RelativePosition EmpiricalRule Chebyshev

Sam

Objectives:

Learn concept of relative position of an element of a data set.
Learn 2 measures of the relative position of a measurement.
1. percentile rank
2. z-score
Learn meaning of quartiles
Learn meaning of the five-number summary (box plot)

Percentile::Percent of data \(\leq\) the number.

1st Q = 25% 2nd Q = 50% (median) 3rd Q = 75%

5-number summary::Includes the 3 quartiles and the 2 extreme values (box plot) IQR::\(Q_3 - Q_1\)

Z-score::distance from the mean in units of standard deviation.

Z = \(\frac{Value - Mean}{St Dev}\)

The Empirical Rule::applies to normal distributions (bell-shaped and symmetric). It provides approximate percentages of data within specific standard deviations of the mean.
Key Points:
68% of the data falls within 1 standard deviation of the mean.
95% of the data falls within 2 standard deviations of the mean.
99.7% of the data falls within 3 standard deviations of the mean.
Use Case: This rule is commonly used when dealing with data that is approximately normally distributed.

Chebyshev’s Theorem::applies to any distribution (not limited to normal distributions). It provides a minimum percentage of data that lies within a specified number of standard deviations from the mean.
Key Points:
For \(k\) standard deviations (where \(k > 1\)), at least \(1 - \frac{1}{k^2}\) of the data lies within \(k\) standard deviations of the mean.
- Example:
- For \(k = 2\), at least \(1 - \frac{1}{2^2} = 75\%\) of the data is within 2 standard deviations.
- For \(k = 3\), at least \(1 - \frac{1}{3^2} = 88.9\%\) of the data is within 3 standard deviations.
Use Case: This theorem is useful when the distribution shape is unknown or not normal.