Z-score, percentile, and empirical rules

A Z-score indicates how many standard deviations an element is from the mean. It is calculated as $$Z = \frac{(X - \mu)}{\sigma}$$ where X is the value, $$\mu$$ is the mean, and $$\sigma$$ is the standard deviation.

To calculate a Z-score, use the formula: $$Z = \frac{(X - \mu)}{\sigma}$$ where X is the value, $$\mu$$ is the mean, and $$\sigma$$ is the standard deviation.

A Z-score of -2 indicates that the value is 2 standard deviations below the mean.

The empirical rule states that for a normal distribution: 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.

The value is 220, calculated as: $$X = \mu + Z \cdot \sigma = 300 + (-2) \cdot 40 = 220$$ .

A percentile is a measure used to indicate the value below which a given percentage of observations in a group of observations falls.

A Z-score of -1.08 corresponds to the 14th percentile.

For a normal distribution, 68% of data falls within 1 standard deviation of the mean. If the mean is $$\mu$$ and the standard deviation is $$\sigma$$ , the range is $$[\mu - \sigma, \mu + \sigma]$$ .

The range is 28,000 km to 32,000 km, calculated as: $$[30000 - 2000, 30000 + 2000]$$ .

The formula is: $$Z = \frac{(X - \mu)}{\sigma}$$ .