Question

You are interning at the weather station. This week the low temperatures were \(4^{\circ} \mathrm{C},-8^{\circ} \mathrm{C}\) \(4^{\circ} \mathrm{C},-1^{\circ} \mathrm{C},-2^{\circ} \mathrm{C},-2^{\circ} \mathrm{C},-2^{\circ} \mathrm{C} .\) Determine the average low temperature for the week.

Short answer

The average low temperature is \(-1^{\circ} \mathrm{C}\).

Step-by-step solution

List the Temperatures

Begin by writing down all the low temperatures for the week: \(4^{\circ} \mathrm{C}, -8^{\circ} \mathrm{C}, 4^{\circ} \mathrm{C}, -1^{\circ} \mathrm{C}, -2^{\circ} \mathrm{C}, -2^{\circ} \mathrm{C}, -2^{\circ} \mathrm{C}\).

Sum the Temperatures

Add all the temperatures together: \(4 + (-8) + 4 + (-1) + (-2) + (-2) + (-2) \).

Calculate the Total Sum

Perform the addition: \(4 - 8 + 4 - 1 - 2 - 2 - 2 = -7\).

Count the Number of Days

Determine how many days there are in the week, which is 7.

Calculate the Average

The formula for the average is the total sum of the temperatures divided by the number of days. So, \( \text{average} = \frac{-7}{7} \).

Simplify the Average

Divide the sum by the number of days to get the average: \( \text{average} = -1^{\circ} \mathrm{C} \).

Key concepts

Understanding Integers

Integers are whole numbers that can be positive, negative, or zero. They do not contain fractions or decimals, which makes them easy to work with when performing arithmetic operations. In our temperature exercise, each day's low temperature is represented by an integer. For example, the temperatures 4°C and -8°C are both integers just as 0 is.
Understanding integers is crucial because they help us describe real-world situations like financial gains and losses, levels below or above sea level, and temperatures above or below freezing point. In mathematics, working comfortably with these numbers lays the foundation for more advanced arithmetic calculations, including those in our day-to-day calculations like finding an average temperature.
It's important to remember that integers are not just numbers on a number line but are used practically to represent real scenarios. When working with integers, think about whether you're dealing with positives or negatives, as they will influence how operations like addition and subtraction are approached.

Addition and Subtraction with Integers

Performing addition and subtraction with integers can be simple once you understand the basic rules. When you add or subtract integers, consider their signs, which tell you if they're positive or negative.

• **Adding Integers**: If both numbers have the same sign, add their absolute values and keep the sign. For example, \(4 + 4 = 8\). If they have different signs, subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value, e.g., \(4 + (-8) = -4\).
• **Subtracting Integers**: Convert the subtraction to addition by adding the opposite (i.e., changing the sign) of the number being subtracted. For instance, \(4 - (-2) = 4 + 2 = 6\).

This process was applied in our weather exercise where temperatures (integers) were added together to get a total of \(-7\), combining positive and negative values. Being adept at these calculations is essential when managing real-life data like temperature records or financial statements.

Calculating the Arithmetic Mean

The arithmetic mean, often referred to as the average, is a common statistical measure that allows us to find the central value of a set of numbers. To find the arithmetic mean, follow these steps:

• **Sum the Numbers**: Add up all the numbers you're working with. This sums the information into a single value. In our example, the sum was \(-7\) after adding all the daily temperatures.
• **Count the Values**: Determine how many values there are in your data set. In the weather example, there were seven days or temperatures to consider.
• **Divide the Sum by the Count**: This gives you the average. For the weekly temperatures, \(\frac{-7}{7} = -1°C\), which means the average temperature for the week was \(-1°C\).

The arithmetic mean provides a quick snapshot of a data set's overall trend. It's widely used across disciplines, from everyday weather reporting to more sophisticated data analyses in scientific research or economics.