Question

Find the coefficient of variation for the given distribution: (1) \(\frac{200 \sqrt{6}}{11}\) (2) \(\frac{200 \sqrt{3}}{11}\) (3) \(\frac{500}{11}\) (4) \(\frac{200 \sqrt{5}}{11}\)

Short answer

Answer: The second distribution has the smallest coefficient of variation, with a value of \(\frac{200 \sqrt{3}}{11}\).

Step-by-step solution

Determine the mean and standard deviation for each distribution

Since we are already given the mean and standard deviation (in the form of expressions), we don't need to explicitly calculate them. We only need to keep them in mind when calculating the coefficient of variation in the next step.

Compute the coefficient of variation for each distribution

The coefficient of variation is calculated as follows: CV = \(\frac{Standard\ Deviation}{Mean}\) Now let's calculate the coefficient of variation for each given distribution: (1) CV1 = \(\frac{200 \sqrt{6}}{11}\) (2) CV2 = \(\frac{200 \sqrt{3}}{11}\) (3) CV3 = \(\frac{500}{11}\) (4) CV4 = \(\frac{200 \sqrt{5}}{11}\)

Compare the coefficients of variation and select the smallest one

In order to identify the distribution with the smallest coefficient of variation, we need to compare the values of CV1, CV2, CV3, and CV4 calculated in Step 2. By comparing the expressions for CV1, CV2, CV3, and CV4, we can see that CV2 (which equals \(\frac{200 \sqrt{3}}{11}\)) has the smallest value, as the square root of 3 is the smallest among all the square root terms. Therefore, the distribution with the smallest coefficient of variation is the second one: \(\frac{200 \sqrt{3}}{11}\).

Key concepts

Standard Deviation

Understanding standard deviation is crucial for analyzing data sets in statistics. It is a measure that tells us how spread out the numbers in a data set are around the mean (average). If the data points are close to the mean, the standard deviation will be small, indicating that there is less volatility. Conversely, a larger standard deviation shows that data points are more spread out and the data set has higher variability. To calculate it, one would subtract the mean from each data point, square the result, average these squares, and take the square root of that average. This process gives insight into the consistency of the data.

Mean

The mean, also known as the average, is a fundamental statistical measure that is found by adding up all the numerical values in a data set and dividing by the number of values. It provides a central value around which the data points are distributed. While the mean can offer insights into the general trend of the data, it can also be affected by outliers – extremely high or low values that could skew the result. It's important for students to remember that the mean is different from the median (the middle value) or the mode (the most frequent value), each serving distinct purposes in the realm of statistics.

Statistical Distribution

In statistics, a statistical distribution describes how the values of a variable are observed to occur, documenting the frequency of each value or range of values. There are various types of distributions, such as normal (bell-curve), uniform, or skewed distributions, each with unique characteristics. Distributions help in understanding the probabilities of different outcomes and are essential for conducting tests of significance. When visualizing distributions, histograms or probability density functions can be used to represent the data in a way that highlights the underlying pattern of variation.

Comparing Coefficients

Comparing coefficients, particularly the coefficient of variation (CV), is a powerful statistical tool used to determine the relative variability or dispersion of data sets. Unlike standard deviation, the CV expresses variability as a percentage of the mean, making it especially useful when comparing the degree of variation from one data series to another, even if the means are drastically different. It is calculated by dividing the standard deviation by the mean, then multiplying by 100 for a percentage. The smaller the CV, the more consistent the data and vice versa. In practical terms, the CV offers a means to evaluate investment risks, test reliability, and assess quality control in various fields, favoring consistency in outcomes and decisions based on data.