Finding the Range from Average of Smallest and Largest Numbers

The question provides four numbers, arranged in ascending order. Let's call them w, x, y, and z, where w < x < y < z.

We are given information about the average of subsets of these numbers. The average is calculated as the sum of the numbers divided by the count of numbers.

Step 1: Write down the given information as equations

The average of the smallest three numbers (w, x, y) is 18.

This can be written as:

\(\frac{w + x + y}{3} = 18\)

Multiplying both sides by 3, we get:

\(w + x + y = 18 \times 3\)

\(w + x + y = 54\) (Equation 1)

The average of the largest three numbers (x, y, z) is 22.

This can be written as:

\(\frac{x + y + z}{3} = 22\)

Multiplying both sides by 3, we get:

\(x + y + z = 22 \times 3\)

\(x + y + z = 66\) (Equation 2)

Step 2: Use the equations to find the relationship between the largest and smallest numbers

We have two equations:

• Equation 1: \(w + x + y = 54\)
• Equation 2: \(x + y + z = 66\)

Notice that both equations share the terms x and y. If we subtract Equation 1 from Equation 2, the terms x and y will cancel out, leaving us with a relationship between z and w.

Subtract Equation 1 from Equation 2:

\((x + y + z) - (w + x + y) = 66 - 54\)

Distribute the negative sign on the left side:

\(x + y + z - w - x - y = 66 - 54\)

Combine like terms. The +x and -x cancel out, and the +y and -y cancel out:

\(z - w = 12\)

Step 3: Determine the Range of the data

The range of a data set is defined as the difference between the largest value and the smallest value in the set.

In this case, the largest number is z and the smallest number is w. The range is \(z - w\).

From our calculation in Step 2, we found that \(z - w = 12\).

Therefore, the range of the data is 12.

Summary of Steps

The range of the data is 12.

Revision Table: Key Concepts

Additional Information on Data Spread

The range is one way to measure the spread or dispersion of data. It gives a simple idea of how far apart the extreme values are. However, it doesn't tell us anything about how the data is distributed between the smallest and largest values. Other measures of spread include:

• Interquartile Range (IQR): The difference between the third quartile (\(Q_3\)) and the first quartile (\(Q_1\)). It measures the spread of the middle 50% of the data.
• Variance: The average of the squared differences from the mean. It gives a measure of how spread out the numbers are from their average value.
• Standard Deviation: The square root of the variance. It is the most commonly used measure of spread and is in the same units as the original data.

Understanding these different measures helps in getting a more complete picture of the data distribution beyond just the average.