Problem 60 - Algebra 2

Question

Find the mean, median, and mode for each set of values. $$ \begin{array}{lllllllllll}{9} & {6} & {8} & {1} & {3} & {4} & {5} & {2} & {6} & {8} & {4} & {9} & {12} & {3} & {4} & {10} & {7} & {6}\end{array} $$

Step-by-Step Solution

Answer

The mean of the data is 5.72, the median is 6, and the modes are 4 and 6.

Step 1: Organize the data in ascending order

By arranging the given set of numbers, \(9, 6, 8, 1, 3, 4, 5, 2, 6, 8, 4, 9, 12, 3, 4, 10, 7, 6\) in ascending order, we get: \(1, 2, 3, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 9, 9, 10, 12\)

Step 2: Calculate the mean

To get the mean, sum up all numbers and divide by the total number of values. So (1+2+3+3+4+4+4+5+6+6+6+7+8+8+9+9+10+12)/18 = 103/18 = 5.72 (approximately)

Step 3: Calculate the median

Our list contains 18 numbers, which is even, so the median is the average of the 9th and 10th value. Therefore, the median is (6+6)/2 = 6

Step 4: Find the mode

The mode is the value that appears most frequently in the data set. Here, the numbers 4 and 6 both appear thrice, so the modes are 4 and 6

Key Concepts

Understanding the Mean

The mean, often referred to as the average, is a central value of a set of numbers. It is calculated by summing up all the individual numbers and then dividing the total by the number of values present. This calculation gives a single value that represents a typical number from the dataset.
For example, in the dataset provided:

• We first add all numbers: 1 + 2 + 3 + 3 + 4 + 4 + 4 + 5 + 6 + 6 + 6 + 7 + 8 + 8 + 9 + 9 + 10 + 12 = 103
• Then we divide this sum by the count of available numbers, which is 18, to get the mean: 103 ÷ 18 = 5.72 (approximately).

This calculation illustrates the distribution of values around this central number.

Finding the Median

The median is a measure of central tendency that represents the middle value in a dataset when it is organized in ascending or descending order.
This calculation offers insight into the midpoint of a dataset, effectively dividing it into two halves.
To find the median:

• First, arrange all values in order from least to greatest.
• Since the dataset has 18 numbers, an even count, you identify the middle two values.
• For our dataset, the two middle values are the 9th and 10th ones: both are 6.
• The median is the average of these two values: (6 + 6) ÷ 2 = 6.

Thus, the median is representative of the center of data distribution.

Identifying the Mode

The mode is the number that appears most frequently in a dataset. Unlike the mean and median, the mode reflects which value shows up the most often.
In some cases, a dataset might have one mode, more than one mode, or no mode at all if all numbers appear with equal frequency.
In our dataset:

• The numbers are: 4 and 6, both appearing three times in the list.
• This makes both numbers the mode of the dataset.

It's important to note that having more than one mode, as in this example, is called being "bimodal." Understanding the mode helps highlight the most common elements in the dataset.

Basic Data Analysis Techniques

Data analysis involves inspecting, cleansing, and modeling data to discover useful information.
It is a science that helps us understand how data behaves and what the underlying patterns might be.
Key techniques include:

• Calculating measurements of central tendency, like the mean, median, and mode, to understand typical values.
• Using these statistics to compare different datasets.
• Visualizing data through graphs and charts for better comprehension.
• Identifying any outliers or unusual patterns.

Basic data analysis creates a foundation for more complex analytical approaches and can reveal trends and relationships in raw data. Each central measurement provides a different perspective; together, they offer a comprehensive picture of the data's tendencies.