Problem 29 - Prealgebra

Question

Find the mean, median, and mode for each set of data. Round to the nearest tenth, if necessary. $$0.9,0.5,0.7,0.4,0.3,0.2$$

Step-by-Step Solution

Answer

Mean: 0.5, Median: 0.5, Mode: None.

Step 1: Calculate the Mean

To find the mean, add all the numbers together and divide by the total number of values. Mean = \( \frac{0.9 + 0.5 + 0.7 + 0.4 + 0.3 + 0.2}{6} = \frac{3.0}{6} = 0.5 \).

Step 2: Determine the Median

First, arrange the data in ascending order: \(0.2, 0.3, 0.4, 0.5, 0.7, 0.9\). Since there is an even number of data points (6), the median is the average of the third and fourth values: Median = \( \frac{0.4 + 0.5}{2} = 0.45 \), which rounds to \(0.5\) to the nearest tenth.

Step 3: Identify the Mode

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

Key Concepts

Mean Calculation

The mean, often referred to as the average, is a measure of central tendency that provides a single value representing the center of a data set. To calculate the mean, you simply need to add up all the values in your data set and then divide this sum by the number of values you have. Let’s break it down:

Imagine you have a set of numbers: 0.9, 0.5, 0.7, 0.4, 0.3, and 0.2. To find their mean:

• Add them up: \(0.9 + 0.5 + 0.7 + 0.4 + 0.3 + 0.2 = 3.0\).
• Count how many numbers there are: 6.
• Divide the sum by the count: \(\frac{3.0}{6} = 0.5\).

This means the mean of this data set is 0.5. This single value tells us where the center of the data points lie. Remember to round your final result if necessary, as this exercise suggests rounding to the nearest tenth.

Median Determination

The median is another type of central tendency, which represents the middle value of a data set when it is ordered from smallest to largest. It’s particularly useful because it is less affected by outliers or very large or small values compared to the mean.

Here is how you find the median:

• First, arrange your numbers in ascending order: 0.2, 0.3, 0.4, 0.5, 0.7, 0.9.
• Count your values. If the number of values is odd, the median is the middle number. If it's even, the median will be the average of the two middle numbers.
• In our data set, we have 6 numbers, an even number, so we take the average of the third and fourth numbers: \(\frac{0.4 + 0.5}{2} = 0.45\).

After rounding 0.45 to the nearest tenth, it becomes 0.5. Thus, the median of this data set is also 0.5. It divides our data into two equal parts, with half of the numbers below and the other half above.

Mode Identification

The mode is the value that appears most frequently in a data set. It is one of the most straightforward measures of central tendency but can sometimes be less descriptive in cases with unique data points.

Finding the mode is simple:

• Look through your data: 0.9, 0.5, 0.7, 0.4, 0.3, 0.2.
• Identify any repeated value.

In this case, each number appears only once, meaning no number is repeated. Thus, the data set has no mode. Sometimes data sets can have more than one mode if multiple numbers appear with the same highest frequency, or none at all, as we see here. This makes the mode particularly useful in identifying the occurrence and frequency of data points.