Problem 7 - Prealgebra

Question

Find the mean, median, and mode for each set of data. Round to the nearest tenth, if necessary. $$41,37,43,43,36$$

Step-by-Step Solution

Answer

Mean: 40, Median: 41, Mode: 43.

Step 1: Arrange Data in Ascending Order

To find the median and mode, it's helpful to first arrange the data in ascending order. Here, the numbers are: 36, 37, 41, 43, 43.

Step 2: Compute the Mean

To calculate the mean, add all the numbers together and divide by the total count of numbers. Mean = \( \frac{36 + 37 + 41 + 43 + 43}{5} \).The sum is 200, so Mean = \( \frac{200}{5} = 40 \).

Step 3: Find the Median

The median is the middle number in a sorted list. Since there are 5 numbers, the middle one is the third number: 41.

Step 4: Identify the Mode

The mode is the most frequently occurring number in the list. Since 43 appears twice and all other numbers appear only once, the mode is 43.

Key Concepts

Understanding Mean

The mean is one of the simplest yet powerful statistical measures. It represents the average of a data set and is calculated by summing up all the individual numbers, then dividing by how many numbers you have. It's like adding up everyone's contributions to a team project and then finding out how much each member should be awarded on average. To calculate the mean in our data set, you take each number—41, 37, 43, 43, 36—and add them all together to get a total of 200.
Divide 200 by the number of values, which is 5. This division gives us a mean of 40.
The mean is particularly useful because it provides a quick snapshot of the center of your data set. However, be cautious because extremely high or low numbers (outliers) can skew the mean and give a misleading impression of the dataset's average.

Finding the Median

The median is like the halfway point of your data set. It's the point where half the numbers are smaller, and half are larger. If you were to line up everyone in class shortest to tallest, the person in the middle is your median student height.
For the median calculation, the first step is arranging the numbers in order from least to greatest. Our dataset becomes 36, 37, 41, 43, and 43.
With five numbers, the third number (41) sits right in the middle, making it the median.
Note that if you have an even set of numbers, you take the average of the two middle numbers to get your median. This figure can be more representative of a typical item in your data set, especially if there are outliers that don't affect it like they might the mean.

Identifying the Mode

In any list of numbers, the mode is the one that appears most frequently. Think of it as the most popular choice—the item that shows up the most in your favorite playlist.
Looking at our data set—36, 37, 41, 43, 43—you'll notice that 43 shows up twice, more than any other number. This repetition makes it our mode.

• A data set can have more than one mode (bimodal) if multiple numbers appear with the same highest frequency.
• If no number repeats, the data set is considered "mode-less."

Modes can be particularly helpful in understanding the most common occurrence or preference within a set of data, especially in value repetition scenarios.