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Charlier's Check


A check which can be used to verify correct computations in a table of grouped classes. For example, consider the following table with specified class limits and frequencies f. The class marks x_i are then computed as well as the rescaled frequencies u_i, which are given by

 u_i=(f_i-x_0)/c,
(1)

where the class mark is taken as x_0=74.5 and the class interval is c=10. The remaining quantities are then computed as follows.

class limitsx_if_iu_if_iu_if_iu_i^2f_i(u_i+1)^2
30-3934.52-4-83218
40-4944.53-3-92712
50-5954.511-2-224411
60-6964.520-1-20200
70-7974.53200032
80-8984.52512525100
90-9994.572142863
total100-20176236

In order to compute the variance, note that

s_u^2=(sum_(i)f_iu_i^2)/(sum_(i)f_i)-((sum_(i)f_iu_i)/(sum_(i)f_i))^2
(2)
=(176)/(100)-((-20)/(100))^2
(3)
=1.72,
(4)

so the variance of the original data is

 s_x^2=c^2s_u^2=172.
(5)

Charlier's check makes use of the additional column f_i(u_i+1)^2 added to the right side of the table. By noting that the identity

sum_(i)f_i(u_i+1)^2=sum_(i)f_i(u_i^2+2u_i+1)
(6)
=sum_(i)f_iu_i^2+2sum_(i)f_iu_i+sum_(i)f_i,
(7)

connects columns five through seven, it can be checked that the computations have been done correctly. In the example above,

 236=176+2(-20)+100,
(8)

so the computations pass Charlier's check.


See also

Variance

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References

Kenney, J. F. and Keeping, E. S. "Charlier Check." §6.8 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 47-48, 81, 94-95, and 104, 1962.

Referenced on Wolfram|Alpha

Charlier's Check

Cite this as:

Weisstein, Eric W. "Charlier's Check." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CharliersCheck.html

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