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Calculating the mean is the first step when analyzing a dataset. It gives us an average value, offering a simple look at the data as a whole. In this example, the mean percentage of defective items is calculated from five different percentages: 2%, 1.4%, 4%, 3%, and 2.2%. Here's how you calculate the mean:1. Add all the percentages together: \( 2 + 1.4 + 4 + 3 + 2.2 = 12.6 \).2. Divide the sum by the total number of observations, which in this case is 5 days: \( \frac{12.6}{5} = 2.52 \% \).The mean percentage of defective items for the week is therefore 2.52%. This figure helps us understand the typical percentage of defects over the observed period.

Standard deviation measures the amount of variation or dispersion in a dataset. A low standard deviation means the data points are close to the mean, while a high standard deviation indicates greater spread. Here's how to calculate it using our example:1. Calculate the deviation of each day's percentage from the mean (2.52%). - Monday: \(2 - 2.52 = -0.52\) - Tuesday: \(1.4 - 2.52 = -1.12\) - Wednesday: \(4 - 2.52 = 1.48\) - Thursday: \(3 - 2.52 = 0.48\) - Friday: \(2.2 - 2.52 = -0.32\)2. Square each deviation to eliminate negative values and emphasize larger deviations: - \((-0.52)^2 = 0.2704\) - \((-1.12)^2 = 1.2544\) - \((1.48)^2 = 2.1904\) - \((0.48)^2 = 0.2304\) - \((-0.32)^2 = 0.1024\)3. Average these squared deviations (this is the variance).4. Finally, take the square root of the variance to get the standard deviation: \( \sqrt{0.8096} \approx 0.8997 \% \).In this case, the standard deviation is approximately 0.8997%, indicating a moderate dispersion around the mean.

Variance provides a numerical representation of how much the values in a dataset differ from the mean. It is an important concept because it sets the stage for calculating standard deviation. To obtain the variance:1. First, calculate the deviation of each data point from the mean. This has already been done, with values like -0.52 and 1.48 as examples.2. Square each of these deviations to ensure all differences are positive, emphasizing larger deviations.3. Sum up these squared deviations: \( 0.2704 + 1.2544 + 2.1904 + 0.2304 + 0.1024 = 4.048 \).4. Divide this sum by the number of observations (5) to find the average of these squared values: \( \frac{4.048}{5} = 0.8096 \).Thus, the variance of the dataset is 0.8096. This shows us how far each percentage point is, on average, from the mean in square percentage terms.

Data analysis encompasses examining datasets to uncover patterns, insights, or anomalies. The goal is to turn information into a format that can be easily understood and used to make decisions. In this context, mean, variance, and standard deviation serve as crucial tools: - **Mean** offers a quick, central value of data, helping us grasp the average performance. - **Variance** details how spread out the data values are around this mean. - **Standard deviation** provides a more tangible metric of that spread, by conveying it in the same units as the original data. By calculating these statistics for defective items, the factory manager gains insights into the weekly data's consistency and variability. If the standard deviation is high, it suggests the percentage of defects varies significantly each day, possibly flagging underlying issues in the production process that need attention. Metrics like these support proactive decision-making in quality control and operational adjustments.