Problem 4 - Thinking Mathematically

Question

In Exercises \(1-8\), find the mean for each group of data items. \(100,100,90,30,70,100\)

Step-by-Step Solution

Answer

The mean of the data set is 81.67

Step 1: Sum all data points

First, we need to add all the data points together. So, \(100 + 100 + 90 + 30 + 70 + 100 = 490\)

Step 2: Count the number of data points

Next, count the number of data points. In our case, there are 6 data points.

Step 3: Calculation of the mean

Divide the sum of all data points by the count to calculate the mean. That is \(490 \div 6 = 81.67\) when rounded to two decimal places.

Key Concepts

Data Points

When we talk about "data points," we're referring to the individual numbers or values in a set of data that are being evaluated. Each data point contributes equally to the overall analysis. In the given example, the data points are: 100, 100, 90, 30, 70, and 100.
These numbers might represent anything such as test scores, sales figures, or temperature readings. Recognizing each data point separately is crucial as they collectively form the dataset.

• In our example, the data points are fixed at six discrete numbers.
• They provide the raw material necessary to proceed with statistical analysis such as calculating the mean.

Understanding what data points are helps set the stage for more advanced concepts like finding the mean.

Sum of Data Points

Adding the data points together gives you what's known as the "sum of data points." This sum acts as a foundation for calculating various statistical measures, including the mean. It's a straightforward concept: simply add all the data point values together. In the example provided, this is conducted as follows:
\[100 + 100 + 90 + 30 + 70 + 100 = 490\]
This total sum of 490 is pivotal for further calculations. By finding the sum, you're aggregating all the data into a single figure that captures the total magnitude.

• This measure is critical before moving ahead to calculate the mean.
• It acts as a stepping stone by summarizing the entire dataset in one number.

Recognizing the importance of the sum of data points is an essential step in every statistical analysis.

Number of Data Points

The "number of data points" is essentially a count of all individual data values included in your set. This simple count gives you an idea about the size of the dataset. In this particular exercise, we are dealing with six data points:

• 100
• 100
• 90
• 30
• 70
• 100

The number of data points offers the divisor in the formula for calculating the mean. By knowing how many data points are there, you understand how to balance the total sum across them. It's also a factor in analyzing the reliability and consistency of your data. Small numbers of data points might lead to less reliable statistics, for instance.

Arithmetic Mean

The "arithmetic mean," often referred to simply as the mean, is a type of average that is calculated by summing all data points and then dividing by the number of those data points. This measure gives you a central value that represents the entire dataset. Here, the steps are simple:

• First, sum the data points: \[490\].
• Next, divide by the number of data points: \[6\].
• This calculation gives \[ \frac{490}{6} = 81.67 \].

Finding the arithmetic mean offers insight into the overall trend and helps in making comparisons. For example, with a mean of 81.67 in a set of test scores, you understand the average performance level. While useful, remember the mean might not always be the best measure when the data has outliers or is skewed. Nonetheless, it's a handy tool for a first glance at the data's tendency.