Problem 13 - Algebra 2 and Trigonometry

Question

The given values represent data for a sample. Find the variance and the standard deviation based on this sample. 1, 3, 5, 22, 30, 45, 50, 55, 60, 70

Step-by-Step Solution

Answer

The sample variance is approximately 648.99, and the standard deviation is approximately 25.47.

Step 1: Find the Mean

To find the mean, sum up all the data points and divide by the number of data points. Add the values: \(1 + 3 + 5 + 22 + 30 + 45 + 50 + 55 + 60 + 70 = 341\). There are 10 data points.Calculate the mean: \(\bar{x} = \frac{341}{10} = 34.1\)

Step 2: Calculate the Deviations

Subtract the mean from each data point to get the deviations.For the values:1: \(1 - 34.1 = -33.1\)3: \(3 - 34.1 = -31.1\)5: \(5 - 34.1 = -29.1\)22: \(22 - 34.1 = -12.1\)30: \(30 - 34.1 = -4.1\)45: \(45 - 34.1 = 10.9\)50: \(50 - 34.1 = 15.9\)55: \(55 - 34.1 = 20.9\)60: \(60 - 34.1 = 25.9\)70: \(70 - 34.1 = 35.9\)

Step 3: Calculate the Squared Deviations

Square each of the deviation values found in step 2. -33.1² = 1095.61 -31.1² = 967.21 -29.1² = 846.81 -12.1² = 146.41 -4.1² = 16.81 10.9² = 118.81 15.9² = 252.81 20.9² = 436.81 25.9² = 670.81 35.9² = 1288.81

Step 4: Sum the Squared Deviations

Add up all the squared deviations to get the sum of squared deviations.\(1095.61 + 967.21 + 846.81 + 146.41 + 16.81 + 118.81 + 252.81 + 436.81 + 670.81 + 1288.81 = 5840.9\)

Step 5: Compute the Variance

Divide the sum of squared deviations by the number of data points minus 1 (since this is a sample). There are 10 data points, so use \(n-1=9\) in the denominator.Calculate the variance: \(\sigma^2 = \frac{5840.9}{9} \approx 648.99\)

Step 6: Calculate the Standard Deviation

Take the square root of the variance to obtain the standard deviation.\(\sigma = \sqrt{648.99} \approx 25.47\)

Key Concepts

Standard Deviation

Understanding standard deviation is key when analyzing data. It tells us how much the values in a data set differ from the mean. In simpler terms, it provides an indication of the spread or variability of the data set. A small standard deviation signifies that the data points are close to the mean, while a large one indicates they are spread out over a large range.

The formula to calculate the standard deviation from a sample is the square root of the variance. Here, our calculated variance was approximately 648.99. By taking the square root of this value, we find the standard deviation to be approximately 25.47. This tells us that the typical deviation of the data points from the mean is about 25.47 units.

Mean Calculation

Calculating the mean is the first step towards understanding a data set's core characteristics. The mean or average is determined by adding up all the data points and dividing by the number of points. In the given example, the sum of all data points is 341, with 10 data points in total.

Thus, the mean is computed as follows:

• Add the values: 1 + 3 + 5 + 22 + 30 + 45 + 50 + 55 + 60 + 70 = 341.
• Divide by the number of values: \( \bar{x} = \frac{341}{10} = 34.1 \).

This mean provides a central value around which the rest of the data is compared when calculating further statistical measures.

Deviation

Deviation shows how much a specific data point differs from the mean. Calculating the deviation for each value helps to understand the spread around the mean. To find the deviation, subtract the mean from each data point.

For example, the mean we calculated is 34.1. The deviation for the first data point (1) would be calculated as follows:

• Deviation for 1: \( 1 - 34.1 = -33.1 \).

This calculation is repeated for each value, and each result reflects whether the value is above or below the mean.

Squared Deviations

Squared deviations are used to eliminate negative values that arise during deviation calculation. By squaring these deviations, we ensure that all values are positive, which is necessary for calculating variance.

In essence, squaring the deviations serves to emphasize larger deviations, as they contribute more to the variance. For example, for a deviation of -33.1, its square is:

• \( (-33.1)^2 = 1095.61 \).

Performing this squaring for all deviations, we eventually sum them up to finalize our variance calculation. This step boosts the impact of any outliers in the data on the resulting variance.