Problem 23 - Thinking Mathematically

Question

In Exercises 17-26, find the standard deviation for each group of data items. Round answers to two decimal places \(1,1,1,4,7,7,7\)

Step-by-Step Solution

Answer

The standard deviation of the data set is 3

Step 1: Calculate the Mean

Add all the data items together and then divide by the number of data items: Mean \(\overline{x} = (1+1+1+4+7+7+7)/7 = 4\)

Step 2: Subtract the Mean and square the Result

For each data point, subtract the mean and square the result: (1-4)^2=9, (1-4)^2=9, (1-4)^2=9, (4-4)^2=0, (7-4)^2=9, (7-4)^2=9, (7-4)^2=9. It can be noticed that all deviations from the mean squared are either 0 or 9.

Step 3: Determine the average squared deviation from the Mean (variance)

Add up the square of all deviations and divide by the total number of data items-1: Variance \(\sigma^2 = \frac{(9+9+9+0+9+9+9)}{7-1} = \frac{54}{6} = 9\)

Step 4: Take the square root of variance

To calculate the standard deviation, take the square root of the variance: \(\sigma = \sqrt{9} = 3\)

Key Concepts

Mean Calculation

The mean, also known as the average, is the central tendency of a data set. Calculating the mean involves a straightforward process. You start by adding up all the data points. In this exercise, we dealt with the numbers

• 1
• 1
• 1
• 4
• 7
• 7
• 7

Adding these numbers gives you a total sum of 28. Next, you divide this sum by the total number of data points, which is 7, resulting in a mean of 4.
This mean value of 4 serves as a crucial component when further analyzing the data, as it represents an average value that all data points can be compared against.

Variance

Variance is a measure of how spread out the numbers in the data set are around the mean. Calculating variance is a critical step to understanding data dispersion.
To find the variance, you first need to calculate the deviations of each data point from the mean. In this case, the deviations are

• (1−4)
• (1−4)
• (1−4)
• (4−4)
• (7−4)
• (7−4)
• (7−4)

These deviations are squared to eliminate negative numbers, resulting in:

• 9
• 9
• 9
• 0
• 9
• 9
• 9

The variance is calculated by averaging these squared deviations. This is done by summing up the squared results (which total 54) and dividing by the number of data points minus one (which is 6 in this case). Thus, the variance comes out to be 9. A higher variance would indicate more variability in the data while a smaller variance suggests lesser spread around the mean.

Data Analysis

Data analysis involves breaking down and interpreting the gathered data to draw conclusions. Standard deviation, a concept closely tied to variance, offers insights about data variability. In this exercise, the standard deviation was calculated by taking the square root of the variance, yielding a result of 3.
This value informs us about the average distance the data points lie from the mean. Understanding the standard deviation helps to comprehend the data set's degree of variation.
In practical terms:

• A smaller standard deviation suggests that data points are nearer to the mean, implying consistency.
• A larger standard deviation points to more scattered data, signaling greater variability.

Using these calculations, you can make informed decisions and predictions based on how much variation exists within the data set being analyzed.