Problem 45 - Basic Statistics for Business and Economics with Student

Question

Plywood, Inc., reported these returns on stockholder equity for the past 5 years: \(4.3,4.9,7.2,6.7,\) and \(11.6 .\) Consider these as population values. a. Compute the range, the arithmetic mean, the variance, and the standard deviation. b Comnare the return on stockholder ea

Step-by-Step Solution

Answer

Range: 7.3, Mean: 6.94, Variance: 6.5944, Standard Deviation: 2.568.

Step 1: Calculate the Range

The range is the difference between the maximum and minimum values in the data set. Identify the smallest and largest numbers. Smallest (min) = 4.3, Largest (max) = 11.6. Thus, the range = 11.6 - 4.3 = 7.3.

Step 2: Calculate the Arithmetic Mean

Sum all the values and divide by the number of values. Arithmetic mean = \( \frac{4.3 + 4.9 + 7.2 + 6.7 + 11.6}{5} = \frac{34.7}{5} = 6.94 \).

Step 3: Calculate the Variance

Variance is the average of the squared differences from the mean. First, compute each difference from the mean, square it, and then average those squares. Differences: \((4.3 - 6.94)^2, (4.9 - 6.94)^2, (7.2 - 6.94)^2, (6.7 - 6.94)^2, (11.6 - 6.94)^2\). Squared differences: \( (2.64)^2, (2.04)^2, (0.26)^2, (0.24)^2, (4.66)^2 \). Squares: \(6.9696, 4.1616, 0.0676, 0.0576, 21.7156\). Variance = \( \frac{6.9696 + 4.1616 + 0.0676 + 0.0576 + 21.7156}{5} = \frac{32.972}{5} = 6.5944 \).

Step 4: Calculate the Standard Deviation

The standard deviation is the square root of the variance.\( \text{Standard deviation} = \sqrt{6.5944} \approx 2.568 \).

Key Concepts

Arithmetical Mean

The arithmetical mean, often referred to as the average, is a central value that represents a set of numbers. It is widely used in business statistics to obtain a general idea of the data's overall level. To calculate the arithmetical mean, you sum up all the values in the dataset and then divide by the number of values.
For example, with the returns on stockholder equity given as 4.3, 4.9, 7.2, 6.7, and 11.6:

• Sum of values: 4.3 + 4.9 + 7.2 + 6.7 + 11.6 = 34.7
• Number of values: 5
• Arithmetic mean: \( \frac{34.7}{5} = 6.94 \)

This mean gives a single value that summarizes the overall performance across these years. It's important because it allows for quick comparison with other periods or datasets. However, it does not show how much the individual data points vary from this average.

Data Variance

Variance measures how much the data is spread out. It tells us how much the individual numbers in a dataset differ from the arithmetical mean. In business statistics, understanding variance can help assess stability or expose volatility in returns.
To find the variance, follow these steps:

• Calculate each value's deviation from the mean. For example, for the value 4.3 and mean 6.94: \(4.3 - 6.94 = -2.64\).
• Square each deviation to avoid negative differences: \((-2.64)^2 = 6.9696\).
• Average these squared differences. For example: \(\frac{(6.9696 + 4.1616 + 0.0676 + 0.0576 + 21.7156)}{5} = 6.5944\).

A higher variance indicates a wider range of values, which means more volatility in the returns. In our example, the variance of 6.5944 suggests the returns have a moderate spread around the mean.

Standard Deviation

Standard deviation is closely related to variance, serving as a measure of spread in the dataset and is the square root of the variance. It provides a way to quantify uncertainty or inconsistency in business statistics.
To compute the standard deviation, you take the square root of the variance:

• Given variance: 6.5944
• Standard deviation: \( \sqrt{6.5944} \approx 2.568 \)

This value is more intuitive because it is expressed in the same units as the original data points. In our case, a standard deviation of approximately 2.568 tells us the average distance each return is from the mean. A lower standard deviation would indicate that the data points tend to be very close to the mean, suggesting more reliability in the returns.

Range in Statistics

The range is the simplest measure of variability in statistics and represents the difference between the highest and lowest values in a dataset. This number provides insight into the span of the data.
To compute the range, identify the minimum and maximum values from the dataset and subtract the former from the latter:

• Minimum value: 4.3
• Maximum value: 11.6
• Range: 11.6 - 4.3 = 7.3

The range tells us how widespread the data values are but does not inform us about their distribution or clustering. A large range implies high variability, while a smaller range suggests that the data points are more clustered around a central value. In our scenario, a range of 7.3 indicates notable dispersion in the return values.