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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 formula \(C V=100\left(\frac{s}{\bar{x}}\right)\). 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 the Mean

First, find the mean (average) of each sample by summing up the dataset and then dividing by the number of elements in the sample. For sample 1, use the formula: \(\bar{x_1} = \frac{1}{n_1}\sum_{i=1}^{n_1} x_{1i}\). Similarly for sample 2, \(\bar{x_2} = \frac{1}{n_2}\sum_{i=1}^{n_2} x_{2i}\)

02

Calculate the Standard Deviation

Next, calculate the standard deviation of each sample, which measures the amount of variation in the weights. For each sample, first subtract the mean from each data point and square the result. Then, add up these squared results and divide by the number of data points. Finally, take the square root of this resulting number. Use the formula \(s_i=\sqrt{\frac{1}{n_i-1}\sum_{k=1}^{n_i} (x_{ik}-\bar{x_i})^2}\) for each sample \(i\).

03

Calculate the Coefficient of Variation

Now calculate the coefficient of variation for each sample. The coefficient of variation (CV) is a measure of relative variability. It is the ratio of the standard deviation to the mean, and it is often expressed as a percentage. The formula to calculate the coefficient of variation is \(CV_i=100\left(\frac{s_i}{\bar{x_i}}\right)\).

04

Interpret the Results

Finally, interpret the results. Compare the coefficients of variation for both samples. The sample with the larger coefficient of variation has more variability when compared to the sample's mean. Analyse if this result aligns with the expectation, based on the nature of the samples (different types of pet food). The surprise element in the results could come from the assumption of higher variability expected in larger quantities (50 lbs bag vs 8 ounces can).

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Key Concepts

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

Understanding Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of values. It is one of the most widely used metrics for understanding how spread out data points are around the mean (average) value of the dataset. To calculate standard deviation, one typically follows these steps: subtract the mean from each data point to find the deviation of each point, square each deviation to make all values positive, average the squared deviations, and then take the square root of that average.

A smaller standard deviation signifies that the data points tend to be closer to the mean, indicating less variability. Conversely, a larger standard deviation indicates more spread out data points and greater variability. This metric provides a context for the mean, giving users a sense of the 'typical' distance data points are from the average value.

Relative Variability and the Coefficient of Variation

Relative variability is an assessment of how significant the standard deviation is in relation to the mean of the dataset. It answers the question: relative to the size of the mean, how large is the variability? This is crucial when comparing variability across datasets with different units or scales. To measure this, we use the coefficient of variation (CV).

The CV is calculated by dividing the standard deviation by the mean and then multiplying by 100 to express it as a percentage: \( CV = 100 \left(\frac{s}{\bar{x}}\right) \) where \( s \) is the standard deviation and \( \bar{x} \) is the mean. The higher the CV, the greater the level of dispersion relative to the mean. When CV is used, we can effectively compare how variable two datasets are, regardless of their absolute values or units, as seen in our exercise with different types of pet food weights.

Statistical Analysis in Practice

Statistical analysis involves collecting, exploring, and presenting large amounts of data to discover underlying patterns and trends. This may involve applying various statistical measures, like mean and standard deviation, to make sense of the data collected.

In the context of the provided exercise, statistical analysis allows us to understand not just the typical weights of pet food, but also how consistent those weights are. This is vital for quality control — ensuring consumers get what they pay for. Through calculation of the mean and standard deviation, then interpreting the coefficient of variation, one gets a comprehensive view of the product’s consistency. This ultimately guides manufacturers and consumers alike in making informed decisions. While often surprising, results such as these shed light on the nature of production processes and help in setting or understanding expectations for variability in different products or contexts.