Measures of Variation by Dr. Bornwell Sikateyo - Department of Medical Education Development - Biostatistics-unza

Measures of variation Dr. Bornwell Sikateyo Department of Medical Education Development Summary Measures Arithmetic Mean Median Mode Describing Data Numerically Variance Standard Deviation Coefficient of Variation Range Interquartile Range Geometric Mean Skewness Central Tendency Variation Shape Quartiles Quartiles • Quartiles split the ranked data into 4 segments with an equal number of values per segment 25% 25% 25% 25% The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger Q2 is the same as the median (50% are smaller, 50% are larger) Only 25% of the observations are greater than the third quartile Q1 Q2 Q3 Quartile Formulas Find a quartile by determining the value in the appropriate position in the ranked data, where First quartile position: Q1 = (n+1)/4 Second quartile position: Q2 = (n+1)/2 (the median position) Third quartile position: Q3 = 3(n+1)/4 where n is the number of observed values (n = 9) Q1 is in the (9+1)/4 = 2.5 position of the ranked data so use the value half way between the 2nd and 3rd values, so Q1 = 12.5 Quartiles Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Example: Find the first quartile Q1 and Q3 are measures of non-central location Q2 = median, a measure of central tendency (n = 9) Q1 is in the (9+1)/4 = 2.5 position of the ranked data, so Q1 = 12.5 Q2 is in the (9+1)/2 = 5th position of the ranked data, so Q2 = median = 16 Q3 is in the 3(9+1)/4 = 7.5 position of the ranked data, so Q3 = 19.5 Quartiles Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Example: (continued) Same center, different variation Measures of Variation Variation Variance Standard Deviation Range Interquartile Range Measures of variation give information on the spread or variability of the data values. Range • Simplest measure of variation • Difference between the largest and the smallest values in a set of data: Range = Xlargest – Xsmallest 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13 Example: • Ignores the way in which data are distributed • Sensitive to outliers 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5 Disadvantages of the Range 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120 Range = 5 - 1 = 4 Range = 120 - 1 = 119 Interquartile Range • Can eliminate some outlier problems by using the interquartile range • Eliminate some high- and low-valued observations and calculate the range from the remaining values • Interquartile range = 3rd quartile – 1st quartile = Q3 – Q1 Interquartile Range Median (Q2) X maximum X minimum Q1 Q3 Example: 25% 25% 25% 25% 12 30 45 57 70 Interquartile range = 57 – 30 = 27 • Quartiles: The values which divide a series of observations, arranged in ascending order, into 4 equal parts. (Thus the 2nd Quartile is the Median). • The Interquartile Range represents the central portion of the distribution and is calculated as the difference between the third quartile and the first quartile. • This range includes about one-half of the observations in the set, leaving one quarter of the observations on each side.