Skewness Formula, Introduction, Interpretation (2012)

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Skewness is the degree of asymmetry or departure from the symmetry of the distribution of a real-valued random variable.

Positive Skewed
If the frequency curve of distribution has a longer tail to the right of the central maximum than to the left, the distribution is said to be skewed to the right or to have positively skewed. In a positively skewed distribution, the mean is greater than the median and the median is greater than the mode i.e. $$Mean > Median > Mode$$

Negative Skewed
If the frequency curve has a longer tail to the left of the central maximum than to the right, the distribution is said to be skewed to the left or to be negatively skewed. In a negatively skewed distribution, the mode is greater than the median and the median is greater than the mean i.e. $$Mode > Median > Mean$$

Measure of Skewness Formulation

In a symmetrical distribution, the mean, median, and mode coincide. In a skewed distribution, these values are pulled apart.

Skewness Formula

Pearson’s Coefficient of Skewness Formula

Karl Pearson, (1857-1936) introduced a coefficient to measure the degree of skewness of distribution or curve, which is denoted by $S_k$ and defined by

\begin{eqnarray*}
S_k &=& \frac{Mean – Mode}{Standard Deviation}\\
S_k &=& \frac{3(Mean – Median)}{Standard Deviation}\\
\end{eqnarray*}
Usually, this coefficient varies between –3 (for negative) to +3 (for positive) and the sign indicates the direction of skewness.

Bowley’s Coefficient of Skewness Formula (Quartile Coefficient)

Arthur Lyon Bowley (1869-1957) proposed a measure of skewness based on the median and the two quartiles.

\[S_k=\frac{Q_1+Q_3-2Median}{Q_3 – Q_1}\]
Its values lie between 0 and ±1.

Moment Coefficient of Skewness Formula

This measure of skewness is the third moment expressed in standard units (or the moment ratio) thus given by

\[S_k=\frac{\mu_3}{\sigma^3} \]
Its values lie between -2 and +2.

If $S_k$ is greater than zero, the distribution or curve is said to be positively skewed. If $S_k$ is less than zero the distribution or curve is said to be negatively skewed. If $S_k$ is zero the distribution or curve is said to be symmetrical.

The skewness of the distribution of a real-valued random variable can easily be seen by drawing a histogram or frequency curve.

The skewness may be very extreme and in such a case these are called J-shaped distributions.

Skewness: J-Shaped Distribution

FAQs about Skewness

  1. What is the degree of asymmetry called?
  2. What is a departure from symmetry?
  3. If a distribution is negatively skewed then what is the relation between mean, median, and mode?
  4. If a distribution is positively skewed then what is the relation between mean, median, and mode?
  5. What is the relation between mean, median, and mode for a symmetrical distribution?
  6. What is the range of the moment coefficient of skewness?

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Correlation Coefficient: A Comprehensive Guide

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The correlation is a measure of the co-variability of variables. It is used to measure the strength between two quantitative variables. It also tells the direction of a relationship between the variables. The positive value of the correlation coefficient indicates that there is a direct (supportive or positive) relationship between the variables while the negative value indicates there is a negative (opposite or indirect) relationship between the variables.

Correlation as Interdependence Between Variables

By definition, Pearson’s correlation is the interdependence between two quantitative variables. The causation (known as) cause and effect, is when an observed event or action appears to have caused a second event or action. Therefore, It does not necessarily imply any functional relationship between the variables concerned. Correlation theory does not establish any causal relationship between the variables as it is interdependence between the variables. Knowledge of the value of Pearson’s correlation coefficient $r$ alone will not enable us to predict the value of $Y$ from $X$.

High Correlation Coefficient does not Indicate Cause and Effect

Sometimes there is a high Relationship between unrelated variables such as the number of births and the number of murders in a country. This is a spurious correlation.

For example, suppose there is a positive correlation between watching violent movies and violent behavior in adolescence. The cause of both these could be a third variable (extraneous variable) say, growing up in a violent environment which causes the adolescents to watch violence-related movies and to have violent behavior.

Correlation Coefficient

Other Examples

  • The number of absences from class lectures decreases the grades.
  • As the weather gets colder, air conditioning costs decrease.
  • As the speed of the train (car, bus, or any other vehicle) is increased the length of time to get to the final point will also decrease.
  • As the age of a chicken increases the number of eggs it produces also decreases.
Statistics Help https://itfeature.com, Correlation Coefficient

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Partial Correlation Coefficient (2012)

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The Partial Correlation Coefficient measures the relationship between any two variables, where all other variables are kept constant i.e. controlling all other variables or removing the influence of all other variables. Partial correlation aims to find the unique variance between two variables while eliminating the variance from the third variable. The partial correlation technique is commonly used in “causal” modeling of fewer variables. The coefficient is determined in terms of the simple correlation coefficient among the various variables involved in multiple relationships.

Assumptions for computing the Partial Correlation Coefficient

The assumption for partial correlation is the usual assumption of Pearson Correlation:

  1. Linearity of relationships
  2. The same level of relationship throughout the range of the independent variable i.e. homoscedasticity
  3. Interval or near-interval data, and
  4. Data whose range is not truncated.

We typically conduct correlation analysis on all variables so that you can see whether there are significant relationships amongst the variables, including any “third variables” that may have a significant relationship to the variables under investigation.

This type of analysis helps to find the spurious correlations (i.e. correlations that are explained by the effect of some other variables) as well as to reveal hidden correlations – i.e. correlations masked by the effect of other variables. The partial correlation coefficient $r_{xy.z}$ can also be defined as the correlation coefficient between residuals $dx$ and $dy$ in this model.

Suppose we have a sample of $n$ observations $(x1_1,x2_1,x3_1),(x1_2,x2_2,x3_2),\cdots,(x1_n,x2_n,x3_n)$ from an unknown distribution of three random variables. We want to find the coefficient of partial correlation between $X_1$ and $X_2$ keeping $X_3$ constant which can be denoted by $r_{12.3}$ is the correlation between the residuals $x_{1.3}$ and $x_{2.3}$. The coefficient $r_{12.3}$ is a partial correlation of the 1st order.

\[r_{12.3}=\frac{r_{12}-r_{13} r_{23}}{\sqrt{1-r_{13}^2 } \sqrt{1-r_{23}^2 } }\]

Partial Correlation Coefficient

The coefficient of partial correlation between three random variables $X$, $Y$, and $Z$ can be denoted by $r_{x,y,z}$ and also be defined as the coefficient of correlation between $\hat{x}_i$ and $\hat{y}_i$ with
\begin{align*}
\hat{x}_i&=\hat{\beta}_{0x}+\hat{\beta}_{1x}z_i\\
\hat{y}_i&=\hat{\beta}_{0y}+\hat{\beta}_{1y}z_i\\
\end{align*}
where $\hat{\beta}_{0x}$ and $\hat{\beta_{1x}}$ are the least square estimators obtained by regressing $x_i$ on $z_i$ and $\hat{\beta}_{0y}$ and $\hat{\beta}_{1y}$ are the least square estimators obtained by regressing $y_i$ on $z_i$. Therefore by definition, the partial correlation between of $x$ and $y$ by controlling $z$ is
\[r_{xy.z}=\frac{\sum(\hat{x}_i-\overline{x})(\hat{y}_i-\overline{y})}{\sqrt{\sum(\hat{x}_i-\overline{x})^2}\sqrt{\sum(\hat{y}_i-\overline{y})^2}}\]

Partial Correlation Analysis

The coefficient of partial correlation is determined in terms of the simple correlation coefficients among the various variables involved in a multiple relationship. It is a very helpful tool in the field of statistics for understanding the true underlying relationships between variables, especially when you are dealing with potentially confounding factors.

Reference
Yule, G. U. (1926). Why do we sometimes get non-sense correlations between time series? A study in sampling and the nature of time series. J. Roy. Stat. Soc. (2) 89, 1-64.

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What is the Measure of Kurtosis (2012)

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Introduction to Kurtosis

In statistics, a measure of kurtosis is a measure of the “tailedness” of the probability distribution of a real-valued random variable. The standard measure of kurtosis is based on a scaled version of the fourth moment of the data or population. Therefore, the measure of kurtosis is related to the tails of the distribution, not its peak.

Measure of Kurtosis

Sometimes, the Measure of Kurtosis is characterized as a measure of peakedness that is mistaken. A distribution having a relatively high peak is called leptokurtic. A distribution that is flat-topped is called platykurtic. The normal distribution which is neither very peaked nor very flat-topped is also called mesokurtic.  The histogram in some cases can be used as an effective graphical technique for showing the skewness and kurtosis of the data set.

Measure of Kurtosis

Data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak.

Moment ratio and Percentile Coefficient of kurtosis are used to measure the kurtosis

Moment Coefficient of Kurtosis= $b_2 = \frac{m_4}{S^2} = \frac{m_4}{m^{2}_{2}}$

Percentile Coefficient of Kurtosis = $k=\frac{Q.D}{P_{90}-P_{10}}$
where Q.D = $\frac{1}{2}(Q_3 – Q_1)$ is the semi-interquartile range. For normal distribution, this has a value of 0.263.

Dr. Wheeler defines kurtosis as:

The kurtosis parameter is a measure of the combined weight of the tails relative to the rest of the distribution.

So, kurtosis is all about the tails of the distribution – not the peakedness or flatness.

A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is considered Leptokurtic. If its kurtosis is less than 3, it is considered Platykurtic.

A large value of kurtosis indicates a more serious outlier issue and hence may lead the researcher to choose alternative statistical methods.

Measure of Kurtosis

Some Examples of Kurtosis

  • In finance, risk and insurance are examples of needing to focus on the tail of the distribution and not assuming normality.
  • Kurtosis helps in determining whether the resource used within an ecological guild is truly neutral or which it differs among species.
  • The accuracy of the variance as an estimate of the population $\sigma^2$ depends heavily on kurtosis.

For further reading see Moments in Statistics

FAQs about Kurtosis

  1. Define Kurtosis.
  2. What is the moment coefficient of Kurtosis?
  3. What is the definition of kurtosis by Dr. Wheeler?
  4. Give examples of kurtosis from real life.

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