What Kind of Data Can Be Corrected?
Sheppard’s correction has some severe restrictions, which make it applicable to only a small range of distributions. The correction should only be made to data with the following characteristics:
- Frequencies should taper to zero in both the positive and negative direction. In other words, frequencies should be symmetrical and gradually taper off (like the behavior you would see in a normal distribution).
- Variables should be continuous. The method is not suited to discrete variables.
- Class intervals should be equal in width.
- Class intervals should be more than 1/20th of the total range.
In addition, Sharma’s Textbook of Sampling and Attributes note that the adjustment should not be made to:
- J-shaped distributions.
- Highly Skewed distributions.
- U-shaped distributions.
- Samples under 1,000 items (with smaller samples, sampling error is going to be a greater issue than grouping errors).
Sheppard’s Correction Formulas
- First moment (mean)—no correction needed.
- Second moment (variance): calculated estimate of the variance (m2 – (i2/12) where i is the class interval width.
- Third moment (skewness) — no correction needed.
- Fourth moment (kurtosis): calculated estimate for kurtosis (m4) – ½i2m2 + (7/240)i4
- m1 = 0
- m2 = 2.64
- m3 = 0.08
- m4 = 28.30
Use Sheppard’s correction for these moments. Note that the class interval for the distribution is 2.
- m1 = 0; no correction needed.
- m2 = m2 – (i2/12) = 2.64 – 22/12 = .33 = 2.3.
- m3 = 0.08; no correction needed
- m4 = (m4) – ½i2m2 + (7/240)i4 = 28.30 – 5.28 + 0.12 = 23.14.
A.K. Sharma. Textbook of Sampling and Attributes.
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