Problem 8 - Algebra 2

Question

Find the variance and standard deviation of each set of data to the nearest tenth. {2.4, 5.6, 1.9, 7.1, 4.3, 2.7, 4.6, 1.8, 2.4}

Step-by-Step Solution

Answer

Variance: 3.1, Standard Deviation: 1.8

Step 1: Calculate the Mean

Find the mean (average) of the data set by summing all the values and dividing by the number of values. The data set is \{2.4, 5.6, 1.9, 7.1, 4.3, 2.7, 4.6, 1.8, 2.4\}. Add these numbers together: \(2.4 + 5.6 + 1.9 + 7.1 + 4.3 + 2.7 + 4.6 + 1.8 + 2.4 = 32.8\). The number of values is 9, so the mean \(\bar{x}\) is \(\frac{32.8}{9} \approx 3.6\).

Step 2: Calculate the Squared Differences

Subtract the mean from each data point and square the result. This gives: \((2.4 - 3.6)^2 = 1.44\), \((5.6 - 3.6)^2 = 4\), \((1.9 - 3.6)^2 = 2.89\), \((7.1 - 3.6)^2 = 12.25\), \((4.3 - 3.6)^2 = 0.49\), \((2.7 - 3.6)^2 = 0.81\), \((4.6 - 3.6)^2 = 1\), \((1.8 - 3.6)^2 = 3.24\), \((2.4 - 3.6)^2 = 1.44\).

Step 3: Calculate the Variance

Find the variance by determining the average of the squared differences. Add up all the squared differences: \(1.44 + 4 + 2.89 + 12.25 + 0.49 + 0.81 + 1 + 3.24 + 1.44 = 27.56\). Divide by the number of data points (9) to find the variance: \(\frac{27.56}{9} \approx 3.1\).

Step 4: Calculate the Standard Deviation

The standard deviation is the square root of the variance. Take the square root of 3.1 to get the standard deviation: \(\sqrt{3.1} \approx 1.8\).

Key Concepts

Mean Calculation

The calculation of the mean, often referred to as the average, is a fundamental step in understanding statistics. It provides a central value for the data set, which is useful for comparison and analysis. To find the mean of a set of numbers, follow these steps:

• Add all the numbers in the data set together.
• Count how many numbers are in the data set.
• Divide the total sum by the count of numbers.

This will give you the arithmetic mean. For example, in the data set \{2.4, 5.6, 1.9, 7.1, 4.3, 2.7, 4.6, 1.8, 2.4\}, the sum is 32.8, and there are 9 numbers. Therefore, the mean (\(\bar{x}\)) is \(\frac{32.8}{9} \approx 3.6\). The mean acts as a balancing point, where the data points are equally distributed around it.

Squared Differences

Calculating squared differences is an important step in finding both variance and standard deviation. Once the mean of the data set is calculated, the next step is to see how much each data point deviates from this mean.

• Subtract the mean from each data point to find the deviation.
• Square each deviation.

The squaring is essential because it ensures that positive and negative deviations do not cancel each other out. Additionally, squaring emphasizes larger deviations, providing a clearer picture of spread than simply summing deviations would allow. In the given example, you calculate deviations like \((2.4 - 3.6) = -1.2\) and then square it to get \((1.2^2 = 1.44)\). You repeat these steps for each data point in your set.

Standard Deviation

The standard deviation provides a measure of the dispersion or spread in a set of data. It is a critical concept in statistics that reflects how varied the data points are from the mean value. To find the standard deviation, first calculate the variance which involves:

• Summing all squared differences from the mean.
• Dividing this sum by the number of data points in the set to get the variance.

Once the variance is calculated, taking the square root of this value yields the standard deviation:

• Standard deviation = \(\sqrt{\text{variance}}\)

From the previous steps, the variance of the data was found to be approximately 3.1, making the standard deviation \(\sqrt{3.1} \approx 1.8\). This value provides insights into how much variation or "spread" exists from the mean, with a lower standard deviation indicating that the data points tend to be close to the mean, while a higher value indicates a larger spread.