Question

If $a$ represents the median of a group of numbers, $b$ is the mode, and $c$ is the range, what is the value of $a+b-c$ for the following group of numbers: $3,7,-4,2,-5,0,-2,7?$

Short answer

The value of $a+b-c$ is -4.

Step-by-step solution

- Arrange the data in ascending order

First, arrange the given group of numbers in ascending order: $$-5,-4,-2,0,2,3,7,7$$

- Find the median $a$

To find the median, select the middle number(s). Since there are 8 numbers, the median is the average of the 4th and 5th numbers: $$a=\frac{0+2}{2}=1$$

- Identify the mode $b$

The mode is the number that appears most frequently. In this set, the number 7 appears twice, more than any other number: $$b=7$$

- Calculate the range $c$

The range is the difference between the largest and smallest numbers. Here, the largest number is 7 and the smallest number is -5: $$c=7-(-5)=7+5=12$$

- Compute $a+b-c$

Now calculate the expression using the values obtained: $$a+b-c=1+7-12=-4$$

Key concepts

median calculation

To understand how to calculate the median, let's dive into what a median really represents. The median is the middle value in a list of numbers sorted in ascending order. If the list has an odd number of entries, the median is the middle number. If the list has an even number of entries, the median is the average of the two middle numbers.

For example, in the set $[-5,-4,-2,0,2,3,7,7]$, there are 8 numbers, so our median will be the average of the 4th and 5th numbers: $\frac{0+2}{2}=1$. This tells us that the median is a value that separates the higher half from the lower half of our data set.

mode identification

Identifying the mode is all about finding the number that appears most frequently in your data set. The mode is useful in understanding the most common or popular value.

Consider our list of numbers: $[-5,-4,-2,0,2,3,7,7]$. The number 7 appears twice, more than any other number in this set, making it the mode. Mode can be particularly helpful in cases where there is a peak or a frequent occurrence of certain values in your data.

range calculation

The range gives you an idea of the spread or dispersion of your data. It is calculated as the difference between the largest and smallest values in the data set.

In our example, the largest number is 7 and the smallest number is -5. To find the range, you calculate $\text{Range}=7-(-5)=7+5=12$. This shows the extent of the values within the list, giving a quick sense of variation.

basic arithmetic operations

Understanding and performing basic arithmetic operations like addition, subtraction, multiplication, and division are fundamental skills. They form the building blocks for more advanced concepts.

In our problem, we are required to compute $\text{a + b - c}$. With the values found: median $\text{a}=1$, mode $\text{b}=7$, and range $\text{c}=12$, the operation becomes $\text{1 + 7 - 12}$. Performing the addition and subtraction step-by-step: $\text{1 + 7 = 8}$ and then $\text{8 - 12 = -4}$. Thus, the final answer is -4.