A

topic 1

Population and Sample

  • Population: all subjects or items of interest; size N

  • Sample: a subset drawn from a population; size n

  • There are many possible samples from a population; number of samples depends on N and n

Terminology

  • Data: observations collected (e.g., measurements, responses)

  • Parameter: characteristic of a population

  • Statistic: characteristic of a sample (sample statistic)

  • The observed value of a statistic estimates a parameter

  • A statistic is unbiased if its sampling distribution mean equals the parameter

Individuals and Variables

  • Individuals: objects described in data (people, animals, plants, things)

  • Variable: property that characterizes an individual; can take different values

Types of Variables

  • Quantitative: numeric values; can report a mean across individuals (e.g., age, blood pressure, leaf length)

  • Categorical: descriptive categories; can report counts or proportions (e.g., gender, blood type, flower color)

Classifying Variables (Brief Example)

  • Example dataset (from slides): Diagnosis (categorical), Age at death (quantitative)

  • Question: What is recorded about each individual? Is each variable quantitative or categorical?

    • Their diagnosis and age at death are being recorded, with the diagnosis being categorial variables and the age at death being the quantitative variables

Visualizing Quantitative Data: Common Graphs

  • Histograms: single-variable overview; show pattern of variability; useful for large data sets

  • Dotplots (or Stem & Leaf): show raw data; useful for small data sets as they describe the pattern of variability

  • Time Series Plots: data points in sequence (e.g., over time); emphasizes changes over time

Histograms

  • Histogram: horizontal axis = class intervals (bins); vertical axis = frequencies (counts)

  • Heights of bars = frequencies; bars are adjacent

  • Relative Frequency Histogram: vertical axis = relative frequencies (percentage)

Making a Histogram

  • Divide range of the quantitative variable into equal-size intervals (bins)

  • Vertical axis = either frequency or relative frequency

  • For each class, draw a column with height = count or percent in that class

Dotplots

  • Dotplot: plot each data value on a scale; dots stack for identical values

Stem-and-Leaf Plot

  • Stem-and-Leaf: data are split into stem (leading digits) and leaf (trailing digits)

Measures of Center

  • Center = representative value indicating where the data cluster

  • Main measures: Mean, Median, Mode

The Mean

  • Definition: mean = arithmetic average

  • Formula: adding values and dividing by total number of values

  • Key:

    • (mu) = Pop. Mean

    • N= # of indv. in a pop.

    • x = sample mean

    • n = sample size


The Median

  • Definition: middle value when data are ordered

  • How to find: sort values; if n is odd, median = middle value; if n is even, median = mean of two middle values divided by 2

The Mode

  • Definition: most frequent value

  • If two values share greatest frequency: bimodal

  • More than two with greatest frequency: multimodal

  • If no value repeats: no mode

The Best Measure of Center

  • Mean is not resistant to skews and outliers

  • Median is resistant to skew and outliers

  • For approximately symmetric data with one mode, mean ≈ median

  • For obviously asymmetric data, report both mean and median

Measures of Variation

  • Variation measures how data values differ from each other

  • Key concepts: range, standard deviation, variance, quartiles

Range and Standard Deviation

  • Range = max − min

  • Standard deviation (sample): s=

  • Notation: \sigma = population SD; s = sample SD

Variance

  • Variance (sample) =

  • Variance = square of SD

  • Notation

  • \sigma = Population standard deviation (sigma)

    x^2 = Population variance

    s = Sample standard deviation

    s2 = Sample variance

    N = Population size

    n = Sample Size

Quartiles and Five-Number Summary

  • Median is Q2 (second quartile), 50th percentile

  • Quartiles divide the data values into 4 equal parts

  • Q1 = 25th percentile; Q3 = 75th percentile

  • Different procedures can yield different quartiles; not universal

  • Five-number summary: min, Q1, Median (Q2), Q3, max

Boxplots and IQR

  • IQR = Q3 − Q1

  • Boxplot shows min, Q1, median, Q3, max; whiskers extend to data range within 1.5 IQR

  • Outliers are values outside typical pattern (suspected outliers) beyond 1.5 IQR from quartiles

IQR and Suspected Outliers

  • Suspected low outlier: value < Q1 − 1.5 × IQR

  • Suspected high outlier: value > Q3 + 1.5 × IQR

How to Draw a Boxplot

  • Steps: compute five-number summary; set scale to include min and max; draw box from Q1 to Q3 with median line; extend whiskers to min and max

Standardization: Z-scores

  • Standardized score (z-score) allows comparison across data sets

    • It’s the number of sd that a given value x is above or below the mean

  • Population: z = \frac{x - \mu}{\sigma}

  • Sample: z = \frac{x - \bar{x}}{s}

Interpreting Histograms (4 characteristics)

  • Shape/Distribution: unimodal, bimodal, symmetric, skewed, irregular

  • Center: approximate midpoint or peak location

  • Spread: range of observed values

  • Outliers: any points that may be outliers

Exploratory Data Analysis (EDA)

  • EDA uses tools to understand center, variation, distribution, outliers, and time

  • Outliers can dramatically affect the mean, SD, and histogram scale

Boxplot Utility (IQR approach)

  • Boxplot highlights central tendency and variability; useful for comparing groups

Quick Reference: Example Use of IQR for Outliers

  • Compute IQR; identify values beyond Q1 − 1.5 IQR or Q3 + 1.5 IQR as possible outliers