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Sheppard’s Correction for Moments | ML

  • Last Updated : 23 Jan, 2020

Prerequisite: Raw and Central Moments

We assume in grouped data that the frequencies are concentrated in the middle part of the class interval. This assumption does not hold true in general and grouping error is introduced. Such an effect can be corrected in calculating the moments by using the information on the width of the class interval.

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Sheppard’s Correction for grouping error is nothing but the adjustment to calculated sample moments for the grouped data or continuous data. Prof. W.F. Sheppard proved that if the frequency distribution is continuous and the frequency tapers off to zero in both directions, the grouping error can be corrected as follows:



Let ‘c’ be the width of the class interval. Then,
Raw Moments
\mu {}'_1_(_c_o_r_r) = \mu {}'_1
\mu {}'_2_(_c_o_r_r) = \mu {}'_2 - c^{2}/12
\mu {}'_3_(_c_o_r_r) = \mu {}'_3 - \mu {}'_1 * c^{2}/4
\mu {}'_4_(_c_o_r_r) = \mu {}'_4 - \mu {}'_2 * c^{2}/2 + c^{4} * 7/240

Central Moments
\mu {}_2_(_c_o_r_r) = \mu {}_2 - c^{2}/12
\mu {}_3_(_c_o_r_r) = \mu {}_3
\mu {}_4_(_c_o_r_r) = \mu {}_4 - \mu {}_2 * c^{2}/2 + c^{4} * 7/240

What Kind of data can be corrected?

  1. This method of correction to moments is only possible for the continuous variables i.e. the continuous data.
  2. The width of the class interval should be equal.
  3. Frequencies should be symmetrical. Frequency should taper off to zero in both directions.

Consider the given distribution of marks.

MarksNumber of Students
0 – 101
10 – 206
20 – 3011
30 – 4017
40 – 5021
50 – 6016
60 – 7013
70 – 807
80 – 905
90 – 1002

For the distribution of marks above, the value for moments are given below:

Raw Moments –
 \mu {}'_r = \frac{1}{n}\sum_{i=1}^{n}f_i x_i^{r} , is the rth raw moment, where f_i is the frequency count and x_i is the mid value of class.

So, using the above formula for the Raw Moment we get following values for moments.
\mu {}'_1 = 48.23
\mu {}'_2 = 2711.87
\mu {}'_3 = 169751.26
\mu {}'_4 = 11515978.53

Sheppard’s correction for Raw Moments –

\mu {}'_1_(_c_o_r_r_) = \mu {}'_1 = 48.23
\mu {}'_2_(_c_o_r_r_) = 2703.53
\mu {}'_3_(_c_o_r_r_) = 168545.51
\mu {}'_4_(_c_o_r_r_) = 11380676.70

Similarly central moments can be corrected using Sheppard’s Correction.




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