Question

The amount of apples (in kg) produced per day by 10 orchards are given below:
218.2 179.5 207.3 224.3 213.7
199.7 184.7 194.4 203.5 185.4
Find the first four moments about the mean and the coefficient of kurtosis for the
above data.

Answer 1

Answer on Question #64972 – Math – Statistics and Probability

Question

The amount of apples (in kg) produced per day by 10 orchards are given below:

218.2 179.5 207.3 224.3 213.7

199.7 184.7 194.4 203.5 185.4

Find the first four moments about the mean and the coefficient of kurtosis for the above data.

Solution

The mean [1, p. 31] is

$$\bar{x} = \frac{\sum x_i}{n}$$

The mean is

$$\frac{218.2 + 179.5 + 207.3 + 224.3 + 213.7 + 199.7 + 184.7 + 194.4 + 203.5 + 185.4}{10} = 201.07$$

The k-th moment about the mean [2] is

$$\mu_k = \frac{\sum (x_i - \bar{x})^k}{n}$$

$1^{\text{st}}$ moment about the mean is

$$\begin{array}{l} \mu_1 = \frac{1}{10} \left[ (218.2 - 201.07) + (179.5 - 201.07) + (207.3 - 201.07) + (224.3 - 201.07) \right. \\ \left. + (213.7 - 201.07) + (199.7 - 201.07) + (184.7 - 201.07) \right. \\ \left. + (194.4 - 201.07) + (203.5 - 201.07) + (185.4 - 201.07) \right] = 0 \end{array}$$

$2^{\text{nd}}$ moment about the mean is

$$\begin{array}{l} \mu_2 = \frac{1}{10} \left[ (218.2 - 201.07)^2 + (179.5 - 201.07)^2 + (207.3 - 201.07)^2 + (224.3 - 201.07)^2 \right. \\ \left. + (213.7 - 201.07)^2 + (199.7 - 201.07)^2 + (184.7 - 201.07)^2 \right. \\ \left. + (194.4 - 201.07)^2 + (203.5 - 201.07)^2 + (185.4 - 201.07)^2 \right] = 206.2461 \end{array}$$

$3^{\text{rd}}$ moment about the mean is

$$\begin{array}{l} \mu_3 = \frac{1}{10} \left[ (218.2 - 201.07)^3 + (179.5 - 201.07)^3 + (207.3 - 201.07)^3 + (224.3 - 201.07)^3 \right. \\ \left. + (213.7 - 201.07)^3 + (199.7 - 201.07)^3 + (184.7 - 201.07)^3 \right. \\ \left. + (194.4 - 201.07)^3 + (203.5 - 201.07)^3 + (185.4 - 201.07)^3 \right] = 126.3489 \end{array}$$

$4^{\text{th}}$ moment about the mean is

$$\begin{array}{l} \mu_ {4} = \frac {1}{10} \left[ (218.2 - 201.07) ^ {4} + (179.5 - 201.07) ^ {4} + (207.3 - 201.07) ^ {4} + (224.3 - 201.07) ^ {4} \right. \\ \left. + (213.7 - 201.07) ^ {4} + (199.7 - 201.07) ^ {4} + (184.7 - 201.07) ^ {4} \right. \\ \left. + (194.4 - 201.07) ^ {4} + (203.5 - 201.07) ^ {4} + (185.4 - 201.07) ^ {4} \right] \\ = 75485.5925 \end{array}$$

The coefficient of kurtosis [3, p. 3] is

$$\frac {\mu_ {4}}{\mu_ {2} ^ {2}}$$

The coefficient of kurtosis is

$$\frac {\mu_ {4}}{\mu_ {2} ^ {2}} = \frac {75485.5925}{206.2461 ^ {2}} = 1.7746$$

References:

1. Saylor Academy open textbook. Introductory statistics textbook. Retrieved from https://www.saylor.org/site/textbooks/Introductory%20Statistics.pdf.

2. Central Moment. An R Introduction to Statistics. Retrieved from http://www.r-tutor.com/elementary-statistics/numerical-measures/moment.

3. Sharon I. O'Donell. Notes – Econ 2370 – Statistics and Probability. Retrieved from http://www.uh.edu/~odonnell/econ2370/moment.pdf.

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