Descriptive Statistics: Presenting and Summarising Data

Review of Fundamental Variable Types and Sampling Methods

• Population vs. Sample:

  • A population encompasses all subjects, items, or observations of interest in a study.

  • A sample is a subset of observations selected from a population.

  • Random Sampling: A sampling design in which every subject in the population has an equal, non-zero probability of being selected into the sample.

• Types of Quantitative Variables:

  • Continuous Variables: Quantitative variables that can take any value within a given numerical interval (e.g., height, weight, distance, temperature, moisture content).

  • Discrete Variables: Quantitative variables whose values are restricted to a finite or countably infinite set of distinct values (e.g., counts of events, number of cells, number of ticks).

• Types of Categorical Variables:

  • Ordinal Variables: Categorical variables composed of ordered or ranked categories (e.g., satisfaction ratings like very dissatisfied, dissatisfied, satisfied, very satisfied).

  • Nominal Variables: Categorical variables with unordered categories that carry no natural quantitative ranking (e.g., eye color, blood type, species name).

Frequency Distributions

• Definition: A tabular summary of data showing the number (frequency) of observations in each of several non-overlapping categories or class intervals.

• Core Requirements for Class Intervals:

  • Mutually Exclusive: Class intervals must not overlap; every single observation fits into exactly one class.

  • All-Inclusive: Class intervals must span the full range of the data; no observation is left unclassified.

  • Number of Class Intervals: As a general rule, a frequency distribution should contain between $6$ and $12$ class intervals.

• Constructing a Frequency Distribution (Step-by-Step):

  1. Identify the sample minimum value ($x_{\text{min}}$) and maximum value ($x_{\text{max}}$).

  2. Determine an appropriate class width and starting point such that the first class includes the minimum value.

  3. Define non-overlapping boundary conditions (e.g., $\ge 21.5\,\text{cm} \text{ to } < 21.9\,\text{cm}$).

  4. Tally and record the count (frequency) of data points falling into each class interval.

• Detailed Example — Vitamin B1 Treatment on Tomato Transplants:

  • Scenario: A plant scientist evaluates the effect of thiamine hydrochloride (vitamin B1) on vegetable transplants. A random sample of $n = 50$ tomato plants is measured $14\,\text{days}$ post-treatment.

• Raw Data (Heights in $\text{cm}$):

  • $21.5, 21.6, 21.8, 21.8, 21.8, 21.9, 21.9, 22.0, 22.1, 22.1$

  • $22.1, 22.2, 22.2, 22.3, 22.4, 22.5, 22.5, 22.5, 22.5, 22.6$

  • $22.6, 22.6, 22.7, 22.7, 22.8, 22.8, 22.8, 22.9, 22.9, 22.9$

  • $22.9, 23.0, 23.0, 23.0, 23.1, 23.2, 23.2, 23.2, 23.2, 23.3$

  • $23.3, 23.4, 23.5, 23.5, 23.6, 23.7, 23.8, 23.9, 24.0, 24.2$

• Constructed Frequency Table (Class Width = $0.4\,\text{cm}$):

  • Class $21.5 - <21.9\,\text{cm}$: Frequency = $5$

  • Class $21.9 - <22.3\,\text{cm}$: Frequency = $8$

  • Class $22.3 - <22.7\,\text{cm}$: Frequency = $9$

  • Class $22.7 - <23.1\,\text{cm}$: Frequency = $12$

  • Class $23.1 - <23.5\,\text{cm}$: Frequency = $8$

  • Class $23.5 - <23.9\,\text{cm}$: Frequency = $5$

  • Class $23.9 - <24.3\,\text{cm}$: Frequency = $3$

• Analytical Summary of Distribution:

  • Most plant heights are concentrated between $22.7\,\text{cm}$ and $23.1\,\text{cm}$.

  • Plant heights vary continuously from a minimum of $21.5\,\text{cm}$ to a maximum of $24.2\,\text{cm}$.

  • Frequencies taper off smoothly toward both extremes, showing no unusually tall or short plants.

Histograms and Distribution Shapes

• Histogram Definition: A visual representation of a continuous frequency distribution consisting of contiguous vertical bars drawn over class intervals.

  • The base of each bar represents the class interval on the horizontal axis.

  • The area of each bar is proportional to the class frequency.

  • When all class intervals have equal width, the height of each bar is directly proportional to its class frequency.

• Adjusting Bar Heights for Unequal Class Widths:

  • When class widths are unequal, bar heights must be adjusted so that bar area accurately represents frequency.

  • Standard Width: Select the narrowest class width in the dataset as the baseline standard width.

  • Width Multiple: Calculate the relative scale factor for each class interval:     $\text{Multiple} = \frac{\text{Class Width}}{\text{Standard Width}}$

  • Adjusted Bar Height (Frequency Density): Compute bar height as:     $\text{Bar Height} = \frac{\text{Frequency}}{\text{Multiple}}$

• Handling Open-Ended Classes:

  • An open-ended class lacks a specified upper or lower boundary (e.g., $\ge 90$).

  • Assign a realistic boundary based on adjacent class widths or physical domain limits (e.g., capping percentage variables at $100\%$).

• Detailed Example — Genome Sizes of $118$ Common Viruses:

  • Standard Width: $10$ (the smallest interval width present).

  • Adjustment Calculations Table:

  • Class $0 - 20$: Width = $20$, Multiple = $2$, Frequency = $12$, Adjusted Height = $\frac{12}{2} = 6$

  • Class $20 - 40$: Width = $20$, Multiple = $2$, Frequency = $16$, Adjusted Height = $\frac{16}{2} = 8$

  • Class $40 - 50$: Width = $10$, Multiple = $1$, Frequency = $20$, Adjusted Height = $\frac{20}{1} = 20$

  • Class $50 - 60$: Width = $10$, Multiple = $1$, Frequency = $24$, Adjusted Height = $\frac{24}{1} = 24$

  • Class $60 - 70$: Width = $10$, Multiple = $1$, Frequency = $18$, Adjusted Height = $\frac{18}{1} = 18$

  • Class $70 - 90$: Width = $20$, Multiple = $2$, Frequency = $20$, Adjusted Height = $\frac{20}{2} = 10$

  • Class $\ge 90$ (Assumed limit $90 - 110$): Width = $20$, Multiple = $2$, Frequency = $8$, Adjusted Height = $\frac{8}{2} = 4$

• Classifying Distribution Shapes:

  • Symmetric Distribution:

  • Left and right halves centered around the midpoint are mirror images.

  • Bell-shaped or unimodal curves.

• Right-Skewed Distribution (Positively Skewed):

  • Data concentration is located on the left side, with a long tail stretching toward higher values on the right.

• Left-Skewed Distribution (Negatively Skewed):

  • Data concentration is located on the right side, with a long tail stretching toward lower values on the left.

Bar Charts for Categorical Data

• Definition: A visual representation of discrete or qualitative data where each category is represented by an isolated rectangular bar.

• Key Features:

  • Bars are drawn separated by gaps to emphasize discrete, non-continuous categories.

  • All bars have equal width.

  • The length or height of each bar is directly proportional to category frequency or percentage.

• Example — Customer Satisfaction Rating ($n = 440$):

  • Very satisfied: Frequency = $105$, Percentage = $23.9\%$

  • Satisfied: Frequency = $134$, Percentage = $30.5\%$

  • Dissatisfied: Frequency = $30$, Percentage = $6.8\%$

  • Very dissatisfied: Frequency = $171$, Percentage = $38.9\%$

Measures of Centrality

• Purpose: Measures of centrality (or location) identify a single typical or central value around which data cluster.

• Mean (Arithmetic Mean):

  • Defined as the sum of all numerical values divided by the total count of observations.

  • Population Mean Formula:     $\mu = \frac{\sum x}{N}$

  • Sample Mean Formula:     $\bar{x} = \frac{\sum x}{n}$

  • Calculation Example: Cell diameters ($n = 10$): $1.2, 2.3, 2.9, 3.4, 3.5, 3.5, 4.0, 4.1, 4.9, 5.0\,\mu m$     $\sum x = 1.2 + 2.3 + 2.9 + 3.4 + 3.5 + 3.5 + 4.0 + 4.1 + 4.9 + 5.0 = 34.8\,\mu m$     $\bar{x} = \frac{34.8}{10} = 3.48\,\mu m$

• Median ($\text{Med}$):

  • Defined as the exact middle observation when data are arranged in ascending order.

  • Positional Index: Located at position index $\frac{n+1}{2}$.

  • Calculation Rules:

  • If sample size $n$ is odd: The median is the single observation in position $\frac{n+1}{2}$.

  • If sample size $n$ is even: There is no single middle value. The median is the average of the two middle observations in positions $\frac{n}{2}$ and $\frac{n}{2} + 1$:       $\text{Med} = x_{(n/2)} + 0.5\left(x_{(n/2+1)} - x_{(n/2)}\right)$

  • Calculation Example (Odd $n = 9$): Canine tick counts: $2, 3, 3, 4, 7, 9, 10, 10, 217$

  • Position = $\frac{9+1}{2} = 5\text{th}$.

  • $\text{Med} = 7$

  • $\bar{x} = \frac{265}{9} = 29.44$

  • Calculation Example (Even $n = 8$): Canine tick counts: $3, 3, 4, 6, 9, 10, 10, 196$

  • Middle positions are $4\text{th}$ ($6$) and $5\text{th}$ ($9$).

  • $\text{Med} = 6 + 0.5(9 - 6) = 7.5$

  • $\bar{x} = \frac{241}{8} = 30.13$

• Robustness and Sensitivity to Distribution Shape:

  • Robustness: The median is robust to extreme outliers; an extreme value change does not alter the middle position value. The mean is sensitive to outliers and gets pulled ("dragged") toward extreme values.

  • Symmetric Distributions: Mean and median are equal ($\bar{x} \approx \text{Med}$).

  • Right-Skewed Distributions: Mean is larger than median ($\bar{x} > \text{Med}$).

  • Left-Skewed Distributions: Mean is smaller than median ($\bar{x} < \text{Med}$).

Measures of Variability and Spread

• Range:

  • Difference between the maximum and minimum values in a dataset:     $\text{Range} = x_{\text{max}} - x_{\text{min}}$

  • Simple to compute, but highly sensitive to outliers as it relies on only two extreme values.

  • Example: Bacteria doubling times ($1, 2, 3, 3, 5, 9, 10, 41\,\text{hours}$):     $\text{Range} = 41 - 1 = 40\,\text{hours}$

• Variance:

  • Measures the mean squared deviation of data points from the sample mean.

  • Population Variance Formula:     $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$

  • Sample Variance Formula:     $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$

  • Worked Example: Butterfly wing lengths ($23, 25, 29, 35, 41, 47, 52\,\text{mm}$):

  • $n = 7$

  • $\bar{x} = \frac{23 + 25 + 29 + 35 + 41 + 47 + 52}{7} = \frac{252}{7} = 36\,\text{mm}$

  • Squared deviations: $(23-36)^2 = 169$, $(25-36)^2 = 121$, $(29-36)^2 = 49$, $(35-36)^2 = 1$, $(41-36)^2 = 25$, $(47-36)^2 = 121$, $(52-36)^2 = 256$

  • Sum of squared deviations: $\sum (x - \bar{x})^2 = 169 + 121 + 49 + 1 + 25 + 121 + 256 = 742$

  • $s^2 = \frac{742}{7 - 1} = \frac{742}{6} = 123.67\,\text{mm}^2$

• Standard Deviation:

  • Positive square root of the variance, returning the dispersion measure to the original unit of measurement:     $s = \sqrt{s^2}$

  • Population Standard Deviation: $\sigma$

  • Sample Standard Deviation: $s$

Quartiles and Percentiles

• Quartiles:

  • Values that partition an ordered dataset into four equal quarters.

  • First Quartile ($Q_1$): 25th percentile. Cutoff below which $25\%$ of observations fall.

  • Second Quartile ($Q_2$): 50th percentile (Median). Cutoff below which $50\%$ of observations fall.

  • Third Quartile ($Q_3$): 75th percentile. Cutoff below which $75\%$ of observations fall.

• Positional Index Formulas in Ordered Data:

  • Index for $Q_1$: $\frac{1}{4}(n+1)$

  • Index for $Q_2$: $\frac{1}{2}(n+1)$

  • Index for $Q_3$: $\frac{3}{4}(n+1)$

  • Index for Percentile $P_x$: $\frac{x(n+1)}{100}$

  • Linear Interpolation Rule: If a positional index is not a whole number, interpolate proportionally between adjacent ordered data values.

• Worked Example — Gibraltar Dolphin Sightings ($n = 14\,\text{days}$):

  • Ordered dataset: $4, 6, 7, 8, 8, 9, 12, 13, 14, 16, 16, 19, 20, 22$

  • Computing $Q_1$:

  • Position = $\frac{1}{4}(14+1) = 3.75\text{th}$

  • $Q_1 = x_{(3)} + 0.75\left(x_{(4)} - x_{(3)}\right) = 7 + 0.75(8 - 7) = 7.75$

  • Computing $Q_2$:

  • Position = $\frac{1}{2}(14+1) = 7.5\text{th}$

  • $Q_2 = x_{(7)} + 0.5\left(x_{(8)} - x_{(7)}\right) = 12 + 0.5(13 - 12) = 12.5$

  • Computing $Q_3$:

  • Position = $\frac{3}{4}(14+1) = 11.25\text{th}$

  • $Q_3 = x_{(11)} + 0.25\left(x_{(12)} - x_{(11)}\right) = 16 + 0.25(19 - 16) = 16.75$

• Interquartile Range (IQR):

  • The range of the middle $50\%$ of the data:     $\text{IQR} = Q_3 - Q_1$

  • Unlike the total range, $\text{IQR}$ is robust to extreme outliers.

Boxplots and Outlier Detection

• Definition: A visual representation based on the Five-Number Summary ($x_{\text{min}}$, $Q_1$, $Q_2$, $Q_3$, $x_{\text{max}}$).

• Anatomy of a Boxplot:

  • Central box spanning from $Q_1$ to $Q_3$ (length equals $\text{IQR}$).

  • Internal vertical/horizontal line drawn at median $Q_2$.

  • Whiskers: Lines extending from box limits to the adjacent values.

  • Lower Adjacent Value: The smallest observation that is $\ge Q_1 - 1.5 \times \text{IQR}$.

  • Upper Adjacent Value: The largest observation that is $\le Q_3 + 1.5 \times \text{IQR}$.

  • Outliers: Observations falling strictly outside lower boundary ($< Q_1 - 1.5 \times \text{IQR}$) or upper boundary ($> Q_3 + 1.5 \times \text{IQR}$). Outliers are plotted individually using symbols such as asterisks (*) or open circles (o).

• Worked Example — Virus Cell Diameters ($n = 23\,\text{nm}$):

  • Raw ordered data: $12, 13, 14, 14, 15, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 21, 21, 22, 23, 23, 25, 32$

  • Quartiles:

  • $Q_1$ position = $\frac{1}{4}(23+1) = 6\text{th}$ value $= 16\,\text{nm}$

  • $Q_2$ position = $\frac{1}{2}(23+1) = 12\text{th}$ value $= 18\,\text{nm}$

  • $Q_3$ position = $\frac{3}{4}(23+1) = 18\text{th}$ value $= 21\,\text{nm}$

  • $\text{IQR}$ and Threshold Boundaries:

  • $\text{IQR} = 21 - 16 = 5\,\text{nm}$

  • $1.5 \times \text{IQR} = 1.5 \times 5 = 7.5\,\text{nm}$

  • Lower Fence = $16 - 7.5 = 8.5\,\text{nm}$

  • Upper Fence = $21 + 7.5 = 28.5\,\text{nm}$

  • Adjacent Values & Outliers:

  • Lower Adjacent Value = $12\,\text{nm}$ (smallest observation $\ge 8.5$).

  • Upper Adjacent Value = $25\,\text{nm}$ (largest observation $\le 28.5$).

  • Outlier = $32\,\text{nm}$ (since $32 > 28.5$).

• Comparative Boxplots:

  • Preparing boxplots side by side on a shared scale enables direct comparison between independent distributions regarding median location, overall dispersion ($\text{IQR}$ and range), skewness, and extreme values.

Applied Practical Scenarios and Exam Exercises

• Dublin ER Hospital Patient Ages Scenario (Summer 2009 Q1b):

  • Sample size $n = 90$. Age classes and frequencies:

  • $15 - 20$: Frequency = $9$ (Width = $5$, Height = $9$)

  • $20 - 25$: Frequency = $13$ (Width = $5$, Height = $13$)

  • $25 - 30$: Frequency = $14$ (Width = $5$, Height = $14$)

  • $30 - 40$: Frequency = $48$ (Width = $10$, Multiple = $2$, Height = $\frac{48}{2} = 24$)

  • $40 - 50$: Frequency = $6$ (Width = $10$, Multiple = $2$, Height = $\frac{6}{2} = 3$)

  • $50 - 80$: Frequency = $6$ (Width = $30$, Multiple = $6$, Height = $\frac{6}{6} = 1$)

  • Distribution Shape Analysis: Strongly right-skewed with concentration between $30$ and $40$ years and a thin upper tail extending to $80$ years.

  • Recommended Measures: Median and Interquartile Range ($\text{IQR}$), because the histogram reveals clear skewness and unequal interval spread.

• West Cork Soil Moisture Content Scenario (Summer 2011 Q2):

  • Sample size $n = 30$ fields in West Cork.

  • Data ($\%$ moisture): $1.50, 3.15, 3.6, 4.1, 5.0, 7.3, 10.0, 10.1, 12.0, 13.4, 15.6, 17.0, 28.5, 30.0, 31.0, 31.0, 31.5, 32.0, 33.0, 33.5, 34.0, 35.5, 36.0, 40.0, 41.0, 42.0, 46.0, 48.0, 54.0, 78.5$

  • Mean & Median Calculations:

  • $\sum x = 808.15$

  • Mean $\bar{x} = \frac{808.15}{30} = 26.94\%$

  • Median position = $15.5\text{th}$, between $31.0$ and $31.0$. Median $\text{Med} = 31.0\%$

  • Quartiles & Interpretation:

  • $Q_1$ position = $\frac{1}{4}(31) = 7.75\text{th}$ value $= 10.0 + 0.75(10.1 - 10.0) = 10.075\%$

  • $Q_2$ position = $\frac{1}{2}(31) = 15.5\text{th}$ value $= 31.0\%$

  • $Q_3$ position = $\frac{3}{4}(31) = 23.25\text{th}$ value $= 36.0 + 0.25(40.0 - 36.0) = 37.0\%$

  • Interpretation: $25\%$ of fields have moisture $\le 10.075\%$, $50\%$ have moisture $\le 31.0\%$, and $75\%$ have moisture $\le 37.0\%$.

  • Boxplot Calculations:

  • $\text{IQR} = 37.0 - 10.075 = 26.925\%$

  • $1.5 \times \text{IQR} = 40.3875$

  • Lower fence = $10.075 - 40.3875 = -30.3125\%$

  • Upper fence = $37.0 + 40.3875 = 77.3875\%$

  • Lower Adjacent Value = $1.50\%$

  • Upper Adjacent Value = $54.0\%$

  • Outlier = $78.5\%$

  • Low Moisture Content Analysis:

  • Low moisture defined as $< 10\%$. Count of fields $< 10\%$ is $6$ ($1.50, 3.15, 3.6, 4.1, 5.0, 7.3$).

  • Percentage = $\frac{6}{30} \times 100\% = 20\%$

  • Expected low moisture fields in a sample of $200$ fields = $200 \times 0.20 = 40$ fields.

• Student University Accommodation Distance Exercise (Summer 2010 Q2):

  • Sample size $n = 29$ full-time students.

  • Data ($\text{km}$): $0.15, 0.25, 0.30, 0.40, 0.45, 1.00, 1.24, 1.30, 1.50, 1.70, 1.75, 2.00, 2.10, 2.30, 2.35, 2.40, 3.00, 3.40, 3.55, 3.60, 3.70, 4.00, 4.20, 4.25, 5.60, 6.10, 6.90, 7.80, 8.50$

  • Mean = $\frac{88.74}{29} = 3.06\,\text{km}$, Median ($15\text{th}$ value) = $2.35\,\text{km}$.

  • Impact of Adding an Extreme Observation ($20\,\text{km}$):

  • New Mean = $\frac{88.74 + 20}{30} = 3.62\,\text{km}$ (increases significantly by $0.56\,\text{km}$).

  • New Median = $\frac{2.35 + 2.40}{2} = 2.375\,\text{km}$ (increases slightly by $0.025\,\text{km}$).

  • Demonstrates the median's robustness against extreme pull compared to the mean.

• Cigarette Nicotine Concentration Exercise (Summer 2008 Q1a):

  • Sample size $n = 98$ cigarette brands.

  • Classes: $0-100$ ($2$), $100-150$ ($10$), $150-200$ ($15$), $200-250$ ($39$), $250-350$ ($14$), $350-500$ ($12$), $500-800$ ($6$).

  • Histogram Shape: Right-skewed distribution with peak at $200 - 250\,\text{mg}$ and a right tail stretching to $800\,\text{mg}$.

  • Comparison: Mean > Median due to rightward tail drag. Median is selected as the superior centrality metric.

• Children Weights Frequency Distribution Exercise (Autumn 2006 Q1c):

  • Dataset of $n = 35$ five-year-old children weights ($\text{kg}$):

  • $23, 20, 21, 20, 19, 16, 24, 23, 22, 23, 19, 23, 29, 21, 20$

  • $16, 17, 21, 21, 24, 24, 20, 23, 20, 21, 19, 18, 23, 20, 21$

  • $31, 21, 26, 23, 24$

  • Class width = $2\,\text{kg}$, first class $\ge 16 \text{ to } < 18\,\text{kg}$.

  • Frequency Distribution Table:

  • Class $16 - <18$: Frequency = $3$

  • Class $18 - <20$: Frequency = $4$

  • Class $20 - <22$: Frequency = $13$

  • Class $22 - <24$: Frequency = $8$

  • Class $24 - <26$: Frequency = $4$

  • Class $26 - <28$: Frequency = $1$

  • Class $28 - <30$: Frequency = $1$

  • Class $30 - <32$: Frequency = $1$