91Ó°ÊÓ

Z-scores provide a way to understand how far a particular value is from the mean in terms of standard deviations. In other words, a Z-score will tell you how many standard deviations away a datapoint is from the average or mean of a data set. To calculate a Z-score, you use the formula: \[ Z = \frac{X - \mu}{\sigma} \] Where:

• \(X\) is the value in the dataset,
• \(\mu\) (mu) is the mean of the dataset,
• \(\sigma\) (sigma) is the standard deviation.

In the original exercise, Z-scores help determine what percentage of monitors fall within specified voltage limits. By transforming these limits into Z-scores, you can use the normal distribution to find probabilities or percentages about how likely monitors are to meet those specifications. Calculating Z-scores for the boundaries of 100 mV and 400 mV gives values of approximately -4.5573 and 3.2552 respectively, encompassing nearly all monitors, about 99.99% under the old specification. For the new specification of 150 mV to 350 mV, the calculated Z-scores are around -3.2552 and 1.9531, corresponding to about 91.02% of monitors meeting these specifications.

The standard deviation is a measure of how spread out the values in a data set are around the mean. When data points are tightly clustered around the mean, the standard deviation is small. Conversely, if data points are spread widely, the standard deviation is large. Mathematically, standard deviation is represented by the symbol \(\sigma\) (sigma) and is calculated using the formula: \[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2} \] Where:

• \(N\) is the number of data points,
• \(X_i\) are the data points,
• \(\mu\) is the mean of the data points.

In the context of the problem, the standard deviation of 38.4 mV tells us about the variability of the monitor mesh tension around the mean value of 275 mV. Understanding standard deviation allows us to utilize the normal distribution by converting boundaries to Z-scores, to find the percentage of monitors meeting certain tension specifications.

The mean, often called the average, is a central measure of a data set. It is calculated by summing up all data points and then dividing by the number of points. In mathematical terms, it is represented as \(\mu\). The formula is: \[ \mu = \frac{1}{N} \sum_{i=1}^{N} X_i \] Where:

• \(N\) is the total number of observations,
• \(X_i\) are the data values.

In this exercise, the mean mesh tension is 275 mV. The mean gives a central value around which other data points are distributed. It acts as a reference point when calculating Z-scores, determining how data points relate to the average tension, which in turn helps us figure out the variability and how common different measurement ranges are within our normal distribution.