Skewness

[Asuka]

In 2nd paragraph on CFA 2014 Text Vol 1 P.396, ther are following 3 statements :

1st statement :

If a distribution is positively skewed with a mean greater than its median, then more than half of the deviations from the mean are negative and less than half are positive.

My question :

1. Is it that the means of all positively skewed distributions shall be greter than their medians ?

2. Why more than half of the deviations from the mean are negative and less than half are positive ?

2nd statement :

In order for the sum to be postive, the losses must be small and likely, and the gains less likely but more likely extreme.

My question :

What does it mean by “for the sum to be positive”?

3rd statement :

Therefore, if the skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations.

1. What does it mean by “the average manitude”?

2. Why is it that if the skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations ?

[S2000magician]

@Asuka

In 2nd paragraph on CFA 2014 Text Vol 1 P.396, there are following 3 statements:

1st statement:

If a distribution is positively skewed with a mean greater than its median, then more than half of the deviations from the mean are negative and less than half are positive.

My question:

1. Is it that the means of all positively skewed distributions shall be greater than their medians?

Yes.

@Asuka

2. Why more than half of the deviations from the mean are negative and less than half are positive?

Half of the values are below the median, and the median is below the mean, so more than half of the values are below the mean.

@Asuka

2nd statement:

In order for the sum to be positive, the losses must be small and likely, and the gains less likely but more likely extreme.

My question:

What does it mean by “for the sum to be positive”?

They mean:

Σ(Xi – X-bar)³ > 0

@Asuka

3rd statement:

Therefore, if the skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations.

1. What does it mean by “the average magnitude”?

The average (Xi – X-bar)

@Asuka

2. Why is it that if the skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations?

There are fewer positive deviations than negative deviations, but the sum above has to be positive, so the average positive deviation has to be bigger than the average negative deviation.

[Asuka]

[quote=“S2000magician”]

@Asuka

In 2nd paragraph on CFA 2014 Text Vol 1 P.396, there are following 3 statements:

@Asuka

2nd statement:

In order for the sum to be positive, the losses must be small and likely, and the gains less likely but more likely extreme.

My question:

What does it mean by “for the sum to be positive”?

They mean: Σ(Xi – X-bar)³ > 0

Then, why the sum has to be positive ?

@Asuka

3rd statement:

Therefore, if the skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations.

1. What does it mean by “the average magnitude”?

The average (Xi – X-bar)

Do you mean :

for “postive” deviations : the average magnitude = (Xi - X-bar) / (Frequency of positive deviations) ?

for “negative” deviations : the average magnitude = (Xi - X-bar) / (Frequency of negative deviations) ?

@Asuka

2. Why is it that if the skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations?

There are fewer positive deviations than negative deviations, but the sum above has to be positive, so the average positive deviation has to be bigger than the average negative deviation.

Is it that, if the skewness is negative, the oppositive shall be true (that is, the average magnitude of “negative” deviations is larger than the average magnitude of “positive” deviations), right ?

[S2000magician]

@Asuka

Is it that, if the skewness is negative, the oppositive shall be true (that is, the average magnitude of “negative” deviations is larger than the average magnitude of “positive” deviations), right?

That is correct.

[Asuka]

[quote=“S2000magician”]

@Asuka

In 2nd paragraph on CFA 2014 Text Vol 1 P.396, there are following 3 statements:

@Asuka

2nd statement:

In order for the sum to be positive, the losses must be small and likely, and the gains less likely but more likely extreme.

My question:

What does it mean by “for the sum to be positive”?

They mean:

Σ(Xi – X-bar)³ > 0

why the sum has to be positive ?

[S2000magician]

Positive skewness.

Here’s the formula: http://financialexamhelp123.com/skewness/.