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# Robust statistics

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Understanding Robust statistics, Courtesy FICCI

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### Robust statistics

1. 1. INTRODUCTION TO ROBUST FICCI QUALITY FORUM STATISTICS 1 Training on assuring quality of test results
2. 2. When there are outliers in a group of data, both the average and the standard deviation get affected. In past there have been methods developed to discard such outliers in order to obtain reliable average and standard deviation. FICCI QUALITY FORUM 2 Training on assuring quality of test results
3. 3. In robust statistics, we estimate ‘Median’ instead of ‘Mean’. Median is not affected by outliers. Similarly, instead of ‘Standard Deviation’, we estimate ‘Inter-quartile Range’ and then normalize it to make it ‘Normalized Inter-quartile Range’ FICCI QUALITY FORUM 3 Training on assuring quality of test results
4. 4. NIQR is deviation. equivalent to standard Like ‘Median’, NIQR is not affected by outliers. Thus, the quality parameters estimated using robust statistics are more reliable when there are outliers in the group of data FICCI QUALITY FORUM 4 Training on assuring quality of test results
5. 5. MEAN ≈ STANDARD DEVIATION MEDIAN ≈ NIQR Estimation of NIQR starts with the understanding of Quartiles After identifying quartiles, we find out the Interquartile Range (IQR), followed by estimation of NIQR FICCI QUALITY FORUM 5 Training on assuring quality of test results
6. 6. QUARTILE Quartile divides the data into 4 equal parts. Q1, Q2, Q3 & Q4.. Q1 = First or Lower quartile Q3 = Third or Higher quartile INTERQUARTILE RANGE : IQR = Q3 - Q1 UNLIKE RANGE, IQR IS UNAFFECTED BY EXTREME VALUES FICCI QUALITY FORUM 6 Training on assuring quality of test results
7. 7. To compute Quartiles - (Example – 8) Data 100.2, 100.5, 100.6, 100.4, 100.3, 100.2, 100.5, 100.6, 100.5, 100.3 Arranged data 100.2, 100.2, 100.3, 100.3, 100.4, 100.5, 100.5, 100.5, 100.6, 100.6 Q2 = Median of data = (100.4 + 100.5)/2 = 100.45 Q1 = Median of data below Q2 = 100.3 Q3 = Median of data above Q2 = 100.5 IQR = Q3 - Q1 = 100.5 - 100.3 = 0.2 FICCI QUALITY FORUM 7 Training on assuring quality of test results
8. 8. To compute Quartiles (Example – 9) Data 25, 23, 24, 34, 28, 22, 31, 35, 32, 30, 33 Arranged data 22, 23, 24, 25, 28, 30, 31, 32, 33, 34, 35 Q2 = Median of data = 30 Q1 = Median of data below Q2 = 24 Q3 = Median of data above Q2 = 33 IQR = Q3 - Q1 = 9 FICCI QUALITY FORUM 8 Training on assuring quality of test results
9. 9. DETERMINING IQR :     Arrange the data in increasing order Find Q2 = Median of the given data Find Q1 = Median of the observations below the location of the Median of all observations (1st Quartile) Find Q3 = Median of the observations above the Location of the Median of all observations (3rd Quartile) Inter Quartile Range Min Value FICCI QUALITY FORUM Q1 Q2 Q3 9 Training on assuring quality of test results Max Value
10. 10. NORMALIZED INTER - QUARTILE RANGE NIQR = 0.7413* IQR Where IQR = Inter - Quartile Range In the Example - 8, NIQR = 0.7413*0.2 = 0.1483 In the Example - 9, NIQR = 6.6726 FICCI QUALITY FORUM 10 Training on assuring quality of test results
11. 11. REASONING BEHIND THE FACTOR 0.7413 The factor comes from the “standard” normal distribution, which has a mean of zero and a standard deviation (SD) equal to one. The interquartile range of such a distribution is [–0.6745, +0.6745] and this is narrower than the familiar ±1 SD interval. FICCI QUALITY FORUM 11 Training on assuring quality of test results
12. 12. So, to convert an IQR into a ±1 SD range, it must be scaled up by the ratio of the interval widths, namely 2/1.3490. To then convert this ±1 SD range (whose width is 2 standard deviations) into an amount equivalent to 1 SD, this range is then halved. Hence the IQR is divided by 1.3490 (or equivalently multiplied by 0.7413) to convert it into an estimate of the standard deviation. FICCI QUALITY FORUM 12 Training on assuring quality of test results
13. 13. Sl. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 FICCI QUALITY FORUM Data 16.37 16.36 16.40 16.34 16.36 16.40 16.37 16.42 16.38 16.39 16.40 16.37 16.41 16.38 Sorted Data 16.34 16.36 16.36 16.37 16.37 16.37 16.38 16.38 16.39 16.40 16.40 16.40 16.41 16.42 Example - A Q1 = Q2 = Q3 = IQR = NIQR = Median = 16.380 Average = NIQR = Std. Dev. = 13 16.37 16.38 16.40 0.030 0.022 16.382 0.0222 0.0222 Training on assuring quality of test results
14. 14. Sl. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 FICCI QUALITY FORUM Data 16.37 16.36 16.40 16.34 16.36 16.40 16.37 16.42 16.38 16.39 16.40 16.37 20.65 16.38 Sorted Data 16.34 16.36 16.36 16.37 16.37 16.37 16.38 16.38 16.39 16.40 16.40 16.40 16.42 20.65 Example - B Q1 = Q2 = Q3 = IQR = NIQR = Median = 16.380 Average = NIQR = Std. Dev. = 14 16.37 16.38 16.40 0.030 0.022 16.685 0.0222 1.1414 Training on assuring quality of test results