lowest scores, subtract $H+L$ from the total sum $S$: Sum of remaining scores = $S-(H+L)$.

Substitute Known Values

Using the expressions from the previous steps, substitute $S=zy$ and $H+L=2x$: Sum of remaining scores = $zy-2x$.

Calculate Average of Remaining Scores

To find the average score of the class excluding the highest and lowest scores, divide the sum of the remaining scores by $z-2$ (since two scores are excluded): Average excluding highest and lowest = $\frac{zy-2x}{z-2}$.

Identify the Correct Answer

Compare the derived expression $\frac{zy-2x}{z-2}$ with the given options. The correct option matches with choice D: D. $\frac{zy-2x}{z-2}$

Key concepts

Average Score Calculation

When calculating the average score of a dataset, we use a simple but powerful formula: the arithmetic mean. Let's break down how this works. The average score, or arithmetic mean, is found by adding up all the scores and then dividing the total by the number of scores.

If we denote the scores as $s_1,s_2,\dots ,s_n$, where $n$ represents the number of scores, the average score ($A$) is calculated as follows: $$A=\frac{s_1+s_2+\dots +s_n}{n}$$

In our problem, the average score of the entire class is denoted by $y$, and the number of students is denoted by $z$. Therefore, the sum of all scores in the class can be expressed as $S=zy$. Next, we will dive into understanding the arithmetic mean, which is a crucial concept used in this exercise.

Arithmetic Mean

The arithmetic mean is one of the fundamental concepts in statistics and quantitative reasoning. It is often referred to as the 'average,' and it provides a central value for a set of numbers. Calculating the arithmetic mean involves two main steps: addition and division.

First, sum all the values in the dataset. Then, divide this total by the number of values. This formula provides a single number that represents the dataset's central tendency.

For example, if we have five test scores: 80, 90, 85, 70, and 95, the arithmetic mean is computed as follows: $$\text{Mean}=\frac{80+90+85+70+95}{5}=\frac{420}{5}=84$$

Understanding the arithmetic mean is essential in solving problems related to averages, such as the one presented, where the mean scores $x$ and $y$ are compared and manipulated to find solutions.

Excluding Data Points

In some scenarios, you may need to exclude certain data points from your calculations to find a more accurate or relevant average. This process is known as 'excluding data points.'

In the given problem, we are asked to exclude the highest and lowest scores from the dataset. Let's understand this concept with the steps already provided:

• First, determine the sum of the entire dataset using the given average formula.
• Next, identify the values you need to exclude. Here, they are the highest and lowest scores.
• Subtract these values from the total sum.

From the solution, we know: $$H+L=2x$$ and $$S=zy$$

So, the sum of the remaining scores after excluding the highest and lowest is: $$S-(H+L)=zy-2x$$.

To find the new average, divide this sum by $(z-2)$, the adjusted number of data points. The formula is: $$\text{New Average}=\frac{zy-2x}{z-2}$$

This process ensures that our new average accurately reflects the remaining data by fairly distributing the sum over the adjusted number of students.