Lesson 4.1: Describing & Analyzing Single-Variable Data

See the Story Behind the Numbers

Use visual displays and summary statistics to understand center, spread, and shape. Identify clusters, gaps, and outliers.

Learning Objectives

Marks each data value; great for small datasets.

Bins data into intervals; shows shape and spread.

Displays Q1, median, Q3, IQR, and potential outliers.

Scores: 85, 78, 92, 88, 90, 75, 82, 86, 95, 80. Find median and IQR.

Sorted: 75, 78, 80, 82, 85, 86, 88, 90, 92, 95

Median = $(85+86)/2=85.5$

Q1 = 80, Q3 = 90 → IQR = $Q_3-Q_1=10$

Quartiles: Q1 is median of lower half, Q3 is median of upper half (exclude the overall median when n is odd).

Outlier rule (Tukey): values outside $[Q_1-1.5\,IQR,\ Q_3+1.5\,IQR]$ are potential outliers.

Robust statistics: median and IQR are resistant to extreme values; mean and standard deviation are not.

Sample SD: $s=\sqrt{\dfrac{\sum (x_i-\bar{x})^2}{n-1}}$ ; Population SD: $\sigma=\sqrt{\dfrac{\sum (x_i-\mu)^2}{n}}$ .

z-score: $z=\dfrac{x-\bar{x}}{s}$ measures how many SDs a point is from the mean.

Chebyshev (any distribution): proportion within k SDs is at least $1-\dfrac{1}{k^2}$ for $k>1$ .

Data (sorted): 4, 5, 6, 7, 8, 12, 13, 14, 30

Median = 8; Q1 = 6; Q3 = 13 → IQR = 7

Fence: $[6-1.5\cdot7,\ 13+1.5\cdot7]=[-4.5,\ 23.5]$

30 is outside → potential outlier

Class scores: $\bar{x}=78$ , $s=6$ . Alice scored 90, Bob scored 68. Compare relative standing.

Alice: $z=\dfrac{90-78}{6}=2$ (2 SDs above mean)

Bob: $z=\dfrac{68-78}{6}=-1.67$ (1.67 SDs below mean)

At least what proportion of data lie within 3 SDs of the mean?

1-\dfrac{1}{3^2}=\tfrac{8}{9}\approx0.8889

1) Compute mean, median, IQR, and SD for [2, 4, 6, 8].

Solution

Sorted same; median 5; Q1=3, Q3=7 → IQR=4;

\bar{x}=5

s=\sqrt{5}

2) For Q1=12, Q3=20, apply IQR rule. Is 35 an outlier?

Solution

IQR=8; fence [12-12, 20+12]=[0,32]; 35>32 → outlier

3) A value has z=2.5 (using class mean & SD). Interpret in context.

Sample Answer

It is 2.5 SDs above average — unusually high relative performance.