Charlier’s Checks for Moments - clearly explained
Charlier’s Checks for Moments helps to verify correct computations of moments in a table of grouped classes to guard against computational mistakes.
𝐹𝑜𝑟 𝑠𝑖𝑚𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑢_𝑖=(𝑥_𝑖−𝑥_0)/ℎ,
𝑥_0=𝐶ℎ𝑜𝑠𝑒𝑛 𝑜𝑟𝑖𝑔𝑖𝑛 𝑝𝑟𝑒𝑓𝑒𝑟𝑎𝑏𝑙𝑦 𝑎𝑟𝑜𝑢𝑛𝑑 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑠𝑢𝑐ℎ 𝑎𝑠 𝑚𝑒𝑎𝑛
ℎ=𝑆𝑢𝑖𝑡𝑎𝑏𝑙𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑝𝑟𝑒𝑓𝑒𝑟𝑎𝑏𝑙𝑦 𝑐𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡ℎ
Charlier’s Checks for Moments uses the technique of comparing the left side of the binomial with the right side of that binomial's expansion form.
If both the sides show the same value, then the means of the scaled deviations are correct, hence computation can be contnued to obtain the moments.
Grouping error: The error occurred due to the assumption that the frequencies are uniformaly distributed with the class intervals.
To have a lesser error, the distribution within a class should be:
1. Symmetrical
2. About the mid-point of the class interval
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Conditions for applying Sheppard’s Corrections for Moments:
1. Observations should relate to continuous variable.
2. Frequency curve of the distributions should be continuous.
3. The class frequencies should decrease towards the two ends of the range smoothly and gradually.
4. The class intervals should not be too small i.e., class count should not be large. Generally, minimum total frequency is 1000 and maximum class count is 20.
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Three types of moments are:
1. Moments about arbitrary point 2. Moments about origin 3. Moments about mean
A Moment about Origin is a special case of a Moment about an arbitrary point where the arbitrary point is set to origin 0.
Moments about Mean (also called Central moments)
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Why Moments are important?
a. Describe the characteristic of a distribution
b. Represent a convenient and unifying method for summarizing several statistical measures such as measures of tendency, variation, skewness and kurtosis.
00:00 Example data
00:36 Class mark or mid-point
03:12 Change of origin and scale
04:26 Raw moment
05:41 Why multiply by h?
05:58 Effect of change of origin and scale
10:00 Technique of Charlier
13:05 Binomial expansion
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