Summary Statistics¶

Definitions¶

The standard deviation for a population: \(\sigma_x = \sqrt{\dfrac{\sum (X_i - \overline{x})^2}{n}}\)
The standard deviation for a sample: \(s_x = \sqrt{\dfrac{\sum(x_i - \overline{x})^2}{n-1}}\)
Variance is the square of the standard deviation, i.e., \(\sigma^2\) and \(s^2\)

Frequency: number of times that a certain value has appeared, denoted by \(f\)
\(\overline{x} = \dfrac{\sum\limits_{i=1}^n f_i x_i}{n}\)
Relative Frequency: \(\dfrac{f}{n}\)

\(x\) is an outlier if \(x < Q1 - 1.5 \cdot IQR\) or \(x > Q3 + 1.5 \cdot IQR\)
\(x\) is an outlier if \(x \geq \mu + 2\sigma\) or \(x \leq \mu - 2\sigma\)

Skewness describes the direction in which a non-symmetrical distribution of data is leaning
- A distribution that has its tail on the right side has positive skew
- A distribution that has its tail on the left side has negative skew
Find the skewness from a boxplot:
- If the median is roughly in the middle of the first and third quartiles, then the distribution is approximately symmetrical
- If the median is closer to Q1, then the distribution has positive skew
- If the median is closer to Q3, then the distribution has negative skew
Find the skewness from the median and the mean
- Positively skewed: median < mean
- Negatively skewed: median > mean