Notes on Descriptive Statistics (Qualitative and Quantitative)

2.1 Graphical and Numerical Methods for Describing Qualitative Data

• Purpose: Describe qualitative observations by categorizing observations into mutually exclusive categories (or classes).

• Data description: Use counts or proportions (relative frequencies) per category.

• Example context: Safety of nuclear power reactors; hazards of energy use.

  • Qualitative variable: cause of fatal energy-related accidents.

  • Data: 62 accidents fall into six categories (causes).

• Graphical descriptions for qualitative data:

  • Bar graphs: height/length proportional to category frequency (or relative frequency).

  • Pie charts: circle divided into slices; central angle of each slice proportional to the category relative frequency.

• Pareto diagram:

  • A frequency bar graph with bars arranged in descending order of height (leftmost the tallest).

  • Named after Vilfredo Pareto (Italian economist).

  • Use: popular in process and quality control to highlight most frequent problems (e.g., defects, accidents, breakdowns).

2.1.1 Applied Exercises

• Exercise 1.1: Social robots

  • Data: random sample of 106 social robots; 63 with legs only, 20 with wheels only, 8 with both legs and wheels, 15 with neither.

  • Tasks:

  • a. Identify the type of graph used to describe the data.

  • b. Identify the variable measured for each of the 106 designs.

  • c. Use the graph to identify the most-used social robot design.

  • d. Compute class relative frequencies for the categories.

  • e. Use the results in (d) to construct a Pareto diagram.

• Exercise 2.1: STEM experiences for girls

  • NSF study: 174 young women in informal STEM programs; geographic location described via a pie chart (urban, suburban, rural).

  • Data: 107 urban, 57 suburban, 10 rural.

  • Tasks: construct the pie chart and interpret the results.

• Exercise: Beach erosional hotspots

  • Data: data collected on six beach hotspots.

  • Tasks:

  • a. Identify each recorded variable as quantitative or qualitative.

  • b. Form a pie chart for the beach condition of the six hotspots.

  • c. Form a pie chart for the nearshore bar condition of the six hotspots.

  • d. Comment on the reliability of using pie charts to infer about all beach hotspots in the country.

• Exercise 3.1: Railway track allocation

  • Data: 53 trains allocated to 11 tracks; table provided.

  • Task: Construct a Pareto diagram for the data and determine if track allocation is evenly distributed; identify underutilized and overutilized tracks.

2.2 Graphical Methods for Describing Quantitative Data

• Descriptive goals: Describe, summarize, and detect patterns in quantitative data using three graphical methods:

  • Dot plots

  • Stem-and-leaf displays

  • Histograms

• Software-friendly focus: Interpretation of these displays rather than their construction.

• Example context: EPA mileage ratings (100 measurements) for a car model (Table 2.2 referenced).

• Dot plot:

  • Horizontal axis: scale for the quantitative variable (e.g., miles per gallon, mpg).

  • Each measurement is represented by a dot at the rounded value (to the nearest 0.5 mpg).

  • Repeated values stack into piles above the same location.

  • Insight: almost all mpg values lie in the 30s; most between 35 and 40 mpg.

• Stem-and-leaf display (MINITAB example, Figure 2.6):

  • Stem: left of the decimal point; Leaf: right of the decimal point.

  • Rows ordered from smallest stem (e.g., 30) to largest (e.g., 44).

  • Interpretation: shows distribution between 30.0 and 44.9 mpg; counts per stem show frequency; e.g., six leaves in stem row 34 indicate six observations in [34.0, 35.0).

  • Advantage: retains original measurements; easy to locate exact values (e.g., two observations at 36.3).

  • Limitation: can be unwieldy for very large data sets due to many stems/leaves.

• Histogram (SPSS example, Figure 2.7):

  • Horizontal axis: mpg intervals (e.g., 30–31, 31–32, …, 44–45).

  • Vertical axis: frequency (count) of observations in each interval.

  • Insight: e.g., about 21 of 100 cars (21%) fall in [37, 38) mpg.

• General notes:

  • Histograms give a good overall visual of the distribution for large data sets but do not display individual measurements.

  • Dot plots and stem-and-leaf displays show individual observations; stem-and-leaf sorts data in ascending order and highlights exact values, but can be large for big data sets.

2.2.1 Applied Exercises

• Exercise 2.1: Sound waves from a basketball (American Journal of Physics, 2010)

  • Data: frequencies of the first 24 resonances (Hz) listed in a table.

  • Task: use a graphical method to describe the distribution of these frequencies.

• Exercise 2.1: Surface roughness of pipe (Oil field pipes)

  • Data: 20 sample surface roughness measurements (in micrometers).

  • Task: describe the sample data with an appropriate graphical method.

2.3 Numerical Methods for Describing Quantitative Data

• Numerical descriptive measures: numbers computed from a data set to describe its relative frequency distribution.

• Three categories:

  • Measures of central tendency (location/center)

  • Measures of variation (spread)

  • Measures of relative standing (position of an observation within the data set)

• Notation:

  • y denotes the observed variable; observations are y1, y2, …, yn.

  • Statistics: numerical descriptive measures computed from sample data.

  • Parameters: corresponding population measures; often denoted by Greek letters (e.g., μ for population mean).

• Key concepts:

  • Sample mean:

  • Population mean: μ

  • Median, mode, and the idea of central tendency as core descriptive statistics.

2.4 Measures of Central Tendency

• Three common measures:

  • Arithmetic mean: balance point of the relative frequency distribution.

  • Median: middle value; splits data into two equal-area halves.

  • Mode: value with the greatest frequency (peak of distribution).

• Visual intuition (Figure 2.9 in text):

  • Mean is the balance point of the relative frequency distribution.

  • Median divides the distribution such that half the area is to the left and half to the right.

  • Mode corresponds to the peak (highest relative frequency).

• Practical considerations:

  • Mean is sensitive to extreme values (outliers) and skewness; can be misleading in skewed data.

  • Median is a resistant measure of central tendency; not affected as much by extreme values; preferred for highly skewed data (e.g., starting salaries).

  • Mode is rarely the best measure unless the mode itself is of specific interest (e.g., modal nail length for a supplier).

• Summary: the best measure depends on the type of descriptive information desired.

2.4.1 Applied Exercises

• Exercise: Highest paid engineers (Electronic Design, 2012)

  • Given population means: software engineering manager mean = 126{,}417; manufacturing/production engineer mean = 92{,}360; assume mound-shaped distributions.

  • Statements (true/false):

  • a. All software engineering managers earn exactly 126{,}417. (False)

  • b. Half of manufacturing/production engineers earn less than 92{,}360. (False)

  • c. A randomly selected software engineering manager will always earn more than a randomly selected manufacturing/production engineer. (False)

• Exercise: Ammonia in car exhaust (Environmental Science & Technology, 2000)

  • Data: eight daily afternoon ammonia concentrations (ppm).

  • Tasks: compute the mean; compute the median; interpret the values.

2.3–2.4 Further Note on Central Tendency and Variability

• Additional context from subsequent materials (e.g., 3.1 Crude oil biodegradation exercise) introduces related calculations for means, medians, and modes on specific data sets; these extend the application of central tendency concepts to real data.

2.5 Measures of Variation

• Rationale: Central tendency alone omits information about dispersion (spread).

• Common measures of variation:

  • Range: difference between maximum and minimum values; simple but insensitive to distribution shape.

  • Variance: average squared deviation from the mean; units are squared of the original units (e.g.,
    units^2).

  • Standard deviation: square root of the variance; same units as the original data; easiest to interpret when used with the mean.

• Relationship with mean:

  • When combined with the mean, standard deviation provides a practical sense of spread around the center.

• Rules for interpretation:

  • Empirical Rule (for mound-shaped distributions):

  • Approximately 68% of observations lie within ar{y} \, ext{±} \, s.

  • Approximately 95% lie within ar{y} \, ext{±} \, 2s.

  • Approximately 99.7% lie within ar{y} \, ext{±} \, 3s.

  • Chebyshev’s Rule (for any distribution):

  • At least 1 - \frac{1}{k^2} portion of observations lie within k standard deviations of the mean for any k > 1.

2.5.1 Applied Exercises

• Exercise: Highest paid engineers (mean = 126{,}417; sd = 15{,}000)

  • Distributions are mound-shaped and symmetric with given sd.

  • Task: sketch the distribution showing intervals ar{y} \pm \sigma, \bar{y} \pm 2\sigma, \bar{y} \pm 3\sigma and estimate the proportion in each interval.

• Exercise: Ammonia in car exhaust

  • Data: eight daily afternoon ammonia concentrations (ppm) as in 2.1.

  • Tasks:

  • a. Find the range.

  • b. Find the variance.

  • c. Find the standard deviation.

  • d. If the std deviation during morning drive-time is 1.45 ppm, which time (morning or afternoon) is more variable?

• Exercise: Bearing strength of FRP strips

  • Data: 10 FRP strip bearing strength measurements (MPa).

  • Task: use the sample data to provide an interval likely to contain the bearing strength.

2.6 Measures of Relative Standing

• Purpose: Describe the location of an observation relative to the rest of the data.

• Percentiles: specify the relative standing; e.g., the 84th percentile means 84% of observations are below that value.

• Key percentiles:

  • 25th percentile = lower quartile (Q1)

  • 50th percentile = median (Q2)

  • 75th percentile = upper quartile (Q3)

• Z-scores:

  • Definition: a z-score describes the location of an observation y relative to the mean in units of the standard deviation.

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

  • Sample form: z =\frac{y - \bar{y}}{s}

• Interpretive guidance:

  • Negative z-scores indicate observations left of the mean; positive z-scores indicate right of the mean.

  • Empirical Rule implications: most observations have |z| < 2, and almost all have |z| < 3.

• 2.6.1 Applied Exercises

  • Exercise: Phosphorous standards in the Everglades

  • Given: 75th percentile of TP distribution is 10 μg/L.

  • Task: interpret this percentile value and justify why it was used as a standard.

  • Exercise: Lead in drinking water (EPA Action Level = 0.015 mg/L)

  • Statement: 90th percentile of a study sample is 0.00372 mg/L.

  • Question: Are customers at risk of unhealthy lead levels? Explain.

2.7 Methods for Detecting Outliers

• Z-score method: A value with a large |z| may be an outlier.

• Box plot method (based on IQR):

  • Define hinges: QL (lower quartile) and QU (upper quartile).

  • Median shown inside the box.

  • Inner fences: at distances 1.5 × IQR from hinges; observations beyond inner fences flagged as potential outliers using markers (e.g., *, 0).

  • Outer fences: at distances 3 × IQR from hinges; outer fences indicated if observations fall beyond them.

  • IQR-based fences are not affected by outliers, whereas z-scores can be inflated by extreme values that inflate the standard deviation.

• Comparison:

  • Both methods provide rule-of-thumb criteria for labeling outliers.

  • Box-plot-based fences use quartiles and IQR, less sensitive to extreme values than standard deviation-based z-scores.

2.7.1 Applied Exercises

• Exercise: Highest paid engineers

  • Setup: Distribution is mound-shaped and symmetric with sd = 15{,}000; salary claim = 180{,}000.

  • Task: Assess believability of the claim and explain.

• Exercise: Barium content of clinkers

  • Given: summary statistics for 200 clinkers: QL = 115, m = 170, QU = 260.

  • Tasks:

  • a. Interpret the median m.

  • b. Interpret QL.

  • c. Interpret QU.

  • d. Compute IQR.

  • e. Endpoints of the inner fence for a box plot.

  • f. If no values fall beyond the inner fences, what does this imply?

• Exercise: Zinc phosphide in sugarcane

  • Data: distribution of ZnP concentration ≈ N(mean = 2.0%, sd = 0.08%).

  • Question: If one batch contains 1.80%, does this indicate too little ZnP in production? Explain.

2.8 Supplementary Exercises

• Exercise: Fate of scrapped tires

  • Data: categories and frequencies for tires’ fate; tasks include identifying variable, classes, class relative frequencies, pie chart, Pareto.

• Exercise: Microbial fuel cells (MFCs)

  • Data: 54 articles categorized by investigation area; graph summarizes areas; tasks:

  • a. Identify the qualitative variable measured for each article.

  • b. Identify the type of graph.

  • c. Convert to a Pareto diagram and identify the dominant investigation area.

• Exercise: Deep-hole drilling

  • Data: frequency histogram for drill chip lengths (50 chips).

  • Tasks:

  • a. Convert to a relative frequency histogram.

  • b. Based on the relative histogram, assess whether you would expect a drill chip of at least 190 mm.

4–6 Contextual and Practical Notes

• Overall understanding: The materials emphasize choosing the right graphical method based on data type, the trade-offs between displaying individual observations vs. distribution summaries, and practical rules for identifying central tendency, variability, and outliers.

• Connections to prior/statistical principles:

  • Concepts of consistency between graphical descriptions and numerical summaries.

  • Foundations for hypothesis testing and quality control (e.g., Pareto diagrams, outlier rules).

• Real-world relevance:

  • Qualitative data handling in safety/hazards contexts, manufacturing quality control, and program evaluation.

  • Quantitative data analysis foundations applied to engineering, environmental science, and industrial operations.

• Ethical/philosophical/practical implications:

  • The choice of descriptive method affects interpretation and decision-making (e.g., misrepresenting spread via inappropriate use of mean or histogram).

  • Outlier handling and data integrity considerations in reporting results.

Key Formulas and Concepts (Summary)

• Relative frequency (proportion) for category i:
  pi = \frac{ni}{n}

• Mean (sample):
  \bar{y} = \frac{1}{n} \sum{i=1}^{n} yi

• Z-score (population):
  z = \frac{y - \mu}{\sigma}

• Z-score (sample):
  z = \frac{y - \bar{y}}{s}

• Interquartile range:
  IQR = QU - QL

• Inner fences (boxplot):
  QL - 1.5 \cdot IQR, \, QU + 1.5 \cdot IQR

• Outer fences (boxplot):
  QL - 3 \cdot IQR, \, QU + 3 \cdot IQR

• Percentiles:

  • 25th = Q1, 50th = Q2 = median, 75th = Q3

• Empirical Rule (approximate for mound-shaped distributions):

  • Within \sigma: ~68%; within 2\sigma: ~95%; within 3\sigma: ~99.7%

• Chebyshev’s Rule (general):

  • At least 1 - \frac{1}{k^2} of observations lie within k\sigma of the mean, for any distribution and k > 1

Note: Throughout, variables are denoted in the text as y (observations), with sample n observations (y1, y2, …, y_n). Population parameters are typically denoted by Greek letters (e.g., μ, σ).