Descriptive Statistics in Biostatistics

Fundamentals of Biostatistics and Learning Outcomes

• Descriptive statistics provides tools and methods to organize, summarize, and display data effectively.
• Key learning objectives for descriptive data analysis include:
  • Understanding how data can be appropriately organized and displayed.
  • Understanding how to reduce large data sets into a few informative descriptive measures.
  • Calculating and interpreting measures of central tendency, including the arithmetic mean, median, and mode.
  • Calculating and interpreting measures of dispersion, including the range, variance, standard deviation, and coefficient of variation.

Data Summarization and Descriptive Measures

• In many practical scenarios, frequency distributions alone are insufficient to convey essential data characteristics, necessitating single-number summaries termed descriptive measures.
• Descriptive measures are computed from data collected either from a sample or an entire population:
  • Statistic: A descriptive measure computed from sample data. Standard Roman letters are typically used to represent statistics (such as $\bar{x}$ for sample mean or $s$ for sample standard deviation).
  • Parameter: A descriptive measure computed from population data. Greek letters are typically used to represent parameters (such as $\mu$ for population mean or $\sigma$ for population standard deviation).
• Descriptive measures are broadly categorized into measures of central tendency and measures of dispersion.

Measures of Central Tendency: The Mean

• A measure of central tendency provides a single numerical value that is considered representative or typical of a dataset as a whole.

• The arithmetic mean (commonly referred to simply as the mean) represents the standard average of a set of values, distinguished from other statistical means (e.g., geometric or harmonic means).

• The mean is calculated by summing all individual observations in a dataset and dividing the total sum by the total number of observations.

• Population Mean Formula:

  • For a finite population containing $N$ values where individual observations are designated by $x_i$ ($x_1, x_2, \dots, x_N$):     $\mu = \frac{\sum_{i=1}^{N} x_i}{N}$
  • Where:
  • $\mu$ = Population mean
  • $N$ = Total number of observations in the population
  • $x_i$ = Value of the $i\text{th}$ observation

• Sample Mean Formula:

  • For a sample containing $n$ observations:     $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
  • Where:
  • $\bar{x}$ = Sample mean
  • $n$ = Total number of observations in the sample

• Sample Subject Age Example:

  • Given a sample set of $10$ subjects with corresponding ages:
  • Subject 1: $50$
  • Subject 2: $50$
  • Subject 3: $58$
  • Subject 4: $71$
  • Subject 5: $79$
  • Subject 6: $57$
  • Subject 7: $61$
  • Subject 8: $53$
  • Subject 9: $61$
  • Subject 10: $56$

• CDC Health-Care Worker Influenza Vaccination Example:

  • According to CDC 2016–2017 reports, influenza vaccination percentages for health-care workers in select western states were:
  • Arizona (AZ): $88.9\%$
  • Colorado (CO): $97.1\%$
  • Nevada (NV): $79.0\%$
  • Idaho (ID): $92.0\%$
  • California (CA): $83.9\%$
  • Oregon (OR): $81.2\%$
  • New Mexico (NM): $86.2\%$
  • Washington (WA): $89.8\%$
  • Calculating the sample mean percentage ($n = 8$):     $\bar{x} = \frac{88.9 + 97.1 + 79.0 + 92.0 + 83.9 + 81.2 + 86.2 + 89.8}{8} = 87.3\%$

• Properties of the Mean:

  • Uniqueness: For any given dataset, there exists one and only one arithmetic mean.
  • Simplicity: It is straightforward to understand and compute.
  • Sensitivity to Outliers: Every observation in the dataset enters into the calculation. Consequently, extreme values (outliers) strongly influence the mean and can distort it, reducing its reliability as a typical value.
  • Example of Limitation due to Outliers: Consider procedural fees charged by five physicians: $\$75$, $\$75$, $\$80$, $\$80$, and $\$280$.     $\bar{x} = \frac{75 + 75 + 80 + 80 + 280}{5} = \$118$     The calculated mean of $\$118$ is higher than four of the five fees and does not represent the central cluster of the data.

Measures of Central Tendency: The Median

• The median is the middle value of an ordered dataset, dividing the observations into two equal halves such that an equal number of values lie above and below it.

• Determining the Median Position:

  • First, all observations must be arranged in order of magnitude (an ordered array).
  • The position of the median is given by the formula:     $\text{Median Position} = \left(\frac{n + 1}{2}\right)\text{th ordered observation}$
  • Odd Sample Size ($n$ is odd): The median is the single middle observation.
  • Example: For $n = 11$ observations, the median is the $\left(\frac{11 + 1}{2}\right) = 6\text{th}$ ordered observation.
  • Example: For 5 test scores ($30, 70, 80, 85, 90$), the middle value is $80$ (two scores are $\le 80$ and two scores are $\ge 80$).
  • Even Sample Size ($n$ is even): There is no single middle observation; the median is defined as the arithmetic mean of the two middle observations.
  • Example: For $n = 12$ observations, the median is the $\left(\frac{12 + 1}{2}\right) = 6.5\text{th}$ position, calculated as the average of the $6\text{th}$ and $7\text{th}$ ordered observations.

• CDC Health-Care Worker Vaccination Sample Median Calculation:

  • Step 1: Order the vaccination percentages for the $8$ states from smallest to largest:     $79.0, 81.2, 83.9, 86.2, 88.9, 89.8, 92.0, 97.1$
  • Step 2: Since $n = 8$ (even), identify the two middle values ($4\text{th}$ and $5\text{th}$ observations), which are $86.2$ and $88.9$.
  • Step 3: Compute the mean of the two middle values:     $\text{Median} = \frac{86.2 + 88.9}{2} = 87.55\% \approx 87.6\%$

• Properties of the Median:

  • Uniqueness: Only one median exists for a given dataset.
  • Simplicity: Easy to determine once data are ordered.
  • Robustness: Resistant to extreme values or outliers, making it preferred over the mean for skewed distributions.

Measures of Central Tendency: The Mode

• The mode is defined as the value that occurs most frequently in a dataset.
• A dataset may have:
  • No Mode: When all observations occur with equal frequency (e.g., dataset $10, 21, 33, 53, 54$ has no mode).
  • Unimodal: A single value occurs with highest frequency.
  • Multimodal: Two or more values share the highest frequency (e.g., ages of 10 lab employees: $20, 21, 20, 20, 34, 22, 24, 27, 27, 27$ has two modes: $20$ and $27$, each occurring 3 times).
• Application to Qualitative/Categorical Data: Mode is the only measure of central tendency that can be used for nominal or qualitative data. For instance, in a mental health clinic recording patient diagnoses (mental retardation, organic brain syndrome, psychosis, neurosis, personality disorder), the diagnosis with the highest count is termed the modal diagnosis.
• Population Age Grid Example:
  • In a population age dataset arranged sequentially, the value $53$ occurs $17$ times, making $53$ the mode for that population.

Shape of Distributions and Skewness

• Symmetric Distribution:
  • A distribution is symmetric if its visual representation (histogram or frequency polygon) can be divided into two mirror-image halves along a vertical axis.
  • In a unimodal symmetric distribution, the mean, median, and mode are all equal ($\text{Mean} = \text{Median} = \text{Mode}$).

• Asymmetric / Skewed Distribution:

  • If a distribution lacks symmetry, it is referred to as skewed.
  • Positive Skew (Right-Skewed):
  • The graph exhibits a long tail extending toward higher values on the right side.
  • The mean is pulled toward the long tail, resulting in: $\text{Mean} > \text{Median} > \text{Mode}$.
  • Negative Skew (Left-Skewed):
  • The graph exhibits a long tail extending toward lower values on the left side.
  • The mean is pulled toward the long tail, resulting in: $\text{Mean} < \text{Median} < \text{Mode}$.

• Mathematical Formula for Sample Skewness:   $\text{Skewness} = \frac{\sqrt{n} \sum_{i=1}^{n} (x_i - \bar{x})^3}{\left(\sum_{i=1}^{n} (x_i - \bar{x})^2\right)^{3/2}} = \frac{\sqrt{n} \sum_{i=1}^{n} (x_i - \bar{x})^3}{(n - 1)\sqrt{n - 1}\,s^3}$

  • Where $s$ is the sample standard deviation.
  • Interpretation of Skewness values:
  • $\text{Skewness} = 0$: Symmetric distribution.
  • $\text{Skewness} > 0$: Positively skewed (right-skewed).
  • $\text{Skewness} < 0$: Negatively skewed (left-skewed).

• Numerical Comparison of Distribution Shapes:   | Summary Measure | No Skew (Symmetric) | Right Skew (Positive) | Left Skew (Negative) |   | :--- | :--- | :--- | :--- |   | Mean | $8.0000$ | $6.6667$ | $8.3333$ |   | Median | $8.0000$ | $6.0000$ | $9.0000$ |   | Mode | $8.00$ | $5.00$ | $10.00$ |   | Skewness Coefficient | $.000$ | $.627$ | $-.627$ |

Measures of Dispersion: Range

• Dispersion (also termed variation, spread, or scatter) quantifies the degree of variability among observations in a dataset.
• If all observations in a dataset are identical, dispersion is zero; as differences among observations increase, dispersion increases.

• Range ($R$):
  • The range is defined as the difference between the largest value ($x_L$) and the smallest value ($x_S$) in a dataset:     $R = x_L - x_S$
  • Example Calculation: In a dataset where the youngest subject is $30\,\text{years}$ old and the oldest subject is $82\,\text{years}$ old:     $R = 82 - 30 = 52\,\text{years}$
  • Limitations of Range:
  • While simple to calculate, the range relies exclusively on two extreme observations and ignores all intermediate values, making it a poor standalone metric of overall dispersion.
  • Range is often more informative when presented as an interval pair $[x_S, x_L]$, such as $[30, 82]$.

Measures of Dispersion: Variance and Standard Deviation

• Variance measures dispersion relative to the scatter of individual observations around their arithmetic mean.

• Sum of Deviations Principle:

  • Raw deviations $(x_i - \bar{x})$ sum to zero ($\sum (x_i - \bar{x}) = 0$) because positive and negative deviations cancel out. To prevent this cancellation, each deviation is squared.

• Sample Variance ($s^2$):

  • The sample variance is the sum of squared deviations from the sample mean divided by the sample size minus 1 ($n - 1$):     $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}$
  • Degrees of Freedom: Dividing by $n - 1$ instead of $n$ provides an unbiased estimate of the population variance, where $n - 1$ represents the number of independent degrees of freedom.

• Population Variance ($\sigma^2$):

  • For a finite population of size $N$:     $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$

• CDC Health-Care Worker Vaccination Variance Calculation:

  • Using vaccination percentages ($n = 8$, mean $\bar{x} = 87.3\%$):     $s^2 = \frac{(88.9 - 87.3)^2 + (97.1 - 87.3)^2 + \dots + (89.8 - 87.3)^2}{8 - 1} = 245.8$

• Standard Deviation:

  • Because variance is expressed in squared units (e.g., $\text{years}^2$, $\%^2$), taking its positive square root converts the measure back into original measurement units.
  • Sample Standard Deviation ($s$):     $s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}$
  • Population Standard Deviation ($\sigma$):     $\sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}$

Coefficient of Variation

• Standard deviation expresses absolute variation in measurement units. However, comparing absolute standard deviations across datasets can lead to fallacious conclusions if:

  • The variables are measured in different units (e.g., serum cholesterol in $\text{mg/100\,mL}$ versus body weight in $\text{pounds}$).
  • The variables have vastly different means (e.g., body weights of first-grade children versus high school freshmen).

• The Coefficient of Variation (C.V.) expresses standard deviation as a percentage of the arithmetic mean, producing a unitless metric of relative variation:   $\text{C.V.} = \frac{s}{\bar{x}} \times 100\%$

• Comparative Weight Variability Example:

  • Consider two male age groups:     | Metric | Sample 1 (25-year-olds) | Sample 2 (11-year-olds) |     | :--- | :--- | :--- |     | Mean Weight | $145\,\text{pounds}$ | $80\,\text{pounds}$ |     | Standard Deviation | $10\,\text{pounds}$ | $10\,\text{pounds}$ |
  • Calculations:
  • For 25-year-olds:       $\text{C.V.} = \frac{10}{145} \times 100\% = 6.9\%$
  • For 11-year-olds:       $\text{C.V.} = \frac{10}{80} \times 100\% = 12.5\%$
  • Interpretation: Although both groups have an identical standard deviation of $10\,\text{pounds}$, relative weight variability is significantly higher in 11-year-olds ($12.5\%$ ) than in 25-year-olds ($6.9\%$ ) relative to their respective body masses.

Percentiles, Quartiles, and Interquartile Range

• Location Parameters: Measures such as the mean, median, percentiles, and quartiles locate specific points along the horizontal scale of a data distribution.

• Percentiles:

  • The $p\text{th}$ percentile ($P$) is defined as the value below which at least $p\,\%$ of observations lie, and above which at most $(100 - p)\,\%$ of observations lie.
  • Subscripts denote specific percentiles (e.g., $P_{10}$ is the 10th percentile, $P_{70}$ is the 70th percentile, $P_{50}$ is the 50th percentile/median).

• Quartiles:

  • Quartiles divide an ordered dataset into four equal segments:
  • First Quartile ($Q_1$): $25\text{th}$ percentile ($P_{25}$).
  • Second Quartile ($Q_2$): $50\text{th}$ percentile ($P_{50}$ or Median).
  • Third Quartile ($Q_3$): $75\text{th}$ percentile ($P_{75}$).
  • Position Formulas for Quartiles (for an ordered array of size $n$):     $Q_1 \text{ position} = \left(\frac{n + 1}{4}\right)\text{th ordered observation}$$Q_2 \text{ position} = \frac{2(n + 1)}{4} = \left(\frac{n + 1}{2}\right)\text{th ordered observation}$$Q_3 \text{ position} = \left(\frac{3(n + 1)}{4}\right)\text{th ordered observation}$

• Interquartile Range (IQR):

  • The IQR measures the spread of the middle $50\%$  of observations and is calculated as:     $\text{IQR} = Q_3 - Q_1$
  • A larger IQR indicates greater variability within the central portion of the dataset.

Kurtosis and Distribution Peakness

• Kurtosis quantifies the degree of peakedness or flatness of a distribution relative to a standard normal distribution.

• Sample Kurtosis Formula:   $\text{Kurtosis} = \frac{n \sum_{i=1}^{n} (x_i - \bar{x})^4}{\left(\sum_{i=1}^{n} (x_i - \bar{x})^2\right)^2} - 3 = \frac{n \sum_{i=1}^{n} (x_i - \bar{x})^4}{(n - 1)^2\,s^4} - 3$

• Classifications of Kurtosis:

  • Mesokurtic: Normal bell-shaped distribution with a excess kurtosis value of $0$ ($\text{Kurtosis} = .000$).
  • Leptokurtic: Possesses a smaller proportion of observations in its tails and a sharper peak than a normal distribution ($\text{Kurtosis} > 0$, e.g., $.608$).
  • Platykurtic: Possesses an excessive proportion of observations in its tails and a flatter top than a normal distribution ($\text{Kurtosis} < 0$, e.g., $-1.158$).

• Numerical Comparison of Kurtosis Types:   | Summary Measure | Mesokurtic | Leptokurtic | Platykurtic |   | :--- | :--- | :--- | :--- |   | Mean | $6.0000$ | $6.0000$ | $6.0000$ |   | Median | $6.0000$ | $6.0000$ | $6.0000$ |   | Mode | $6.00$ | $6.00$ | $6.00$ |   | Kurtosis | $.000$ | $.608$ | $-1.158$ |