Data Summarization and Descriptive Statistics

Data are facts or figures, numerical or otherwise, collected with a defined purpose.

Qualitative Data:
- Nominal Data:
  - Examples include gender (women, men), hair color (blonde, brown), and ethnicity (Hispanic, Asian).
- Ordinal Data:
  - Examples include first, second, and third positions, letter grades (A, B, C), and economic status (low, medium).
Quantitative Data:
- Discrete Data:
  - Examples include the number of students in a class, the number of workers in a company, and the number of home runs in a baseball game.
- Continuous Data:
  - Examples include the height of children, the square footage of a two-bedroom house, and the speed of cars.

Process:
- Expectation → Transcribe → Organize → Check → Validate → Summarize → Reality
Variables include:
- Object
- Pollutants
- Organisms
- Populations
- Temperature
- Rainfall/Precipitation
- Relative Humidity (RH)
- Color
- Sweetness
- Weight

Examples:
- Observational Unit: Butterfly
  - Variable: Cases
  - Data: ΣΣΣΣ (6 cases)
- Observational Unit: Quadrat
  - Variable: Cases
- Observational Unit: Nest
  - Variable: Number of colonies
  - Data: 12 cases with varying numbers (e.g., 4, 10, 6, 8, 4, 14, 3, 9, 11, 10).

Descriptive statistics are methods used to describe, show, and summarize data in a meaningful way.

Variable: Height of hedgehogs in cm
- Example data points: 25.40, 28.99, 24.64, 24.86, 27.86, …, 18.73, 21.55
- Total: 196 cases

Size Class, Frequency, and Relative Frequency
- 5-9: Frequency 8, Relative Frequency 0.041 (4.1%)
- 9-13: Frequency 25, Relative Frequency 0.127 (12.7%)
- 13-17: Frequency 37, Relative Frequency 0.188 (18.8%)
- 17-21: Frequency 52, Relative Frequency 0.265 (26.5%)
- 21-25: Frequency 38, Relative Frequency 0.194 (19.4%)
- 25-29: Frequency 28, Relative Frequency 0.143 (14.3%)
- 29-33: Frequency 8, Relative Frequency 0.041 (4.1%)
The area of the bins represents the percentage or proportion of values within the limits of the bins’ width.

Provide information about where the center of the set of data is and indirectly about its shape.
1. The Mean:
- Standardized sum
- Example calculation: \frac{25.4 + 28.9 + 24.64 + 24.86 + 27.86}{5} = \frac{131.75}{5} = 26.35
- Mean value represents a typical height for a hedgehog.

2. The Median:
- The value at the middle; 50% of the values are below and 50% are above.
- Example: 24.64 | 24.86 | 25.4 | 27.86 | 28.9. The median is 25.4.

3. The Mode:
- The most frequent value.
- Example: 24.64 | 24.86 | 25.4 | 27.86 | 24.64. The mode is 24.64.

Provide information about the variability of our data.
1. The Range:
- Difference between the minimum and maximum values.
- Example: 33.66 – 5.70 = 27.96
2. Quartiles:
- Q1 (25th Percentile)
- Q2 (50th Percentile, Median)
- Q3 (75th Percentile)
- IQR (Inter-Quartile Range)

3. The Variance and Standard Deviation:
- Summarize how close each value is to the mean.

Statistics allow us to use a reduced number of values – or even one – to describe the features of a variable.
Location:
- Mean: 19.18
- Median: 18.98
- Mode: 18.19
Spread:
- Variance: 37.88
- SD (Standard Deviation): 6.11
- IQR (Inter-Quartile Range): 8.63

A measure of location should always be reported along with one measure of spread.
- Example 1: Mean = 19.80, Standard Deviation = 6.11 (19.80 ± 6.11)
- Example 2: Median = 16.05, IQR = 8.90 (16.05 ± 8.90)
Recommended size of hole calculations:
- Based on mean: Upper limit = 19.8 + 6.11 ≈ 26 cm
- Based on median = 25 cm
- Recommended size of hole with median = 18 cm

Skewness: Asymmetry relative to the center of a distribution.
Kurtosis: Determines the “peak” of the distribution (how flat).
- Right Skewed: Mode < Median < Mean
- Symmetric: Mode = Median = Mean
- Left Skewed: Mean < Median < Mode
Skewness focuses on the spread (tails) of normal distribution, Kurtosis focuses more on the height of the distribution curve.

Statistics is concerned with collecting, describing, and using data.
Descriptive statistics encompass a set of procedures to summarize data from a variable of interest.
The method to summarize and describe data depends on the scale of measurement of that variable of interest.
The measures of location provide information about the values of the majority of the cases (what is a typical value?).
The measures of spread provide information about the variability or how much our values differ from each other and from the mean (how typical is typical?).
We always report a measure of location accompanied by the appropriate measure of spread or variability.