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Standard deviation measures how spread out the numbers in a data set are. It tells us how much variation or dispersion there is from the average (mean). For example, in manufacturing, if you produce bolts that should be 8 cm on average, the standard deviation helps you understand how much individual bolt lengths vary from this average.
Mathematically, standard deviation is represented as \[ \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2} \] where \( \sigma \) is the standard deviation, \( N \) is the number of data points, \( x_i \) represents each data point, and \( \mu \) is the mean. The larger the standard deviation, the more spread out the data points are.
In our exercise, the standard deviation of the bolt lengths is 0.05 cm, indicating minor variation from the mean length of 8 cm.

Mean calculation is a fundamental concept in statistics. The mean is the average of a set of numbers. It’s calculated by adding all the values together and then dividing by the number of values. For the bolts, if you want to know the average length, you sum up the lengths of all bolts and divide by the number of bolts.
Mathematically, this is expressed as \[ \mu = \frac{1}{N}\sum_{i=1}^{N}x_i \] where \( \mu \) is the mean, \( N \) is the number of data points, and \( x_i \) represents each data point. In our example, the mean length of the bolts is given as 8 cm. This mean value helps to set a standard for comparison with individual bolt lengths.

Determining the acceptable range is crucial in quality control to decide which products to keep. Here's how you do it: First, you need the mean and standard deviation of your data, which are 8 cm and 0.05 cm for our bolts. The quality control policy specifies that bolts differing by more than 2 standard deviations from the mean will be destroyed.
To calculate the acceptable range:

• First, find the total deviation allowed by multiplying the standard deviation (0.05 cm) by 2: \(2 \times 0.05 = 0.1\) cm.
• Next, determine the lower limit by subtracting the total deviation from the mean: \(8 - 0.1 = 7.9\) cm.
• Then, find the upper limit by adding the total deviation to the mean: \(8 + 0.1 = 8.1\) cm.

Therefore, any bolt shorter than 7.9 cm or longer than 8.1 cm will be destroyed.

Statistical quality control (SQC) involves using statistical methods to monitor and control production processes. It ensures products are manufactured to specific standards consistently. A major part of SQC is setting acceptable ranges and identifying products that do not meet these standards.
In our example, the manufacturer uses the mean and standard deviation to set tolerance levels. By detecting whether bolts fall within 2 standard deviations of the mean, the manufacturer can ensure most bolts are within the desired length.
SQC tools like control charts, process capability analysis, and histograms can help in analyzing data and sustaining quality. Employing SQC helps to reduce waste, minimize defects, and maintain product reliability and customer satisfaction.