Biostatistics: Measures of Central Tendency and Variance in Medical Laboratory Settings

Module 5

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Objectives

• Define:– Mode– Mean– Median– Confidence limits– Gaussian curve– Standard Deviation– Coefficient of Variation

Upon completion of this lesson you will be able to:

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• Prepare a frequency distribution table

• Calculate mean, standard deviation and %CV

• Identify median and mode

• Discuss how the values of mean, median and mode influence the validity of statistical data

• Evaluate data to determine if normal vs. non-normal distribution curve

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Objectives

Upon completion of this lesson you will be able to:

Statistics Used with Laboratory Data

• Measures of central tendency– Mean– Median– Mode

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• Variability of measurements

• Measures of Variation: – Standard Deviation, (s)– Coefficient of Variation (CV)

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Statistics Used with Laboratory Data

• Ideal: repeat analysis of a sample would produce the same value each time

• Real world: there will always be a certain amount of variability in repeated measurements

• Variability in measurements caused by • Heterogeneity of the sample over time• Variation in the technique of the analyst• Heterogeneity of reagents over time• Instrument variation

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Variability of Measurements

• Visualized with a bar graph

• Frequency Distribution table of repeated measurements– Reflects how easy it was to repeat the measurement

and obtain the same value

• Want distribution plot to display central tendency and a bell shaped curve– Bell shaped curve also called Gaussian Curve– Represents normal distribution pattern of values

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Variability of Measurements

Frequency Distribution of Values

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Comparison

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Statistics Used to Measure Central Tendency

• Mean

• Median

• Mode

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• Mean = average of all data points

• Mean = sum of data points = (xi) number of results N

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Statistics Used to Measure Central Tendency

• Median = the middle data point observed once the data are arranged in descending or ascending order

• Mode = the value that occurs with the greatest frequency

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Statistics Used to Measure Central Tendency

Normal Gaussian Distribution

• Symmetric about the mean

• Obtained when the Mean = Median = Mode– The Frequency distribution graph makes a bell-

shaped curve.

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Example

• Determine the mean, median and mode for the following values:– 6,11,8,5,6,7,9,10,11,8,6

• We will use that information to determine if this data is normally distributed.

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6,11,8,5,6,7,9,10,11,8,6• Mean =

• Median =

• Mode =

• Normal or skewed distribution?

• Let’s take time to perform this.

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Example

6,11,8,5,6,7,9,10,11,8,6• Mean = 87/11 = 8

• Median = 8

• Mode = 6

• Normal (nearly normal) distribution

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Measures of Variation

• Desirable to have repeated measure data show a slim distribution about the mean, reflecting low variability and low random error

• Standard Deviation (SD or s)

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Standard Deviation

• Standard Deviation (SD or s)– Measurement statistic that describes the

average distance each data point in a normal distribution is from the mean

– Expressed with same units as the measured analyte

– SD or s = √∑(xi-mean)2

N – 1

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• A large standard deviation: – Large variation in data– Wide bell shaped curve of frequency

distribution

• A small standard deviation:– Small variation in data– Narrow bell-shaped curve of frequency

distribution

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Standard Deviation

Variance

• Standard deviation squared is variance.

• What is the variance of the following?– Standard Deviation = 4– Variance = ? 16

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• One SD unit approx 34% total distance of the x-axis on a normal distribution curve

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Standard Deviation

• CV is the standard deviation (SD or s) expressed as a percentage of the mean

• CV = s X 100 mean

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Coefficient of Variation (CV)

• Used to evaluate precision or reproducibility of repeated measures

• Allows comparison without influence from magnitude of data base

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Coefficient of Variation (CV)

• A low CV value indicates the distribution of values about the mean is tight rather than broad

• Acceptable: CV <5% – Modern instrumentation CV <3%– Manual methods CV ~8-10%– Other methods CV >10%

• Can be used to monitor personnel pipetting technique

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Coefficient of Variation (CV)

• Which of the following two methods is more precise (reproducible) showing the least amount of variability and thus the least amount of random error? First we must calculate both CVs.

Glucose Method A Glucose Method BMean = 500 mg/dl Mean = 100 mg/dlSD = 20 mg/dl SD = 6 mg/dlCV= CV =4 6

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Example

• Given the CV we calculated which of the following two methods is more precise (reproducible) showing the least amount of variability and thus the least amount of random error?

Glucose Method A Glucose Method BMean = 500 mg/dl Mean = 100 mg/dlSD = 20 mg/dl SD = 6 mg/dlCV = 4 CV = 6

• Method A is more precise

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Example

Confidence Intervals

• Confidence Intervals also referred to as– Acceptable range– Established limits– Confidence limits

• Defined as the limits between which we expect a specified proportion or percentage of a population of values to lie

• Most of the data in a normal distribution lies close to the mean

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• Confidence limits are the standard deviations expressed as percentages – 68%– 95.5%– 99.7%

• Indicate the percentage of values falling within that area of the curve

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Confidence Intervals

Mean - 1 SD = 34.1%Mean +1 SD = 34.1%

Mean + 1 SD = 68.2% of data

This is the sum of % from above

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Confidence Limits

Mean - 2 SD = 34.1 + 13.65 = 47.75 %

Mean +2 SD = 34.1 + 13.65 = 47.75 % Mean + 2 SD = 95.5 % of data

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Confidence Limits

Mean - 3 SD = 47.75 + 2.1 = 49.85 %

Mean +3 SD = 47.75 + 2.1 = 49.85 % Mean + 3 SD = 99.7 % of data

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Confidence Limits

Summary

• Distribution of Values

• Mean

• Median and Mode

• Standard Deviation

• Coefficient of Variation

• Confidence intervals

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