Problem 1 - Prealgebra

Question

Find the mean, median, and mode for each set of data. If necessary, round to the nearest tenth. $$4,5,7,3,9,11,23,37$$

Step-by-Step Solution

Answer

Mean: 12.4, Median: 8, Mode: None.

Step 1: Understanding the Mean

The mean is the average of a set of numbers. To calculate it, add all the numbers in the set and then divide by the number of numbers. Here, the numbers are 4, 5, 7, 3, 9, 11, 23, and 37. We first add them: \[ 4 + 5 + 7 + 3 + 9 + 11 + 23 + 37 = 99 \]Next, divide the sum by the number of values (8) to find the mean: \[ \frac{99}{8} = 12.375 \]Rounding to the nearest tenth, the mean is 12.4.

Step 2: Understanding the Median

To find the median, first arrange the numbers in ascending order: 3, 4, 5, 7, 9, 11, 23, 37. Since there are 8 numbers, the median will be the average of the 4th and 5th numbers:\[ \text{4th number: } 7, \text{ 5th number: } 9 \]Find the median by calculating:\[ \frac{7 + 9}{2} = 8 \]Thus, the median is 8.

Step 3: Understanding the Mode

The mode is the number that appears most frequently in a data set. In this set, each number appears only once: 4, 5, 7, 3, 9, 11, 23, and 37. Since there is no repetition, we conclude that this data set has no mode.

Key Concepts

Understanding the Mean

The mean is a statistical concept that helps us find the average value of a dataset. It's like trying to find the center point to balance all the numbers. To get the mean, follow these steps:

• Add all the numbers in the dataset, just like adding apples in a basket.
• Then, divide the total sum by the number of apples or numbers you have.

In our case, we had the numbers: 4, 5, 7, 3, 9, 11, 23, and 37. Adding them gives us 99. Dividing 99 by 8 (since there are 8 numbers) gives us 12.375. By rounding this to the nearest tenth, we get a mean of 12.4. So, the mean tells us that each number in this dataset, on average, is about 12.4.

Understanding the Median

The median is all about finding the middle value in a dataset. It helps to understand where the half-point mark is. This is how you can find it:

• Arrange your numbers in order from smallest to largest.
• Identify the middle position, or if there are two middle numbers (like in even-numbered lists), find their average.

In our data: 3, 4, 5, 7, 9, 11, 23, 37, we see there are 8 numbers. Thus, the middle numbers are 7 and 9, which occupy the 4th and 5th positions. Calculating their average by \[ \frac{7 + 9}{2} = 8 \]we find the median to be 8. The median gives us the value where half the numbers are greater and half are less, helping to show the dataset's center point more practically.

Understanding the Mode

Mode is the concept that tells us which value appears most frequently in a dataset. It's like knowing which pair of shoes you wear most often in your collection. Here's how you determine it:

• Check each number to see how many times it appears.
• The number that appears the most is the mode.

In our dataset of 4, 5, 7, 3, 9, 11, 23, and 37, you'll notice each number occurs just once. Since no number repeats, there's no mode in this dataset. Sometimes, datasets can have multiple modes, such as several numbers appearing with the highest frequency, or none at all, like here. Knowing the mode helps to identify the most common or popular value in your data.