91Ó°ÊÓ

Variance is a key concept in understanding how data points in a probability distribution are spread out. When calculating variance, we use the formula \[\sigma_{x}^{2} = \sum (x - \mu_{x})^{2} \cdot p(x)\]to determine how far each data point \(x\) is from the mean \(\mu_{x}\). This distance is then squared for each data point, and these squared values are multiplied by their corresponding probabilities \(p(x)\). This results in a weighted sum, which gives us the variance. The variance tells us, in a statistical sense, how much the values of \(x\) deviate from the mean. If the variance is large, there is a large spread in the values, indicating more variability. Conversely, a smaller variance suggests that the values are closely clustered around the mean. In this exercise, the given distribution of defective tires uses this formula to achieve the variance output. By plugging in the values of \(x\) from 0 to 4 and correlating probabilities, we arrive at the variance.

Standard deviation is a measure that helps us understand the spread of the data in a more intuitive way than variance. Mathematically, it is the square root of the variance:\[\sigma_{x} = \sqrt{\sigma_{x}^{2}}\]Taking the square root of the variance converts the units back to the original units of the data, making it easier to interpret and compare.

• If the standard deviation is small, most data points fall very close to the mean.
• If it is large, data points are more spread out, indicating greater variability.

In practical terms, the standard deviation provides a clearer view of the distribution's spread by showing how much typical values deviate from the mean. In the context of our exercise, this means understanding how consistent the number of defective tires is in comparison to the average value. Calculating the standard deviation is straightforward once the variance is known, by simply taking its square root.

The mean value, often referred to as the average, is central to summary statistics. In a probability distribution, the mean gives us a central point or a balancing point of the distribution. It is calculated by the formula\[\mu_{x} = \sum x \cdot p(x)\]This uses each potential value \(x\) multiplied by its probability \(p(x)\) and sums up these products to give the mean. The mean indicates what the 'typical' or 'expected' value is when considering the entire distribution. For the example in this exercise, the mean value was provided as \(\mu_{x} = 1.2\). This tells us that, on average, a car selected at random typically has 1.2 defective tires. Understanding the mean is important as it serves as a reference point when evaluating measures of variability like variance and standard deviation.