Summary Statistics¶

Definitions¶

Parameters & Statistics¶

• $\overline{x}$: The sample mean
• $\mu$: the population mean
• $S$: Sample standard deviation
• $\sigma$: Population standard deviation
• $X^2$: Sample chi-square value
• $\chi^2$: Population chi-quare value
• $\hat{\rho}$: Sample proportion
• $p$: Population proportion

Quartiles¶

• Q1: One quarter (25%) of the data is less than or equal to Q1
• Q3: Three quarters (75%) of the data is less than or equal to Q3

Interquartile Range (IQR)¶

• IQR = Q3 - Q1

Standard Deviation & Variance¶

• 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¶

• 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: $\dfrac{f}{n}$

Outliers¶

• $x$ 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 $x \leq \mu - 2\sigma$

Five-Number Summary¶

• min value
• first quartile
• median
• third quartile
• max value

Boxplots¶

• A box is used to represent the middle 50% of the data
• The width of the box represents IQR
• Two whiskers (horizontal lines) represent min and max value
• Outliers are represented with a cross and are outside of the whiskers

Skewness¶

• 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

Data Comparison¶

• Compare numerical values
• Describe meaning in real life