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SKEWNESS, KURTOSIS,

BOXPLOTDESCRIBING SHAPES AND PATTERNS

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SKEWNESSSymmetry of Distribution

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MOTIVATION:

In both histograms, the mean is 6 and the

variance is roughly around 8.7. But, the shape

of distribution of values are not necessarily thesame. Hence, the measures of skewness and

kurtosis are summary measures that describe

the shape of the distribution.

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SKEWNESS

•  A measure of symmetry/asymmetry of a

distribution of values in a data.

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SKEWNESS

POSITIVELY SKEWED or SKEWED TOTHE RIGHT - the concentration of the

values are on the left side of the distribution,

with a tail that is tapering off on the right

side.

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SKEWNESS

NEGATIVELY SKEWED or SKEWED TOTHE LEFT - the concentration of the values

are on the right side of the distribution, with

a tail that is tapering off on the left side.

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SKEWNESS

• the distribution is symmetric, if the mean =

median = mode.

• the distribution is skewed to the right or

positively skewed, if the mean > median >

mode.

• the distribution is skewed to the left or

negatively skewed, if the mean < median <

mode.

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SKEWNESS

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SKEWNESS

′

1  −

′

2      −

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SKEWNESS

Interpretation:if the coefficient of skewness is less than 0,

then the distribution is skewed to the left or

negatively skewed.

If the coefficient of skewness is greaterthan 0, then the distribution is skewed to

the right or positively skewed.

If the coefficient of skeweness is 0, then the

distribution is symmetric.

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KURTOSISConcavity of Distribution

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KURTOSIS

• It describes the “hump” of the

distribution of the values relative to

the “hump” of a normal distribution of

the same amount of variability.

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KURTOSIS

• The Normal Distribution:

 – It is a bell-shaped curve.

 – It is symmetric about its mean.

 – It has tails that extend on both ends but it doesn’ttouch the x-axis

 – The area below the curve and above the x-axis is 1.

 – 68% of observations lie within one standard deviation

from the mean, 95% of observations lie within twostandard deviations from the mean, and at least 99%

of observations lie within three standard deviations

from the mean.

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KURTOSIS

• The Normal Distribution

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KURTOSIS

• Coefficient of kurtosis:

4

4

 

=1   

  4

4

interpretation:4

4  3 < 0 →

4

4  3 > 0 → 4

4  3 0 →

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KURTOSIS

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KURTOSIS

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BOXPLOTExploratory-Data-Analysis Tool

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BOXPLOT

Box-and-whisker plot – a

graphical tool for assessing

the shape of distribution.Features:

Location

Spread

Symmetry

Extremes

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BOXPLOT

OUTLIERS!

3rd Quartile

1st Quartile

2nd Quartile

Or Mean

AXIS

Q3+1.5(IQR)

or max

Q1-1.5(IQR)

or min

Range of

the

middle50% of

the data

IQR=Q3-Q1

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 ASSESSING SKEWNESS

USING A BOXPLOT

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DESCRIBE THE SKEWNESS

OF EACH BOXPLOT

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 /:END:/