Question

Use the data set \(\\{1,3,14,28,2,18\), \(27,86,34,45,44,36,21,11,51,23,37,52,29,41,33\), \(19,24,38,15,87\\}\). What is the range of the data set? Write your answer in the blank.

Short answer

The range of the data set is 86.

Step-by-step solution

- Identify the Largest Value

To find the range, first identify the largest value in the data set. Scan the given numbers and find the highest one. In this case, the largest number is 87.

- Identify the Smallest Value

Next, identify the smallest value in the data set. Scan the given numbers again to find the lowest one. The smallest number in this case is 1.

- Calculate the Range

To calculate the range, subtract the smallest value from the largest value using the formula: \[ \text{Range} = \text{Largest value} - \text{Smallest value} \] Substituting the values, we get: \[ \text{Range} = 87 - 1 \]

- Solve the Equation

Perform the subtraction to find the range. \[ 87 - 1 = 86 \] Therefore, the range of the given data set is 86.

Key concepts

Data Set

In statistics, a data set is a collection of numbers or values that relate to a specific subject or experiment. Imagine a list of all your test scores over the year—this list is your data set.
For example, the data set provided in this exercise is: \[1, 3, 14, 28, 2, 18, 27, 86, 34, 45, 44, 36, 21, 11, 51, 23, 37, 52, 29, 41, 33, 19, 24, 38, 15, 87\].
Each number represents one observation. Data sets can vary in size; they can have just a few numbers or thousands.
The larger your data set, the more information you have, but it may also be more challenging to analyze.

Largest Value

The largest value in a data set is crucial because it helps in various statistical calculations like finding the range.
To identify the largest value, scan through each number and compare them. In our example data set, after comparing, we find the largest number is 87.
Knowing the largest value can also provide insights into the data's tendencies and potential anomalies. For instance, an unusually high number might indicate an error or an exceptional case worth investigating.

Smallest Value

Just as the largest value is important, so is the smallest value. It represents the lowest number in your data set.
To find the smallest value, you do the same process but look for the lowest number. In our example, the smallest number is found to be 1.
Spotting the smallest value can help you understand the lower limit of your data. This is useful for understanding the range of behaviors or observations your data set includes.

Subtraction

Subtraction is the process used in calculating the range of a data set. It involves taking one number (the smallest value) away from another (the largest value).
The formula for range is simple: \[ \text{Range} = \text{Largest value} - \text{Smallest value} \]
So, if the largest value is 87 and the smallest value is 1, you carry out the subtraction as follows: \[ 87 - 1 = 86 \]
Therefore, the range of our data set is 86. Subtraction helps you understand the spread or variability of your data, showing how far apart the values in your data set are from each other.