Basic Statistics - Statistics for Data Analyst

Skewness Formula, Introduction, Interpretation (2012)

Sep 21, 2024Jul 11, 2012 by Muhammad Imdad Ullah

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. $$Mode > Median > Mean$$

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

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.

FAQs about Skewness

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

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What is the Measure of Kurtosis (2012)

Sep 21, 2024Jul 2, 2012 by Muhammad Imdad Ullah

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.

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.

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

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

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Measures of Central Tendency: A Comprehensive Guide

Sep 6, 2024Apr 5, 2012 by Muhammad Imdad Ullah

Question: What is a measure of central tendency and what are the common measures of central tendency? Also, when is the median preferred over the mean?

A measure of central tendency is the single numerical value considered most typical of the values of a quantitative variable.

The most common measure of central tendency is the mode (i.e., the most frequently occurring number)

The median (i.e., the middle point or fiftieth percentile), and the mean (i.e., the arithmetic average).

The median is preferred over the mean when the numbers are highly skewed (i.e., non-normally distributed).

Since, measures of central tendency condense a bunch of information into a single, digestible value that represents the center of the data, this makes measures of central tendencies important for several reasons:

Summarizing data: Instead of listing every data point, one can use a central tendency measure to get a quick idea of what is typical in the data set.
Comparisons: By computing central tendency measures for different groups or datasets, one can easily compare them to see if there are any differences.
Decision making: Central tendency measures can help to make wise decisions. For instance, knowing the average income in an area can help set prices. Imagine an organization is analyzing customer purchases. Knowing the average amount spent can help them tailor promotions or target specific customer groups.
Identifying trends: Measures of central tendencies may help in observing the trend over time. This can be done by using different visualizations to see if there are any trends, like a rise in average house prices.

However, it is very important to understand these Measures of Central Tendency (mean, median, mode). Each measure of central tendency has its strengths and weaknesses. Choosing the right measure of central tendency depends on the kind of data and what one’s interest is to extract from and try to understand.

Statistics Help measures of central tendency

Constructing Frequency Tables (2012)

Mar 23, 2024Mar 18, 2012 by Muhammad Imdad Ullah

A frequency table is a way of summarizing a set of data. It is a record of each value (or set of values) of the variable in the data/question. In this post, we will learn about the ways of Constructing Frequency Tables for discrete and continuous data.

A grouping of qualitative data into mutually exclusive classes showing the number of observations in each class is called a frequency table. The number of values falling in a particular category/class is called the frequency of that category/class denoted by $f$.

If data of continuous variables are arranged into different classes with their frequencies, then this is known as continuous frequency distribution. If data of discrete variables is arranged into different classes with their frequencies then it is known as discrete distribution or discontinuous distribution.

Discrete Frequency Distribution Table Example

Car Type	Number of Cars
Local	50
Foreign	30
Total Cars	80

Constructing Frequency Tables

Constructing Frequency tables (distributions) may be done for both discrete and continuous variables. A discrete frequency distribution can be converted back to original values, but for continuous variables, it is not possible.

The following steps are taken into account while constructing frequency tables for continuous data.

Calculate the range of the data. The range is the difference between the highest and smallest values of the given data.
\[Range = Highest Value – Lowest Value\]
Decide the number of Classes. The maximum number of classes may be determined by the formula
Number of classes $C = 2^k$ OR Number of classes $(C) = 1+3.3 log (n)$
Note that: Too many classes or too few classes might not reveal the basic shape of the data set.
Determine the Class Interval or Width
The class all taken together should cover at least the distance from the lowest value in the data up to the highest value, which can be done by this formula \[I=\frac{Highest Value – Lowest Value}{Number of Classes}=\frac{H-L}{K}\]
Where $I$ is the class interval, $H$ is the highest observed value, $L$ is the lowest observed value and $K$ is the number of classes.
Generally, the class interval or width should be the same for all classes.
In particular interval size is usually rounded up to some convenient number, such as a multiple of 10 or 100. Unequal class intervals present problems in graphically portraying the distribution and in doing some of the computations. Unequal class intervals may be necessary for certain situations such as to avoid a large number of empty or almost empty classes.
Set the Individual Class Limits
Class limits are the endpoints in the class interval. State clear class limits so that you can put each of the observations into one and only one category i.e. you must avoid overlapping or unclear class limits. Class intervals are usually rounded up to get a convenient class size, and cover a larger than necessary range.
It is convenient to choose the endpoints of the class interval so that no observation falls on them. It can be obtained by expressing the endpoints to one more place of decimal than the observations themselves, i.e. limits are converted to class boundaries to achieve continuity in data.
Tally the Observation into the Classes
Count the Number of Items in each Class
The number of observations in each class I called the class frequency. Note that totaling the frequencies in each class must equal the total number of observations. After following these steps, we have organized the data into a tabulation form which is called a frequency distribution, which can be used to summarize the pattern in the observation i.e., the concentration of the data.

Note: Arranging/organizing the data into a tabulation or frequency distribution results in a loss of detailed information as the individuality of observations vanishes i.e. in frequency distribution we cannot pinpoint the exact value, and we cannot tell the actual lowest and highest values of the data. However, the lower limit of the largest, class conveys some essentially the same meaning. So in constructing the frequency tables, the advantages of condensing the data into a more understandable and organized form are more than offset this disadvantage.

Further Reading

Frequency Distribution Tables

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Skewness Formula, Introduction, Interpretation (2012)

Table of Contents

Pearson’s Coefficient of Skewness Formula