1. **Stating the problem:** We have ALT parameter data with a kit insert mean of 42 and SD of 2.5. We have 30 daily results and need to analyze them using Westgard rules, then calculate new mean, SD, CV, and bias. 2. **Westgard rules formulas and explanation:** - 1-2S: One control measurement exceeds mean \pm 2SD. - 2-2S: Two consecutive control measurements exceed mean \pm 2SD on the same side. - 1-3S: One control measurement exceeds mean \pm 3SD. - 10X: Ten consecutive control measurements fall on one side of the mean. - R4-1S: Range between two control measurements exceeds 4SD. Systematic errors cause consistent bias; random errors cause unpredictable variation. 3. **Calculate control limits:** Mean = 42, SD = 2.5 - 1-2S limits: 42 \pm 2\times2.5 = 42 \pm 5 = [37, 47] - 1-3S limits: 42 \pm 3\times2.5 = 42 \pm 7.5 = [34.5, 49.5] 4. **Check each data point against 1-2S and 1-3S:** Data points below 37 or above 47 violate 1-2S. Data points below 34.5 or above 49.5 violate 1-3S. 5. **Identify violations:** - 1-2S violations: Values below 37 are on days 2 (37 borderline), 5 (35), 8 (36), 10 (36), 20 (37 borderline), 21 (37 borderline), 29 (37 borderline). - 1-3S violations: None (lowest is 35). 6. **Check 2-2S rule:** Two consecutive values beyond 2SD on same side. Days 5 (35) and 8 (36) are not consecutive, so no 2-2S violation. 7. **Check 10X rule:** Check if 10 consecutive points are all above or below mean 42. From day 1 to 10, values fluctuate around mean, no 10 consecutive on one side. 8. **Check R4-1S rule:** Range between two measurements exceeds 4SD = 4 \times 2.5 = 10. Max difference between any two points is 43 - 35 = 8, less than 10, so no R4-1S violation. 9. **Classify errors:** - 1-2S violations are random errors. - No systematic errors detected. 10. **Phase two: Calculate new mean, SD, CV from 30 data points:** Sum of data = 1140 Mean = 1140 / 30 = 38 Calculate SD: $$\text{SD} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$$ Calculate each squared deviation and sum: (40-38)^2=4, (37-38)^2=1, (39-38)^2=1, (41-38)^2=9, (35-38)^2=9, (38-38)^2=0, (38-38)^2=0, (36-38)^2=4, (37-38)^2=1, (36-38)^2=4, (38-38)^2=0, (39-38)^2=1, (39-38)^2=1, (39-38)^2=1, (39-38)^2=1, (38-38)^2=0, (39-38)^2=1, (39-38)^2=1, (40-38)^2=4, (37-38)^2=1, (37-38)^2=1, (41-38)^2=9, (39-38)^2=1, (40-38)^2=4, (39-38)^2=1, (38-38)^2=0, (43-38)^2=25, (38-38)^2=0, (37-38)^2=1, (38-38)^2=0 Sum = 91 SD = sqrt(91 / 29) = sqrt(3.138) = 1.77 CV = (SD / Mean) \times 100 = (1.77 / 38) \times 100 = 4.66% 11. **Phase three: Calculate bias (d%)** Bias = \frac{|New Mean - Kit Mean|}{Kit Mean} \times 100 Bias = \frac{|38 - 42|}{42} \times 100 = \frac{4}{42} \times 100 = 9.52%