Problem 3 - Algebra Form and Function

Question

Find the mean and standard deviation of the data set. $$ 81,57,14,98,20,20,6 $$

Step-by-Step Solution

Answer

Question: Calculate the mean and standard deviation of the dataset \(81, 57, 14, 98, 20, 20, 6\). Answer: The mean of the dataset is approximately \(42.29\) and the standard deviation is approximately \(33.69\).

Step 1: Find the Mean

To find the mean, we will add up all the data points and divide by the total number of data points. Mean = \(\frac{81+57+14+98+20+20+6}{7}\) Mean = \(\frac{296}{7}\) Mean ≈ \(42.29\)

Step 2: Subtract the Mean and Square the Differences

For each data point, we'd subtract the mean and then square the result: \( (81-42.29)^2 \approx 1500.05\) \( (57-42.29)^2 \approx 216.30\) \( (14-42.29)^2 \approx 804.96\) \( (98-42.29)^2 \approx 3106.30\) \( (20-42.29)^2 \approx 494.47\) \( (20-42.29)^2 \approx 494.47\) \( (6-42.29)^2 \approx 1316.05\)

Step 3: Find Mean of Squared Differences

Now we need to find the mean of these squared differences. Mean of Squared Differences = \(\frac{1500.05+216.30+804.96+3106.30+494.47+494.47+1316.05}{7}\) Mean of Squared Differences = \(\frac{7932.60}{7}\) Mean of Squared Differences = \(1133.23\)

Step 4: Take the Square Root

Finally, we will take the square root of the mean of squared differences to find the standard deviation. Standard Deviation = \(\sqrt{1133.23}\) Standard Deviation ≈ \(33.69\) So the mean of the dataset is approximately \(42.29\) and the standard deviation is approximately \(33.69\).

Key Concepts

Mean Calculation

The first step in understanding descriptive statistics is mastering the concept of mean calculation. This gives us the average value of a data set. To find the mean, you must first add all the numbers in your data set. Then, you divide that sum by how many numbers there are in total. In the case of our data set, which includes the numbers 81, 57, 14, 98, 20, 20, and 6, we first sum them up:

• Total Sum: 81 + 57 + 14 + 98 + 20 + 20 + 6 = 296

After obtaining the total, divide by the number of data points:

• Number of Data Points: 7
• Mean = \(\frac{296}{7} = 42.29\)

Hence, the mean or average of our data is approximately 42.29. This value represents a central tendency of our data and is a foundational measure in statistics. The mean can give us a quick glimpse into the general size or magnitude of the numbers in our dataset.

Standard Deviation Calculation

Calculating the standard deviation is crucial for understanding how spread out the numbers within a data set are. It tells us how much variation from the mean exists. Here's how to find it:First, subtract the mean from each data point and square the result. This captures the deviation of each point from the mean:

• \((81 - 42.29)^2 \approx 1500.05\)
• \((57 - 42.29)^2 \approx 216.30\)
• \((14 - 42.29)^2 \approx 804.96\)
• \((98 - 42.29)^2 \approx 3106.30\)
• \((20 - 42.29)^2 \approx 494.47\)
• \((20 - 42.29)^2 \approx 494.47\)
• \((6 - 42.29)^2 \approx 1316.05\)

Next, find the mean of these squared differences:

• Sum of Squared Differences: 7932.60
• Mean of Squared Differences: \(\frac{7932.60}{7} = 1133.23\)

Finally, take the square root of that mean to find the standard deviation:

• Standard Deviation: \(\sqrt{1133.23} \approx 33.69\)

Thus, the standard deviation tells us that the data points tend to fall about 33.69 units away from the mean, providing insight into the dataset's variability.

Data Analysis

Analyzing data involves summarizing and understanding large sequences of data points to gain insights. Descriptive statistics, like mean and standard deviation, are a great starting point. They help us interpret the initial state of the dataset and derive usable information.
The mean gives an idea of the overall level of the dataset, acting as a measure of central tendency. It's like finding the balance point of the data.
Meanwhile, the standard deviation offers an understanding of the spread of the data. Higher standard deviation means more spread out data points. It can influence how we view the stability or variability of the dataset.
Together, these metrics allow us to draft a comprehensive picture of our dataset's behavior. This analysis shows that while the mean (42.29) centralizes our dataset, its high standard deviation (33.69) signals widespread data points, indicating potential outliers or diversities within the data.