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The standard deviation alone does not measure relative variation. For example, a standard deviation of \(\$ 1\) would be considered large if it is describing the variability from store to store in the price of an ice cube tray. On the other hand, a standard deviation of \(\$ 1\) would be considered small if it is describing store-to-store variability in the price of a particular brand of freezer. A quantity designed to give a relative measure of variability is the coefficient of variation. Denoted by CV, the coefficient of variation expresses the standard deviation as a percentage of the mean. It is defined by the \(C V=100\left(\frac{s}{\bar{x}}\right)\) formula Consider two samples. Sample 1 gives the actual weight (in ounces) of the contents of cans of pet food labeled as having a net weight of 8 ounces. Sample 2 gives the actual weight (in pounds) of the contents of bags of dry pet food labeled as having a net weight of 50 pounds. The weights for the two samples are \(\begin{array}{lrrrrr}\text { Sample 1 } & 8.3 & 7.1 & 7.6 & 8.1 & 7.6 \\ & 8.3 & 8.2 & 7.7 & 7.7 & 7.5 \\ \text { Sample 2 } & 52.3 & 50.6 & 52.1 & 48.4 & 48.8 \\ & 47.0 & 50.4 & 50.3 & 48.7 & 48.2\end{array}\) a. For each of the given samples, calculate the mean and the standard deviation. b. Compute the coefficient of variation for each sample. Do the results surprise you? Why or why not?

Step by step solution

01

Calculate Mean

Add all values in the sample and then divide by the count of values. Do this for both samples.

02

Calculate Standard Deviation

Subtract the mean from each value, square the results, add those, divide by how many values minus one, then take the square root. Do this for both samples.

03

Calculate Coefficient of Variation

Take the standard deviation, divide it by the mean, and multiply by 100. Do this for both samples.

04

Analyzing Results

Discuss which sample has a higher coefficient of variation, and what this tells us about the spread of the data.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation

The standard deviation is a statistical measure that captures the extent of spread or variability in a set of data. It tells us how much the individual data points in a sample deviate from the mean (average) of the data set. To calculate it, you need to follow a few steps:
First, subtract the mean from each data point in your sample. Then, square each of these differences. Next, add all of these squared differences together. Divide this sum by one less than the number of data points in the sample (this is called degrees of freedom). Finally, take the square root of this quotient.
This entire process gives you the standard deviation, which can provide a valuable insight into the data's variability. For example, a small standard deviation suggests that the data points are close to the mean, indicating low variability. Conversely, a large standard deviation suggests a wider range of data points.

Mean Calculation

Calculating the mean is often the first step in understanding your data set. The mean gives you the average value of the data, and it's quite straightforward to compute. To find the mean, simply sum up all the data points and then divide by the number of data points.
Let's say you have five data points like in Sample 1: 8.3, 7.1, 7.6, 8.1, and 7.6. Add these numbers up to get a total, and then divide by 5, which is the count of data points. This simple calculation provides you with the mean, which serves as a reference point for further analysis, such as standard deviation and coefficient of variation.
Knowing the mean is crucial since it allows you to compare other statistical measures, providing a foundation for deeper analysis.

Data Variability

Data variability quantifies the extent to which data values in a set diverge from the average (mean) value. It's an essential aspect of statistical analysis because it gives insights into the reliability and consistency of your data.
High variability implies that the data points are widely spread out, and this could signify inconsistency or diverse conditions affecting the observation. On the other hand, low variability suggests that data points are closely clustered around the mean, indicating stability and consistency.
Understanding variability is important when comparing data from different samples or conditions, as it contextualizes the results. For instance, even if two data sets have the same mean, their variability could offer starkly different stories.

Statistical Analysis

Statistical analysis involves using mathematical methods to make sense of data. This includes summarizing the data, identifying patterns or trends, and making conclusions based on the data. Key elements of statistical analysis include measures of central tendency, like the mean, and measures of dispersion, such as variance and standard deviation.
One useful tool in statistical analysis is the coefficient of variation (CV). It is calculated by dividing the standard deviation by the mean and multiplying by 100 to express it as a percentage. The CV helps compare the degree of variability between different samples, even if their means are vastly different.
By using concepts like these, statistical analysis enables researchers to derive meaningful insights, test hypotheses, and make decisions based on data. It's a vital skill in many fields, from biology and economics to engineering and social sciences.