Shape of Distribution

Skewness

Skewness is the measurement of the lack of symmetry of the distribution. The shape can be Symmetric or skewed

The Same Positions of the Mean, Median, and Mode: Symmetric Distribution

Mode < Median < Mean

Right-Skewed

(Longer tail extends to right)

The coefficient of skewness can range from -3.00 up to 3.00 when using the following formula:

Mean < Median < Mode

Left-Skewed

(Longer tail extends to left)

MeanMedianMode

Symmetrical Distribution (Sk = 0)

Skewness is measured with the help of COEFFICIENT OF SKEWNESS. If the value of coefficient of skewness is zero, the data is symmetric. If the value of coefficient of skewness is less than zero, the data is negatively skewed. If the value of coefficient of skewness is greater than zero, the data is positively skewed.

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MedianXsk 3

1. Karl Pearson

sModeXsk

2. Bowley

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3. α 3

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Curtosis Kurtosis is a measure of whether the data are

peaked or flat relative to a normal distribution.

Kurtosis measures how peaked the histogram is

The kurtosis of a normal distribution is 0

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ns

xxkurtosis

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Platykurtic– When the kurtosis < 3, the frequencies throughout the curve are closer to be equal (i.e., the curve is more flat and wide)

Leptokurtic– When the kurtosis > 3, there are high frequencies in only a small part of the curve (i.e, the curve is more peaked)

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Consider the following data representing the distribution of salaries of 3 companies A, B and C each having 76 employees:

Class Mark

Salary Number of employees

X (in RM) Company A

Company B

Company C

249,5 200-299 3 3 3… 300-399 8 8 9… 400-499 16 15 13… 500-599 22 23 23… 600-699 16 18 20… 700-700 8 6 5… 800-899 3 2 2… 900-999 0 1 1

Total 76 76 769

Exercise

From the data we have the following results about mean and deviation:

Which company has the most skewness and peaked about the employee’s salary?

Remember, Skewness means lack of symmetry.

Company A Company B Company CMean Salary … … …S.D. … … …

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