Quality Assurance in the Clinical Laboratory: Basic Statistical Concepts

Chapter 12: Quality Assurance in the Clinical Laboratory: Basic Statistical Concepts

Objectives of the Chapter

• At the end of this chapter, the reader should be able to perform the following tasks:
  • Define key statistical terms, which include:
  • Mean: Average of a group of numbers.
  • Median: Central number in an ordered group of numbers.
  • Mode: Most frequently occurring number in a group.
  • Gaussian Distribution: A symmetrical, bell-shaped distribution of data.
  • Variance: A measure of how far a set of numbers is spread out.
  • Standard Deviation (SD): A measure of precision in a data set, derived from variance.
  • Coefficient of Variation (CV): A normalized measure of dispersion of a probability distribution.
  • Calculate the mean of a group of numbers.
  • Calculate the median of a group of numbers.
  • Calculate the mode of a group of numbers.
  • Calculate the variance and standard deviation for a group of numbers.
  • Calculate the coefficient of variation for a group of numbers.
  • Compare multiple groups of numbers to determine the group with the highest precision.
  • Define confidence limits and calculate them for a given data set.

Mean

• Defined as the average of a group of numbers.
  • Symbol: ☑
  • It serves as an indicator of "central tendency," representing how data is distributed around a central value.

Example of Calculating Mean

1. A student obtains five replicate absorbance values: 0.425, 0.430, 0.435, 0.432, and 0.428.
2. First, sum the five absorbance values:
   $0.425 + 0.430 + 0.435 + 0.432 + 0.428 = 2.15$
3. Then, divide the sum by the total number of measurements (n = 5):
   ext{Mean value} = rac{2.15}{5} = 0.430

Median

• The median is an indicator of central tendency that represents the middle number in a sequentially ordered set of values.
  • It is defined such that there is an equal quantity of numbers greater than and less than the median value.
  • The median may or may not coincide with the mean.

Example of Calculating Median

1. For the absorbance values: 0.425, 0.430, 0.432, 0.435, 0.428, they are ranked in increasing order: 0.425, 0.428, 0.430, 0.432, 0.435.
2. The total number of values is odd (5):
   • To find the median, add 1 to the total number of values (5):
     $5 + 1 = 6$
   • Divide by 2:
     $rac{6}{2} = 3$
3. The third value on the ordered list (0.430) is the median.

Mode

• The mode is defined as the number that occurs most frequently in a group of data.

Example of Calculating Mode

• Given the following glucose values in mg/dL: 75, 74, 72, 70, 76, 73, 72, 71.

1. Arrange the data from lowest to highest: 70, 71, 72, 72, 73, 74, 75, 76.
2. Frequency of numbers:
   • 70: 1
   • 71: 1
   • 72: 2 (Most frequent)
   • 73: 1
   • 74: 1
   • 75: 1
   • 76: 1
3. The mode for this group is 72 mg/dL.

Gaussian Distribution

• When the mean, median, and mode of a dataset are identical, the dataset is said to have a Gaussian distribution (normal distribution).
  • Characteristics include a bell-shaped curve where there are equal numbers of results above and below the peak.

Accuracy vs. Precision

• Accuracy: Reflects how closely the measured result conforms to the true value (target).
• Precision: Refers to the consistency of repeated measurements, where the same result is achieved upon repetition.
  • It is desired to achieve results that are both accurate and precise in laboratory settings.

Variance

• Variance is an indicator of the precision of a group of numbers.
  • Symbol: $s^2$
  • A large variance indicates more spread in the data while a small variance denotes that the numbers are closely clustered.

Formula for Variance

• Variance formula is given by: s^2 = rac{ ext{∑}(xd - ar{x})^2}{n}
  • Where:
  • $s^2$ = variance
  • $ext{∑}$ = sum of the numbers within parentheses
  • $xd$ = individual data point
  • ar{x} = mean of the group
  • $n$ = total quantity of numbers

Example of Variance Calculation

1. Given glucose values: 94, 93, 92, 100, 101, 93, 105, 98, 99, 87 (mg/dL).
2. First, calculate the sum:
   $87 + 92 + 93 + 93 + 94 + 98 + 99 + 100 + 101 + 105 = 962$
3. Find the mean (n = 10):
   ext{Mean} = rac{962}{10} = 96.2
4. Subtract the mean from each value and square the result:
   • For 87: $87 - 96 = -9; (-9)^2 = 81$
   • For 92: $92 - 96 = -4; (-4)^2 = 16$
   • For 93: $93 - 96 = -3; (-3)^2 = 9$
   • Repeat for all values…
5. Sum of squared differences = 254
6. Substitute into the variance formula for the final result:
   s^2 = rac{254}{10} = 25.4

Standard Deviation

• Standard deviation (SD) is the most frequently used measure of precision, represented by
  • Symbol: $s$
  • It is calculated as the square root of the variance.

Example of Standard Deviation Calculation

• For the glucose values used previously:

1. Given variance: $s^2 = 28.2$
2. Then calculate standard deviation:
   $s = ext{sqrt}(28.2) = 5.3 ext{ mg/dL}$

Probabilities Associated with Standard Deviation

• The following probabilities are statistically valid concerning SD:
  • 68.2%: Results fall within ±1 SD of the mean.
  • 95.5%: Results fall within ±2 SD of the mean.
  • 99.7%: Results fall within ±3 SD of the mean.

Example Probability Scenario

• If the mean glucose control material value is 100 mg/dL and SD is 5 mg/dL:
  • 68.2% of analyses yield results between:
    100 - 5 < ext{Result} < 100 + 5 ext{ which are } 95 ext{ to } 105 ext{ mg/dL}

Establishing Standard Deviation Ranges and Confidence Intervals

• Confidence intervals are determined by establishing ranges using standard deviations from the mean:
  • 1 SD range: ext{Mean} - SD < ext{value} < ext{Mean} + SD
• For example:
  • Mean Sodium value = 140 mEq/L; SD = 3 mEq/L.
  • The 1 SD range would be:
    $140 - 3 ext{ to } 140 + 3 <br /> ightarrow 137 ext{ to } 143 mEq/L$
  • The ranges for 2 SD and 3 SD can be calculated similarly.

Coefficient of Variation

• The Coefficient of Variation (CV) is useful for comparing two or more groups to identify the one with the highest precision.
  • The formula for CV is:
    CV = rac{s}{ar{x}} imes 100 ext{%}
  • Where:
  • $s$ = standard deviation
  • ar{x} = mean

Example Calculation of Coefficient of Variation

• For the previously analyzed glucose values:

1. Mean = 96 mg/dL
2. SD = 5.3 mg/dL
3. Calculate CV:
   CV = rac{5.3}{96} imes 100 = 5.5 ext{%}