Problem 50 - Basic Statistics for Business and Economics with Student

Question

Do the following: a. Compute the sample variance. b. Determine the sample standard deviation. The sample of eight companies in the aerospace industry, referred to in Exercise \(36,\) was surveyed as to their return on investment last year. The results are: \(10.6,12.6,14.8,18.2,12.0,14.8,12.2,\) and \(15.6 .\)

Step-by-Step Solution

Answer

Sample variance is 6.0182 and standard deviation is 2.453.

Step 1: Calculate the Mean

To find the mean, add together all the values of the sample data and then divide by the number of values. The returns on investment are \(10.6, 12.6, 14.8, 18.2, 12.0, 14.8, 12.2,\) and \(15.6\). Summing these gives \(110.8\). Divide by the number of companies, which is 8.\[ \text{Mean} = \frac{110.8}{8} = 13.85. \]

Step 2: Compute Deviations from the Mean

Subtract the mean from each of the sample values to find the deviations: \(10.6-13.85, 12.6-13.85, 14.8-13.85, 18.2-13.85, 12.0-13.85, 14.8-13.85, 12.2-13.85, 15.6-13.85\). The deviations are \(-3.25, -1.25, 0.95, 4.35, -1.85, 0.95, -1.65, 1.75\).

Step 3: Square the Deviations

Square each deviation value calculated in the previous step: \((-3.25)^2, (-1.25)^2, 0.95^2, 4.35^2, (-1.85)^2, 0.95^2, (-1.65)^2, 1.75^2\). The squared deviations are \(10.56, 1.56, 0.9025, 18.92, 3.42, 0.9025, 2.7225, 3.0625\).

Step 4: Compute the Sample Variance

Sum the squared deviations and divide by \(n-1\), where \(n\) is the number of companies (8-1=7). The sum of squared deviations is \(42.1275\). Thus, the sample variance is:\[ s^2 = \frac{42.1275}{7} = 6.0182 \]

Step 5: Determine the Sample Standard Deviation

Take the square root of the sample variance to find the standard deviation: \[ s = \sqrt{6.0182} = 2.453 \]

Key Concepts

Sample Standard Deviation

Ever wondered how to quantify the spread of data points in a sample? That's where the sample standard deviation comes into play. It is a measure that tells us how much the data points in a sample deviate from the mean.

Here's how to calculate it:

• First, find the sample variance (more on this in the sample variance section).
• Once you have the variance, take its square root. This step transforms the measurement back to the original units of the data, making it easier to interpret.

In the provided exercise, the sample variance was calculated as 6.0182. Therefore, the sample standard deviation is calculated as \( s = \sqrt{6.0182} \approx 2.453 \).

Understanding the standard deviation helps identify how much variation exists from the average. A larger standard deviation indicates more spread out data, while a smaller one points to data that is closely clustered around the mean.

Mean Calculation

Calculating the mean, or average, of a dataset is a fundamental step in understanding the data as it provides the central value around which other data points cluster. To find the mean:

• Add all the values together.
• Divide the total by the number of data points.

In our specific example with the companies' returns on investment, we sum values such as 10.6, 12.6, 14.8, 18.2, and so forth, to get 110.8. Then, by dividing by 8 (the number of companies), the mean return is calculated as \( \text{Mean} = \frac{110.8}{8} = 13.85 \).

This mean serves as the foundation for further calculations like deviations and the standard deviation. Always remember, the mean represents a central or typical value in a dataset.

Deviation Calculation

Understanding deviations is key to comprehending variability in data. A deviation indicates how much a single data point differs from the mean of the dataset.

• Subtract the mean from each data point to find the deviation.
• Each result shows how far and in which direction each point lies from the mean.

In the example, the mean was 13.85. Hence, each company's return was adjusted by subtracting this mean, yielding deviations like \(-3.25, -1.25, 0.95, 4.35,\) etc.

These deviations are vital for understanding the overall variance and standard deviation, as they provide insight into the spread of the dataset.

Squared Deviations

After calculating deviations, the next step involves squaring these deviations. But why square them? This process removes any negative signs and emphasizes larger deviations more.

• Square each deviation result.
• Summing these squared values leads to calculating the variance.

For the given dataset, deviations like \(-3.25\) become (after squaring) 10.56, and similar operations follow for other deviations such as 0.95 being squared to become 0.9025.

Squaring deviations plays a critical role in computing the sample variance. By working with squared deviations, we prevent negative differences from canceling out positive ones, ensuring an accurate measure of spread in the data.