Problem 60 - Algebra 2

Question

PREREQUISITE SKILL Find the mean, median, mode, and range for each set of data. Round to the nearest hundredth, if necessary. (Pages 759 and 760 ) $$ 4.8,5.7,2.1,2.1,4.8,2.1 $$

Step-by-Step Solution

Answer

Mean: 3.6, Median: 3.45, Mode: 2.1, Range: 3.6.

Step 1: Calculate the Mean

To find the mean, we add up all the numbers in the data set and then divide by the number of items. First, sum the numbers: \((4.8 + 5.7 + 2.1 + 2.1 + 4.8 + 2.1) = 21.6\). There are 6 numbers in the set, so we divide: \(21.6 \div 6 = 3.6\). Thus, the mean is 3.6.

Step 2: Find the Median

To find the median, we first need to order the numbers from least to greatest: \(2.1, 2.1, 2.1, 4.8, 4.8, 5.7\). The median is the middle number. Since there are 6 items, the median will be the average of the third and fourth numbers: \((2.1 + 4.8) \div 2 = 3.45\). Therefore, the median is 3.45.

Step 3: Determine the Mode

The mode is the number that appears most frequently in the data set. Looking at our ordered list: \(2.1, 2.1, 2.1, 4.8, 4.8, 5.7\), the number 2.1 appears three times, more than any other number. Thus, the mode is 2.1.

Step 4: Calculate the Range

To find the range, subtract the smallest number in the set from the largest number. From our ordered set, the smallest value is 2.1 and the largest is 5.7. So, the range is \(5.7 - 2.1 = 3.6\).

Key Concepts

Mean

The mean, commonly referred to as the average, is a way to find the central value of a data set. To calculate the mean, you simply add up all the numbers and then divide by the total number of values. This gives you an idea of the overall "level" of your data.
To illustrate, consider this data set: 4.8, 5.7, 2.1, 2.1, 4.8, 2.1. First, sum all these numbers:

• 4.8 + 5.7 + 2.1 + 2.1 + 4.8 + 2.1 = 21.6

Next, count the numbers in the data set, which in this case is 6.
Finally, divide the sum by the number of items:

• 21.6 ÷ 6 = 3.6

Therefore, the mean is 3.6, representing the average of these values.

Median

The median is the middle value in an ordered data set. It's a different way of identifying the central point of your data and can be useful, especially if your data contains outliers. To find the median, you must first arrange the numbers from smallest to largest. Using our example data set, you get: 2.1, 2.1, 2.1, 4.8, 4.8, 5.7.

Here, the total number of values is 6. When there's an even number of values, the median is calculated by averaging the two middle numbers. In this case:

• Median = (2.1 + 4.8) ÷ 2
• Median = 6.9 ÷ 2 = 3.45

So, the median of this data set is 3.45, showing the middle ground of your ordered data.

Mode

The mode is the value that appears most frequently in a data set. If no number repeats, the set is considered to have no mode. However, a set can also be multimodal, meaning it has more than one mode if several values tie for highest frequency.
In our given example of the data set: 4.8, 5.7, 2.1, 2.1, 4.8, 2.1, after organizing in ascending order, 2.1, 2.1, 2.1, 4.8, 4.8, 5.7, you observe that 2.1 appears three times. This is more frequent than any other number, so 2.1 is the mode.
The mode gives insight into the most "popular" or common value in your data.

Range

The range gives you an idea of how spread out the values in your data set are. It shows the difference between the highest and lowest values. This can help you understand the variability and spread of the data.
To find the range, identify the smallest and largest numbers in your data set. In our example of
2.1, 2.1, 2.1, 4.8, 4.8, 5.7, the smallest number is 2.1, and the largest is 5.7. Subtract the smallest from the largest to find the range:

• Range = 5.7 - 2.1 = 3.6

The range of 3.6 points to how much variation there is among the numbers, giving you a sense of their spread.