Summary of Descriptive Statistics Exam Practice

## Introduction Descriptive statistics summarize and describe the main features of a data set so we can understand patterns at a glance. In this material we use a short student test-score data set to practice core measures: range, mean, median, mode, standard deviation, and frequency representation. > **Definition:** Descriptive statistics are numerical or graphical methods that summarize and organize the characteristics of a collection of data. ## The data The student scores are: $12$, $18$, $20$, $22$, $18$, $16$, $20$, $18$, $22$, $16$, $20$, $18$. ### Step 1 — Organize the data Sort the data in ascending order to make some measures easier to find: $$12, 16, 16, 18, 18, 18, 18, 20, 20, 20, 22, 22$$ ## Range The **range** is the difference between the maximum and minimum values. > **Definition:** Range is $\text{max} - \text{min}$. Compute: $$\text{Range} = 22 - 12 = 10$$ ## Mean (arithmetic average) The **mean** is the sum of observations divided by the number of observations. > **Definition:** For data $x_1,\dots,x_n$, the mean is $\bar{x} = \dfrac{1}{n}\sum_{i=1}^n x_i$. Compute the sum and count. There are $n=12$ scores. Sum them: $$\sum x_i = 12 + 18 + 20 + 22 + 18 + 16 + 20 + 18 + 22 + 16 + 20 + 18 = 220$$ Then the mean is: $$\bar{x} = \dfrac{220}{12} = \dfrac{110}{6} = \dfrac{55}{3} \approx 18.333\ldots$$ Interpretation: The average score is about $18.33$. ## Mode The **mode** is the most frequently occurring value(s) in the data. > **Definition:** Mode is the value(s) with highest frequency. From the sorted list counts: $12(1)$, $16(2)$, $18(4)$, $20(3)$, $22(2)$. The highest frequency is 4 for $18$. Mode: $18$. ## Median The **median** is the middle value when data are ordered. For even $n$ take the average of the two central values. > **Definition:** Median is the $\left(\dfrac{n+1}{2}\right)$th value for odd $n$, or the average of the $\dfrac{n}{2}$th and $\dfrac{n}{2}+1$th values for even $n$. Here $n=12$ (even). The central positions are $6$ and $7$ in the ordered list. Values at those positions are both $18$. $$\text{Median} = \dfrac{18 + 18}{2} = 18$$ ## Standard deviation We compute the **population** standard deviation (unless sampling context requires sample sd). We'll show both interpretations and compute the population version as default. > **Definition:** Population variance is $\sigma^2 = \dfrac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2$ and population standard deviation is $\sigma = \sqrt{\sigma^2}$. Sample variance uses $\dfrac{1}{n-1}$. Compute squared deviations using $\bar{x} = \dfrac{55}{3} \approx 18.333\ldots$. It's clearer to work with exact fractions where possible. For each unique value with frequency $f$ compute contribution $f( x - \bar{x})^2$. Unique values and frequencies: $12(1)$, $16(2)$, $18(4)$, $20(3)$, $22(2)$. Deviations (exact with fraction $\bar{x}=\dfrac{55}{3}$): - For $12$: $12 - \dfrac{55}{3} = \dfrac{36 - 55}{3} = -\dfrac{19}{3}$ so squared is $\left(-\dfrac{19}{3}\right)^2 = \dfrac{361}{9}$. Contribution: $1\cdot \dfrac{361}{9} = \dfrac{361}{9}$. - For $16$: $16 - \dfrac{55}{3} = \dfrac{48 - 55}{3} = -\dfrac{7}{3}$ so squared is $\dfrac{49}{9}$. Contribution: $2\cdot \dfrac{49}{9} = \dfrac{98}{9}$. - For $18$: $18 - \dfrac{55}{3} = \dfrac{54 - 55}{3} = -\dfrac{1}{3}$ so squared is $\dfrac{1}{9}$. Contribution: $4\cdot \dfrac{1}{9} = \dfrac{4}{9}$. - For $20$: $20 - \dfrac{55}{3} = \dfrac{60 - 55}{3} = \dfrac{5}{3}$ so squared is $\dfrac{25}{9}$. Contribution: $3\cdot \dfrac{25}{9} = \dfrac{75}{9}$. - For $22$: $22 - \dfrac{55}{3} = \dfrac{66 - 55}{3} = \dfrac{11}{3}$ so squared is $\dfrac{121}{9}$. Contribution: $2\cdot \dfrac{121}{9} = \dfrac{242}{9}$. Sum contributions: $$\sum_{i=1}^n (x_i - \bar{x})^2 = \dfrac{361 + 98 + 4 + 75 + 242}{9} = \dfrac{780}{9} = \dfrac{260}{3}$$ Population variance: $$\sigma^2 = \dfrac{1}{n} \sum (x_i - \bar{x})^2 = \dfrac{1}{12} \cdot \dfrac{260}{3} = \dfrac{260}{36} = \dfrac{65}{9} \approx 7.222\ldots$$ Popula