PSBE Chapter 1.1–1.4 Study Notes (Data, Graphs, Describing Distributions, Normal Distributions)

Data basics and getting started with distributions (PSBE Chapter 1.1)

• Data and variables
  • Data describe objects, people, places, or situations.
  • Goals when studying data:
  • Collect and organize the data.
  • Start investigating with a graph.
  • Compute numerical summaries to describe the data.
  • Look for overall patterns and deviations from the pattern.
  • Use a statistical model as appropriate.
• Key terms (1 of 2)
  • Cases: the objects described by a set of data (e.g., customers, cities, patients, cars).
  • Variable: a characteristic of a case (e.g., profit, duration of a service call, number of customers, gender).
  • Different cases can have different values for the variables.
• Example: Real estate firm (1.1)
  • Cases: Clients
  • Variable: Referral source
  • Values: Previous client, vendor, friend of realtor or staff, Internet advertisement, yard sign
• Types of variables
  • Quantitative variable: takes numerical values with arithmetic; examples: age, credit card balance, number of employees, time until served.
  • Categorical variable: places a case into categories (ex: gender, brand, own a home yes/no).
• Example: Credit card spending study (1.1)
  • Population: 21- to 25-year-old cardholders with $1000 limit; sample size: 100
  • For each person: items recorded
  • Items and expected variable types (credit card specific):
  • Average balance over last year
    • Type: Quantitative
    • Possible values: $0.00 through $1000.00
  • Ever late payments
    • Type: Categorical
    • Possible values: Yes, No
  • Day of week most used
    • Type: Categorical
    • Possible values: Sunday, Monday, …, Saturday
  • Age (in years)
    • Type: Quantitative
    • Possible values: integers 21, 22, 23, 24, 25
• Quick questions to understand a data set
  • Who? What cases do the data describe? How many cases?
  • What? How many variables? What is the exact definition and unit of each variable?
  • Why? What is the purpose and what questions are being asked? Are the variables suitable?

Displaying distributions with graphs (PSBE Chapter 1.2)

• Objectives and overview
  • Display distributions for categorical data: bar graphs, pie charts.
  • Display distributions for quantitative data: histograms, stemplots, time plots.
  • Interpret histograms and contrast with stemplots.
• Important terms (2 of 2)
  • Explanatory data analysis: examining and describing features of a data set.
  • Distribution of a variable: values the variable takes and how often it takes them.
  • Distribution of a categorical variable: lists categories and shows counts or percentages.
  • Distribution of a quantitative variable: often shows ranges and frequencies.
• Displaying categorical data
  • Purpose: summarize the data so characteristics of the distribution are clear.
  • Process: list categories and give counts or percents per category.
  • Methods: Bar graphs, Pie charts.
  • Example: Marital status (categories: Married, Never married, Divorced, Widowed).
  • Data summarized in table form (example counts in millions).
  • Ordering categories is flexible (alphabetical, by value, by year, etc.).
• Examples
  • Online research: Locations given by college students as their favorite source for online research.
• Ways to chart quantitative data
  • Histograms and stemplots: single-variable summaries.
  • Time plots: measurements over time; line emphasizes change.
• Histograms
  • Construct by dividing the value range into equal-width classes and counting observations per class.
  • Steps to create:
  • Divide range into equal-width classes.
  • Count observations in each class.
  • Mark x-axis with class widths; scale y-axis; draw a bar for each class.
  • Class size: start with 5–10 classes and adjust; there is no single perfect choice.
• Stemplots vs histograms
  • Stemplots are quick, hand-done summaries that show actual data values; useful for rough calculations.
  • They are less common in publications.
• Stemplot construction and considerations
  • Group data by leading digits (stems) and leaves (final digits).
  • Steps: split leading digits, write stems, place leaves in increasing order to the right.
  • Advantages: shows actual data values; quick pattern checks.
  • Limitations: not ideal for large data sets; digits can be rounded; stems can be split for many observations.
• Interpreting histograms
  • Look for overall pattern: shape, center, spread.
  • Common patterns: skewed right, skewed left, symmetric; many shapes can be complex.
  • Outliers: deviations that lie far from the main pattern.
  • Example note: Alaska and Florida may show unusual elderly representations; large gaps can indicate outliers.
• Time plots
  • Time on x-axis; the variable of interest on y-axis.
  • Look for trend (persistent rise/fall) and seasonal variation (regular intervals).
  • Scales matter: axis scaling can affect interpretation of the graph.
• Practical tips
  • A picture helps, but hard numbers matter; check the scales to avoid misinterpretation.

Describing distributions with numbers (PSBE Chapter 1.3)

• Measures of center
  • Mean (arithmetic average):
  • Formula: ar{x} = rac{1}{n}
    left( ext{sum of all values}
    ight)
  • Example: May include a calculation where the mean is 16.292 days (from a sum of 391 over 24 cases).
• Median
  • Definition: the midpoint where half the observations are below and half above.
  • The median is resistant to skew and outliers; the mean is not.
• Comparing mean and median
  • Symmetric distributions: mean and median are close.
  • Skewed distributions: the mean is pulled toward the tail; the median remains closer to the center.
  • Outliers: the mean can be heavily influenced by outliers, while the median largely resists them.
  • Example visuals: symmetric vs right-skewed vs skewed with outliers (illustrative descriptions).
• When to report mean vs median
  • Realtor example: home prices produce mean and median values; discuss which is more attractive to buyers vs sellers; often report both.
  • Middletown income example: mean for total tax base; median for typical living standards; suggests choosing based on purpose.
• Measuring spread: percentiles and quartiles
  • Percentiles: arrange data, determine position corresponding to a percentage; there may not be an exact observation at the exact percentile.
  • Quartiles: Q1 is the 25th percentile; Q3 is the 75th percentile; defined as medians of lower/upper halves (excluding the overall median).
• Five-number summary and boxplots
  • Five-number summary: min, Q1, median (M), Q3, max.
  • Boxplots visually display the five-number summary and can reveal symmetry/skewness.
• Boxplots and comparisons
  • Side-by-side boxplots compare distributions across groups.
• Outliers and the 1.5 IQR rule
  • Suspected outliers can be flagged if they fall more than 1.5 times the IQR above Q3 or below Q1:
  • Rule: an observation is a suspected outlier if it lies beyond $Q<em>3 + 1.5 \times IQR$ or below $Q</em>1 - 1.5 \times IQR,$ where $IQR = Q<em>3 - Q</em>1.$
• The standard deviation (spread around the mean)
  • Definition: measures the average distance of observations from the mean.
  • Formula (sample): $s = ext{standard deviation} = \sqrt{\frac{1}{n-1} \sum<em>{i=1}^n (x</em>i - \bar{x})^2}$
  • Steps: compute the variance $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$, then take the square root.
  • Example (hourly wages):
  • Mean = $\bar{x} = 16.33$ (dollars per hour, in the example)
  • Sum of squared deviations = 199.99
  • Degrees of freedom: $df = n-1 = 8$
  • Variance: $s^2 = \frac{199.99}{8} = 25.00$
  • Standard deviation: $s = \sqrt{25.00} = 5.00$
• Properties and usage of standard deviation
  • Use s to describe spread when the mean is the chosen center.
  • s is not resistant to outliers and skew; it has the same units as the data.
  • s = 0 only when all observations are identical (no spread).
• Choosing measures of center and spread
  • If the data are fairly symmetric with no outliers, use mean and standard deviation.
  • If skew or outliers are present, report the five-number summary (min, Q1, median, Q3, max) and consider a boxplot.
  • In practice, report both when appropriate to let the reader decide.
• Example contrasts and guidance
  • Real estate or income data examples illustrate how the mean can be sensitive to extreme values while the median remains robust.
• Practical note on reporting
  • Always consider the distribution shape and the presence of outliers when choosing summary statistics.

The Normal distributions (PSBE Chapter 1.4)

• Density curves and key properties
  • A density curve is a model for a distribution with total area under the curve equal to 1.
  • The area under the curve over a range gives the proportion of observations in that range.
  • The mean and median of a density curve: the mean is the balance point; the median is the equal-areas point.
  • For symmetric density curves, mean = median.
  • For skewed curves, the mean is pulled toward the long tail.
• Normal distributions
  • Family: $X \sim N(\mu, \sigma)$ with density shaped like a bell.
  • Common constants: $e \approx 2.71828…$, $\pi \approx 3.14159…$
  • The 68-95-99.7 rule: approximately
  • $P(|X-\mu| \le \sigma) \approx 0.68$
  • $P(|X-\mu| \le 2\sigma) \approx 0.95$
  • $P(|X-\mu| \le 3\sigma) \approx 0.997$
• The standard normal distribution
  • Standardization: convert any normal to the standard normal $Z \sim N(0,1)$ via
  • $Z = \frac{X - \mu}{\sigma}$
  • This allows comparison across different normal distributions.
• Example: women’s heights
  • Heights follow $X \sim N(\mu=64.5, \sigma=2.5)$ inches.
  • Probability that a woman is shorter than 67 inches: P(X < 67).
  • Compute z for 67: $z = \frac{67 - 64.5}{2.5} = 1.$
  • By the 68-95-97.7 rule or Table A, P(X < 67) ≈ 0.84 (more precisely 0.8413).
• Using Table A (the standard normal table)
  • The area to the left of a z-value gives the cumulative probability up to that z.
  • For z = 1.00, area to the left ≈ 0.8413.
• Inverse normal calculations (finding x given a proportion)
  • Process: locate the desired proportion in Table A (the area to the left); read the corresponding z-value; then unstandardize:
  • Formula: $x = \mu + z<em>p \sigma$ where $z</em>p$ is the z-value with area p to the left.
• Tips for Table A and common calculations
  • Because the normal distribution is symmetric, to find the area to the right of a z-value, either use 1 minus the left-area or use symmetry.
  • Area between two z-values: compute left areas for each and subtract: $\text{area}(z<em>1 \text{ to } z</em>2) = \text{area left of } z<em>1 - \text{area left of } z</em>2$
• Real-world examples and applications
  • NCAA SAT qualifiers example (top-level): require a score threshold; given a mean and sd for SAT, compute the proportion above a threshold by converting to z and using Table A or standard normal calculations.
  • Example: NCAA threshold requires a combined SAT of at least 820 for a partial qualifier; using normal approximation you can compute the proportion in that range.
  • SAT Verbal example: distribution approximates N(505, 110). To be in the top 10%, z = 1.28; solving for x gives x ≈ 646.
• Practical notes on standardization and comparing distributions
  • Standardizing allows comparing distributions with different centers and spreads on a common scale.
• TI-84 and Table A usage notes (summary)
  • For normal calculations, you can use TI-84 or other calculators/software to find probabilities and z-values.
• Inverse normal practice examples
  • Example: SAT Verbal distribution N(505, 110); to be in the top 10%, z = 1.28; unstandardize to obtain score ≈ 646.
• Summary one-liners
  • The normal distribution is a cornerstone model for many natural phenomena due to its properties under central limit tendencies.
  • Standardization (z-scores) enables cross-distribution comparisons and facilitates inverse-probability calculations.