Basic Data Visualization Notes (L03)

L03.1 Boxplots for Outliers and Z Scores

• Purpose: identify unusual observations in a numeric data set using boxplots and Z-scores.
• Boxplot method for outliers:
  • Outlier fences are determined from the IQR (interquartile range).
  • IQR = Q3 − Q1.
  • Upper fence = Q3 + 1.5 · IQR.
  • Lower fence = Q1 − 1.5 · IQR.
  • Observations outside these fences are typically labeled as outliers.
• How to compute and interpret in practice:
  • Boxplots visually show the spread of the middle 50% of data (between Q1 and Q3) and possible outliers outside the fences.
• Z-score method for outliers:
  • Compute standardized values (Z-scores) to assess how far observations are from the mean in units of standard deviation.
  • Observations with a Z-score outside ±3 are considered outliers.
• Z-score calculation (two common forms):
  • Population/overall standardization: Z = \frac{X - \mu}{\sigma}
  • Sample standardized value: Z = \frac{X - \bar{X}}{s}
• Minitab steps for boxplots and outliers:
  • Boxplot: Graph → Boxplot → One Y/Simple → Select Vars → OK.
  • Z-score (outliers): Minitab path might involve Calc → Standardize → Select Variable and location; or Descriptive Statistics path: Stat → Basic Statistics → Display Descriptive Statistics → Select Variable → Statistics → Make Selections to view Z Scores outside ±3.
• Practical notes:
  • Boxplot fences help distinguish typical spread from extreme values; outliers can affect measures of central tendency and spread.
  • Z-scores depend on mean and standard deviation; they assume roughly normal distribution in interpreting distance from the mean.

L03.2 Skewness and Kurtosis

• Skewness:
  • Symmetric distribution: skewness ≈ 0.
  • Left-skewed: skewness < 0.
  • Right-skewed: skewness > 0.
• Kurtosis (peakedness/heaviness of tails):
  • Normal distribution: kurtosis ≈ 0 (excess kurtosis).
  • Peakier distribution: kurtosis > 0.
  • Flatter (less peaked) distribution: kurtosis < 0.
• How to compute in Minitab:
  • Stat → Basic Statistics → Display Descriptive Statistics → Select Variable → Statistics → Make Selections.
• Significance and interpretation:
  • Skewness indicates asymmetry around the mean and can affect parameter estimates and tests that assume normality.
  • Kurtosis informs about tail heaviness and peakness relative to a normal distribution; influences tail risk and outlier expectations.
• Notes on use:
  • Skewness and kurtosis are descriptive measures; use them alongside histograms and normality tests for a fuller picture.

L03.3 Visualizing One Numeric Variable: Histogram

• Purpose: visualize the distribution of a single numeric variable.
• Histogram in Minitab:
  • Graph → Histogram → Simple → OK.
• What the histogram communicates:
  • Shape of the distribution (symmetric, skewed, multimodal).
  • Central tendency via the location of the bulk of data and spread via bin width and height.
• Practical considerations:
  • Choice of bin width affects the apparent shape; too few bins can mask features, too many bins can introduce noise.
• Connections:
  • Complements boxplots and descriptive statistics to understand distribution.

L03.4 Scatter Plot: The Relationship Between Two Numeric Variables

• Purpose: assess the relationship between two numeric variables.
• Example data: Restaurant2.mtw; examine whether “Popular Index” changes with “Cost.”
• Scatter plot steps in Minitab:
  • Graph → Scatter Plot → Simple → OK.
  • Optional: Graph → Scatter Plot → With Regression → OK to add a regression line.
• Interpretation questions:
  • Do the two variables move together (positive relationship) or in opposite directions (negative relationship)?
  • Is the relationship strong, moderate, or weak (based on pattern and, if using regression, the R-squared concept)?
• Practical implications:
  • Scatter plots reveal trends, clusters, or potential nonlinear patterns that may warrant further modeling.
• Notes:
  • Regression line (if added) summarizes linear association; correlation coefficient (r) quantifies strength/direction (not shown in transcript but commonly used).

L03.5 Visualizing Categorical Data: One Categorical Variable

• Visualizing a single categorical variable involves understanding relative frequencies.
• Relative frequency table:
  • Stat → Tables → Tally Individual Variables…
• Bar charts:
  • Graph options: Bar Chart → Simple (or Other types such as Cluster, Stack).
  • Bars represent counts of unique values (frequencies) for each category.
  • Example categories shown in transcript: Restaurant, Food, Decor, Service, Cost, Popularity Index, Cuisine, Region, Z Cost, etc.
• Bar Chart: Simple
  • Used to display counts of each category for a single categorical variable.
  • Data representation: height of each bar corresponds to the frequency or count.
• Key features:
  • Helpful for comparing category frequencies at a glance.
  • Can switch to alternative layouts (Cluster, Stack) to compare multiple variables or groupings within the same plot.
• Textual notes from example:
  • Variable labels included: Cuisine, Cost, Region, etc.
  • Example data shows counts like 0, 3, 8, etc. for category frequencies.

L03.5 Visualizing Categorical Data: Pie Chart

• Pie chart aims to show proportional shares of a categorical variable.
• Pie chart setup in Minitab:
  • Graph → Pie Chart → Pie Chart.
  • Chart counts of unique values; Chart values from a table.
  • Categorical variable highlighted: Cuisine.
  • Pie Options…, Labels…, Data Options… to adjust appearance and labeling.
• Example from transcript:
  • Categories listed under Cuisine: American (New), Chinese, French, Indian, Italian, Japanese, Mexican, etc.
  • The chart displays the proportion of each cuisine category among restaurants.
• Considerations:
  • Pie charts are most effective when there are a small number of categories with meaningful shares; hard to compare many slivers.
  • Use relative frequencies for intuitive understanding of shares.

L03.6 A Contingency Table: Visualizing Two Categorical Variables

• Purpose: organize and summarize the relationship between two categorical variables.

• Contingency table concept:

  • Displays counts for combinations of two categorical factors (e.g., Cuisine × Region).
  • Enables analysis of dependence or association between the variables.

• Chi-square based approach (from transcript):

  • Stat → Tables → Cross Tabulation and Chi-Square.

• Key formula (general context):

  • Expected counts: E{ij} = \frac{(Ri)(Cj)}{N} where Ri is the row total for row i, C_j is the column total for column j, and N is the grand total.
  • Chi-square statistic: \chi^2 = \sumi \sumj \frac{(O{ij} - E{ij})^2}{E{ij}} where O{ij} are observed counts.

• Practical notes:

  • Cross-tabulations help identify patterns such as whether certain cuisines are more common in particular regions.
  • Chi-square test assesses whether there is a statistically significant association between the two categorical variables (not detailed in transcript, but implied by the method name).

• Connections to foundational principles:

  • Descriptive statistics summarize data; visualizations (histograms, scatter plots, bar charts, pie charts, contingency tables) convey distribution, relationships, and composition.
  • Basic data visualization practices rely on understanding data types (numeric vs categorical), distribution shape, and the appropriate summary metrics.

• Practical tips and cautions:

  • When interpreting skewness/kurtosis, consider sample size and the presence of outliers which can distort these measures.
  • For histograms, choose bin widths thoughtfully to avoid masking or exaggerating features.
  • In bar charts and pie charts, ensure category labels are clear and totals sum appropriately to avoid misinterpretation.
  • In contingency tables, ensure adequate expected counts for chi-square validity; consider alternative tests if counts are low.

Summary of key formulas and commands

• Boxplot outlier fences:
  • IQR = Q3 − Q1
  • Upper fence = Q3 + 1.5 · IQR
  • Lower fence = Q1 − 1.5 · IQR
• Z-scores:
  • Z = \frac{X - \mu}{\sigma} or Z = \frac{X - \bar{X}}{s}
  • Outliers: |Z| > 3
• Skewness interpretation:
  • Symmetric: skewness ≈ 0
  • Left skewed: skewness < 0
  • Right skewed: skewness > 0
• Kurtosis interpretation (excess):
  • Normal: kurtosis ≈ 0
  • Peakier: kurtosis > 0
  • Flatter: kurtosis < 0
• Histograms: no fixed equation; visualize distribution shape.
• Scatter plots: assess pattern and strength of linear association; regression line optional.
• Contingency table and chi-square:
  • Expected counts: E{ij} = \frac{Ri C_j}{N}
  • Chi-square: \chi^2 = \sumi \sumj \frac{(O{ij} - E{ij})^2}{E_{ij}}