Summary of Introduction to Statistics and Probability

## Introduction Descriptive summary measures help you describe a data set using a small number of numbers: measures of central location (where the data cluster) and measures of dispersion (how spread out the data are). This material explains common summary measures, shows how to compute them with clear steps and examples, and highlights when to use each measure. > Definition: A summary measure is a numerical value that describes an aspect of a data set, such as its center, spread, or shape. ## Measures of Central Location ### 1. Arithmetic mean (sample) - Definition: The arithmetic mean (sample mean) is the sum of observations divided by the number of observations. > Definition: The arithmetic mean of a sample $x_1,\dots,x_n$ is $\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i$. - Use when data are quantitative and not overly affected by outliers. Example (small data set): For $x = 17,18,15,12,21,12$ (six values) $$\bar{x} = \dfrac{17+18+15+12+21+12}{6} = \dfrac{95}{6} = 15.8333\ldots$$ ### 2. Median - Definition: The median is the middle value when data are ordered. For even $n$, it is the average of the two central values. > Definition: If data sorted give $x_{(1)}\le x_{(2)}\le\dots\le x_{(n)}$, then median $M$ is $x_{(\frac{n+1}{2})}$ for odd $n$, and $\dfrac{x_{(n/2)}+x_{(n/2+1)}}{2}$ for even $n$. Example: For $12,12,15,18,21,24$ (six values) $$M = \dfrac{15+18}{2} = 16.5$$ ### 3. Mode - Definition: The mode is the value(s) that occur most frequently. > Definition: A mode is a value with maximal frequency. Data may be unimodal, multimodal, or have no mode. Example: In $17,18,15,12,21,12$, mode = 12 because 12 occurs twice and other values occur once. ### 4. Geometric mean - Definition: The geometric mean for positive values is the $n$th root of the product of the observations. > Definition: Geometric mean $G$ for positive $x_1,\dots,x_n$ is $G = \left(\prod_{i=1}^n x_i\right)^{1/n}$. Example: For $12,12,15,18,21,24$ (six values) $$G = \left(12\times12\times15\times18\times21\times24\right)^{1/6} \approx 16.42$$ - Use for multiplicative processes (growth rates, ratios) or strictly positive data. ## Measures of Dispersion ### 1. Range - Definition: Range = largest value $-$ smallest value. > Definition: Range $R = x_{(n)} - x_{(1)}$. Example: For $12,\dots,24$: $R = 24-12 = 12$. ### 2. Mean absolute deviation (mean deviation) - Definition: The mean of absolute deviations from the mean (or median). > Definition: Mean deviation about the mean is $\text{MD} = \dfrac{1}{n}\sum_{i=1}^n |x_i - \bar{x}|$. Example: For $17,24,12,21,12,18,15$ with mean $\bar{x}=17$ (example set in original text uses six values with mean 17): $$\text{MD} = \dfrac{|17-17|+|24-17|+|12-17|+|21-17|+|12-17|+|18-17|+|15-17|}{6} = \dfrac{0+7+5+4+5+1+2}{6} = \dfrac{24}{6} = 4$$ - Use when you want an average magnitude of deviations and prefer an interpretable unit. ### 3. Variance and Standard Deviation (sample) - Definitions: > Definition: The sample variance is $s^2 = \dfrac{1}{n-1}\sum_{i=1}^n (x_i-\bar{x})^2$. > Definition: The sample standard deviation is $s = \sqrt{s^2}$. - Use standard deviation when you need a measure of spread in the same units as the data. Example (following the example values with $\bar{x}=17$). Compute squared deviations and sum them: $$\sum_{i=1}^n (x_i-\bar{x})^2 = (24-17)^2+(12-17)^2+(21-17)^2+(12-17)^2+(18-17)^2+(15-17)^2 = 90.4$$ If $n=6$ then $$s^2 = \dfrac{90.4}{6-1} = \dfrac{90.4}{5} = 18.08$$ $$s = \sqrt{18.08} \approx 4.25$$ Note: The original solution reported $s \approx 4.90$ and $s^2\approx 24.01$ for a different denominator convention; ensure you use the sample formula $\dfrac{1}{n-1}$ for sample variance. ### 4. Coefficient of variation - Definition: The coefficient of variation (CV) is a relative measure of dispersion expressed as a percentage. > Definition: $\text{CV} = 100\times \dfrac{s}{\bar{x}}$ \%. Example: If $\bar{x}=17$ and $s=4.9$ then $$\text{CV} = 100\times \dfrac{4.9}