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By Professor Omar Hasan Kasule Sr.
Learning Objectives:
· Definition, properties, advantages and disadvantages of common measures of variation: variance, standard deviation, and z-score
· Definition and use of quartiles and percentiles
· Relation among percentile, standard deviation, and area under a normal curve
Key Words and Terms:
· Range
· Standard deviation
· Percentile
· Variation, biological variation
· Variation, inter-subject variation
· Variation, intra-subject variation
· Variation, measurement variation
· Variation, observer variation
· Variation, seasonal variation
· Variation, temporal variation
Unit Outline
INTRODUCTION
A. Variation: Concept and Classification
B. Biological Variation
C. Measurement Error
D. Temporal Variation
E. Measures of Variation
MEASURES OF VARIATION BASED ON THE MEAN
A. Mean Deviation
B. Variance
C. Standard Deviation
D. Normalized Standardized Score (Z-Score)
E. Others
MEASURES OF VARIATION BASED ON QUANTILES
A. Quantiles: General Aspects
B. Quartiles
C. Deciles
D. Percentiles
E. Box and Whisker Plot:
OTHER MEASURES OF VARIATION
A. Range
B. Numerical Rank
C. Percentile Rank
D. Inter-Quartile Range
E. Percentile Range
UNIT SYNOPSIS
INTRODUCTION
Variations are biological, measurement, or temporal. Time series analysis relates biological to temporal variation. Analysis of variance (ANOVA) relates biological variation (inter- or between subject) to measurement variation (intra- or within subject) variation. Biological variation is more common than measurement variation. Temporal variation is measured in calendar time or in chronological time. Measures of variation can be classified as absolute (range, inter-quartile range, mean deviation, variance, standard deviation, and quantiles) or relative (coefficient of variation and standardized z-score). Some measures are based on the mean (mean deviation, the variance, the standard deviation, z score, the t score, the stanine, and the coefficient of variation) whereas others are based on quantiles (quartiles, deciles, and percentiles).
MEASURES OF VARIATION BASED ON THE MEAN
Mean deviation is the arithmetic mean of absolute differences of each observation from the mean. It is simple to compute but is rarely used because it is not intuitive and allows no further mathematical manipulation. The variance is the sum of the squared deviations of each observation from the mean divided by the sample size, n, (for large samples) or n-1 (for small samples). It can be manipulated mathematically but is not intuitive due to use of square units. The standard deviation, the commonest measure of variation, is the square root of the variance. It is intuitive and is in linear and not in square units. The standard deviation,s, is from a population but the standard error of the mean, s, is from a sample with s being more precise and smaller than s. The relation between the standard deviation, s, and the standard error, s, is given by the expression s = s /(n-1) where n = sample size.
The percentage of observations covered by mean +/- 1 SD is 66.6%, mean +/- 2 SD is 95%, and mean +/- 4 SD virtually 100%. The standard deviation has the following advantages: it is resistant to sampling variation, it can be manipulated mathematically, and together with the mean it fully describes a normal curve. Its disadvantage is that it is affected by extreme values. The standardized z-score defines the distance of a value of an observation from the mean in SD units. The coefficient of variation (CV) is the ratio of the standard deviation to the arithmetic mean usually expressed as a percentage. CV is used to compare variations among samples with different units of measurement and from different populations.
MEASURES OF VARIATION BASED ON QUANTILES
Quantiles (quartiles, deciles, and percentiles) are measures of variation based on division of a set of observations (arranged in order by size) into equal intervals and stating the value of observation at the end of the given interval. Quantiles have an intuitive appeal. Quartiles are based on dividing observations into 4 equal intervals. Deciles are based 10, quartiles on 4, and percentiles on 100 intervals. The inter-quartile range, Q3 - Q1, and the semi interquartile range, ½ (Q3 - Q1), have the advantages of being simple, intuitive, related to the median, and less sensitive to extreme values. Quartiles have the disadvantages of being unstable for small samples and not allowing further mathematical manipulation. Deciles are rarely used. Percentiles, also called centile scores, are a form of cumulative frequency and can be read off a cumulative frequency curve. They are direct and very intelligible. The 2.5th percentile corresponds to mean - 2SD. The 16th percentile corresponds to mean - 1SD. The 50th percentile corresponds to mean + 0 SD. The 84th percentile corresponds to mean + 1SD. The 97.5th percentile corresponds to mean + 2SD. The percentile rank indicates the percentage of the observations exceeded by the observation of interest. The percentile range gives the difference between the values of any two centiles.
THE RANGEOTHER MEASURES OF VARIATION:
The full range is based on extreme values. It is defined by giving the minimum and maximum values or by giving the difference between the maximum and the minimum values. The modified range is determined after eliminating the top 10% and bottom 10% of observations. The range has several advantages: it is a simple measure, intuitive, easy to compute, and useful for preliminary or rough work. Its disadvantages are: it is affected by extreme values, it is sensitive to sampling fluctuations, and it has no further mathematical manipulation. The numerical rank expresses the observation's position in counting when the observations are arranged in order of magnitude from the best to the worst. The percentile rank indicates the percentage of the observations exceeded by the observation of interest.