CRP Level 2 Data – dDescriptive Statistics and Levey-Jennings Labeling

Data set and basic information

• Data points (n = 10) for CRP level 2 controls, unit: mg/dL:
  2.3, 2.3, 2.4, 2.1, 1.9, 2.1, 2.3, 2.3, 2.4, 2.6
• These values were collected over 10 days at a local clinic
• Data used for descriptive statistics and Levey-Jennings chart labeling

Mean, median, and mode

• Sum of data points:
  $\sum x_i = 22.7$
• Number of observations:
  $n = 10$
• Mean (average):
  $\mu = \frac{1}{n} \sum x_i = \frac{22.7}{10} = 2.27\ \text{mg/dL}$
• Median (middle value for sorted data):
  • Sorted data: 1.9, 2.1, 2.1, 2.3, 2.3, 2.3, 2.3, 2.4, 2.4, 2.6
  • For even n, median = average of the 5th and 6th values:
    $\text{Median} = \frac{2.3 + 2.3}{2} = 2.3\ \text{mg/dL}$
• Mode (most frequent value):
  • 2.3 mg/dL occurs 4 times (highest frequency) → unique mode = $2.3\ \text{mg/dL}$

Standard deviation and 2SD range

• We treat this as a sample for SD calculation (n−1 in denominator)

• Deviations from the mean and squared deviations:

  • 2.3: $d = 2.3 - 2.27 = 0.03\quad d^2 = 0.0009$
  • 2.3: $d = 0.03\quad d^2 = 0.0009$
  • 2.4: $d = 2.4 - 2.27 = 0.13\quad d^2 = 0.0169$
  • 2.1: $d = -0.17\quad d^2 = 0.0289$
  • 1.9: $d = -0.37\quad d^2 = 0.1369$
  • 2.1: $d = -0.17\quad d^2 = 0.0289$
  • 2.3: $d = 0.03\quad d^2 = 0.0009$
  • 2.3: $d = 0.03\quad d^2 = 0.0009$
  • 2.4: $d = 0.13\quad d^2 = 0.0169$
  • 2.6: $d = 0.33\quad d^2 = 0.1089$

• Sum of squared deviations (SSE):
  $\sum (x_i - \mu)^2 = 0.3410$

• Variance (sample):
  $s^2 = \frac{1}{n-1} \sum (x_i - \mu)^2 = \frac{0.3410}{9} \approx 0.0379$

• Standard deviation (sample):
  $s = \sqrt{s^2} = \sqrt{0.0379} \approx 0.195\ \text{mg/dL}$

• 1SD range: $[\mu - s, \mu + s] = [2.27 - 0.195,\ 2.27 + 0.195] \approx [2.08,\ 2.47]\ \text{mg/dL}$

• 2SD range: $[\mu - 2s, \mu + 2s] = [2.27 - 2(0.195),\ 2.27 + 2(0.195)] \approx [1.88,\ 2.66]\ \text{mg/dL}$

• Summary: Standard deviation ≈ $0.195\ \text{mg/dL}$; 2SD range ≈ $[1.88,\ 2.66]\ \text{mg/dL}$

Are there any outliers?

• Using the 2SD rule (values outside μ ± 2s are potential outliers):
  • All data points lie within $[1.88, 2.66]$; none exceed ±2SD
• Therefore: NO outliers present in this data set

Levey-Jennings labeling of the 10 values

• Levey-Jennings chart basics:
  • Center line: mean $\mu = 2.27\ \text{mg/dL}$
  • SD lines: ±1SD at $\mu \pm s = [2.08, 2.47]$
  • SD lines: ±2SD at $\mu \pm 2s = [1.88, 2.66]$
• Data points relative to the mean (in SD units, z-scores with $s \approx 0.195$):
  • 2.3: $z \approx \frac{2.30 - 2.27}{0.195} \approx +0.15$ (between mean and +1SD)
  • 2.3: $z \approx +0.15$
  • 2.4: $z \approx \frac{2.40 - 2.27}{0.195} \approx +0.67$ (between +1SD and +2SD)
  • 2.1: $z \approx \frac{2.10 - 2.27}{0.195} \approx -0.87$ (between mean and -1SD)
  • 1.9: $z \approx \frac{1.90 - 2.27}{0.195} \approx -1.90$ (close to -2SD)
  • 2.1: $z \approx -0.87$
  • 2.3: $z \approx +0.15$
  • 2.3: $z \approx +0.15$
  • 2.4: $z \approx +0.67$
  • 2.6: $z \approx +1.69$ (between +1SD and +2SD)
• How to label the graph (textual guidance):
  • Mark the center line at $2.27\ \text{mg/dL}$
  • Draw horizontal bands at $2.08\leq x \leq 2.47$ (1SD) and $1.88\leq x \leq 2.66$ (2SD)
  • Plot each data point with its relative SD position (as above) and color-code by SD region (optional)
  • Attach axis labels: x-axis = CRP level (mg/dL), y-axis = events or sample order; include date if available
  • Indicate zone labels: +1SD, -1SD, +2SD, -2SD on the chart for quick reference

Calculation steps and foundational formulas

• Step 1 (Mean):
  $\mu = \frac{1}{n} \sum<em>{i=1}^{n} x</em>i$
• Step 2 (Deviations):
  $d<em>i = x</em>i - \mu$
• Step 3 (Squared deviations):
  $d_i^2$
• Step 4 (Sum of squared deviations):
  $\sum d_i^2$
• Step 5 (Variance):
  • Population variance: $\sigma^2 = \frac{1}{n} \sum (x_i - \mu)^2$
  • Sample variance: $s^2 = \frac{1}{n-1} \sum (x_i - \mu)^2$
• Step 6 (Standard deviation):
  • Population SD: $\sigma = \sqrt{\sigma^2}$
  • Sample SD: $s = \sqrt{s^2}$
• Steps to obtain 1SD and 2SD ranges:
  • 1SD range: $[\mu - s, \mu + s]$
  • 2SD range: $[\mu - 2s, \mu + 2s]$
• Practical note:
  • When analyzing QC data, common practice is to use the sample SD (n−1 in denominator) to estimate dispersion from the control measurements
  • Outlier or warning flags are typically considered relative to these SD bands (e.g., outside ±2SD or ±3SD depending on the rule used)

Practical interpretation and real-world relevance

• Descriptive statistics summarize a lab control dataset to assess precision and consistency
• A mean near 2.27 mg/dL with a small SD (~0.195) indicates tight clustering around the target value
• No outliers means the 10-day set is consistent and within expected analytical performance for this control
• Levey-Jennings labeling helps visually verify that all data fall within defined warning/alert zones; consistent data within ±2SD implies stable control
• Ethical/practical implications:
  • Accurate calculation of mean, SD, and SD ranges is essential for patient safety; miscalculations could misclassify a control run as acceptable or out-of-control
  • When decisions hinge on CRP levels (e.g., inflammation assessment), incorrect interpretation of data dispersion could lead to false reassurance or unnecessary alarm
  • Documentation of the calculation method (sample vs population SD) is crucial for reproducibility and regulatory compliance