Problem 58 - 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 ) $$ 298,256,399,388,276 $$

Step-by-Step Solution

Answer

Mean: 323.4, Median: 298, Mode: None, Range: 143.

Step 1: Find the Mean

To find the mean, sum up all the numbers in the data set and then divide by the number of data points. \[ \text{Mean} = \frac{298 + 256 + 399 + 388 + 276}{5} = \frac{1617}{5} = 323.4 \].

Step 2: Find the Median

To find the median, first arrange the numbers in ascending order: 256, 276, 298, 388, 399. The median is the middle number, which, in this case, is 298, because it is the third number out of five.

Step 3: Find the Mode

The mode is the number that appears most frequently in the data set. Here, each number appears only once, so there is no mode.

Step 4: Find the Range

To find the range, subtract the smallest number in the set from the largest number. \[ \text{Range} = 399 - 256 = 143 \].

Key Concepts

Understanding the Mean

The mean, often referred to as the average, is a measure of central tendency that helps us understand the "central" value of a data set. Calculating the mean is straightforward, and it provides a useful summary of the data you are examining.
To calculate the mean, you start by adding all the numbers in your data set together. This sum is then divided by the total number of values in the set. For the calculation provided in the exercise, the mean is:

• Add up all the numbers: 298 + 256 + 399 + 388 + 276 = 1617
• Count the numbers you added, which in this case is 5
• Divide the total sum by the number of values: \[\text{Mean} = \frac{1617}{5} = 323.4\]

The mean provides a mathematical center of the data and is particularly useful when comparing different data sets or when seeking a single value to represent all data points.

Identifying the Median

The median represents the middle value of a data set when it is ordered from smallest to largest. Unlike the mean, the median is less affected by extreme values, which makes it a useful measure for understanding the central location of a data set.
To find the median, follow these steps:

• Order the data from smallest to largest: 256, 276, 298, 388, 399
• Identify the middle number, which is the third number in this ordered list: 298

If there were an even number of values, you would take the average of the two middle numbers to get the median. The median is a strong representation of the data's center when the distribution is skewed or has outliers.

Determining the Mode

The mode is the value that appears most frequently in a data set. It's possible for a data set to have one mode, more than one mode, or no mode at all if all values occur with the same frequency.
To find the mode in any given data set, you should:

• Observe the frequency of each number appearing in the dataset
• If a number appears more than once, it is the mode. If no number repeats, like in the exercise provided, then the data set has no mode

The mode is particularly useful in categorical data where we wish to identify the most common category. In our numerical example, however, since all numbers appear equally, there's no mode to report.

Calculating the Range

The range provides a simple way to understand the spread or dispersion of a data set. It gives an idea of how varied the numbers in your set are, by showing the difference between the highest and lowest values.
Here's how you calculate the range:

• Identify the largest and smallest numbers in the set: 399 (largest) and 256 (smallest)
• Subtract the smallest number from the largest: \[\text{Range} = 399 - 256 = 143\]

While the range gives a quick snapshot of variability, it can be somewhat limited because it only considers two data points. Nonetheless, it's a great starting point for understanding the overall distribution characteristics in a data set.