Problem 19 - Thinking Mathematically

Question

In Exercises 17-26, find the standard deviation for each group of data items. Round answers to two decimal places \(7,9,9,15\)

Step-by-Step Solution

Answer

The standard deviation for the given data set \(7, 9, 9, 15\) can be found by following the above steps. Replace \(\mu\) with the calculated value in Steps 2 and 3 and evaluate the expressions to get \(\sigma\), the standard deviation. This value should be rounded to two decimal places.

Step 1: Calculate the Mean

Add all the data points together and divide by the total number of points. i.e. \(\mu = \frac{(7 + 9 + 9 + 15)}{4}\)

Step 2: Subtract the Mean and Square the Result

For each data point, subtract the mean found in step 1, then square the result. This gives us the squared variance for each data point. \( (7 - \mu)^2, (9 - \mu)^2, (9 - \mu)^2, (15 - \mu)^2 \)

Step 3: Calculate Variance

Add the squared differences found in step 2 together, then divide by \(n - 1\) where \(n\) is the number of data points: \(\frac{(7 - \mu)^2 + (9 - \mu)^2 + (9 - \mu)^2 + (15 - \mu)^2}{4 - 1}\)

Step 4: Calculate Standard Deviation

Take the square root of the variance calculated in step 3. This is our final answer.

Key Concepts

Mean Calculation

Calculating the mean is the first step in determining the standard deviation. Think of the mean as the average of a set of numbers. To find the mean, you add together all the data points. Then, divide the sum by the total number of data points. For example, with the numbers 7, 9, 9, and 15, add them up to get 40. Next, divide 40 by 4, because there are four data points: \[ \mu = \frac{7 + 9 + 9 + 15}{4} = 10 \]. In this case, the mean is 10. The mean helps us understand the central tendency of the data, which is essential for further calculations like variance and standard deviation. Remember to break problems into smaller parts like this to make the process easier.

Variance Calculation

Variance measures how spread out the numbers in a data set are. It's the second step in finding the standard deviation. Start by subtracting the mean from each data point. This gives you the difference of each point from the average. For example, subtract 10 (the mean) from each data point:

• 7 - 10 = -3
• 9 - 10 = -1
• 9 - 10 = -1
• 15 - 10 = 5

Now, square each of these differences to eliminate negative numbers:

• (-3)^2 = 9
• (-1)^2 = 1
• (-1)^2 = 1
• (5)^2 = 25

Next, add these squared differences together: \[ 9 + 1 + 1 + 25 = 36 \]. Finally, divide this sum by \(n - 1\), which is 3 in this exercise since there are 4 data points. This gives us \[ \text{Variance} = \frac{36}{3} = 12 \]. Variance shows the data's spread around the mean. A larger variance indicates more spread out data.

Statistical Analysis

Statistical analysis involves the collection, exploration, and interpretation of data. It's crucial in fields like education, medicine, and business. Understanding fundamental concepts like mean and variance enables more complex statistical techniques. These numerical calculations help us make sense of large amounts of data, spot trends, or test hypotheses. After knowing the variance, the next step would often be calculating the standard deviation, providing insights into data reliability and stability. In the daily world, statistical analysis helps in:

• Making business decisions
• Predicting trends and behaviors
• Understanding scientific research findings

In essence, good statistical skills can turn raw data into powerful information for informed decision-making.

Educational Mathematics

Educational mathematics plays a key role in understanding data analysis. It equips students with the skills needed to tackle problems like calculating standard deviation. Mastering these steps aids comprehension and boosts confidence in handling real-world scenarios. Mathematics education focuses on:

• Building problem-solving skills
• Enhancing critical thinking
• Facilitating understanding of mathematical concepts, such as mean and variance

By embracing educational mathematics, students learn to apply mathematical concepts practically. Programs often integrate technology, making learning interactive and engaging. This encourages a deeper understanding of concepts beyond procedures, fostering a lifelong appreciation of mathematics.