Basic Statistics

Introduction to statistics

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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Measures of Central Tendency: A Comprehensive Guide

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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).

Measures of Central Tendency

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

Read more about measures of Central Tendency

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Constructing Frequency Tables (2012)

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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 TypeNumber 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.

  1. Calculate the range of the data. The range is the difference between the highest and smallest values of the given data.
    \[Range = Highest Value – Lowest Value\]
  2. 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.
  3. 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.
  4. 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.
  5. Tally the Observation into the Classes
  6. 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.
Constructing Frequency Tables

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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The Word Statistics Meaning and Use

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The post is about “The Word Statistics Meaning and Use”.

The word statistics was first used by German scholar Gottfried Achenwall in the middle of the 18th century as the science of statecraft concerning the collection and use of data by the state.

The word statistics comes from the Latin word “Status” or Italian word “Statistia” or German word “Statistik” or the French word “Statistique”; meaning a political state, and originally meant information useful to the state, such as information about sizes of the population (human, animal, products, etc.) and armed forces.

itfeature.com The word Statistics

According to pioneer statistician Yule, the word statistics occurred at the earliest in the book “The Element of universal erudition” by Baron (1770). In 1787 a wider definition was used by E.A.W. Zimmermann in “A Political Survey of the Present State of Europe”. It appeared in the Encyclopedia of Britannica in 1797 and was used by Sir John Sinclair in Britain in a series of volumes published between 1791 and 1799 giving a statistical account of Scotland. In the 19th century, the word statistics acquired a wider meaning covering numerical data of almost any subject and also interpretation of data through appropriate analysis.

The Word Statistics Now a Day

Now statistics are being used with different meanings.

  • Statistics refers to “numerical facts that are arranged systematically in the form of tables or charts etc. In this sense, it is always used as a plural i.e. a set of numerical information. For instance statistics on prices, road accidents, crimes, births, educational institutions, etc.
  • The word statistics is defined as a discipline that includes procedures and techniques used to collect, process, and analyze numerical data to make inferences and to reach an appropriate decision in a situation of uncertainty (uncertainty refers to incompleteness, it does not imply ignorance). In this sense word statistic is used in the singular sense. It denotes the science of basing the decision on numerical data.
  • The word statistics refers to numerical quantities calculated from sample observations; a single quantity calculated from sample observations is called statistics such as the mean. Here word statistics is plural.

“We compute statistics from statistics by statistics”

The first place of statistics is plural of statistics, in second place is plural sense data, and in third place is singular sense methods.

In another way, the word Statistics has two meanings:

  • The science of data:
    In this sense, statistics deals with collecting, analyzing, interpreting, and presenting numerical data. Therefore, statistics helps us to understand the world around us by making sense of large amounts of information. Statisticians use a variety of techniques to summarize data, identify patterns, and draw wise conclusions.
  • Pieces of data:
    Statistics also refers to the actual numerical data itself, for example, averages, percentages, or other findings from a study. The real-life examples of statistics are: (i) unemployment statistics or (ii) crime statistics.

Most Common Uses of Statistics

The following are the most common uses of Statistics in various fields of life.

Business and Economics

  • Market Research: Understanding consumer behaviour, satisfaction, preferences, and trends.
  • Operations Management: Optimizing processes, inventory control, and quality control.
  • Financial Analysis: Evaluating investments, risk management, and financial performance.

Healthcare

  • Clinical Trials: Compare and Evaluate the effectiveness and safety of new treatments.
  • Epidemiology: Studying the occurrence and distribution of diseases.
  • Public Health: Identifying health risks and developing prevention strategies.

Social Sciences

  • Sociology: Studying social phenomena, such as inequality, crime, and education.
  • Psychology: Understanding human behaviour, personality, and cognition.
  • Political Science: Analyzing political behaviour, public opinion, and election outcomes.

Government

  • Policy Development: Making informed decisions based on data and evidence.
  • Economic Planning: Forecasting economic growth and trends.
  • Public Administration: Improving efficiency and effectiveness of government services.

Education

  • Educational Research: Evaluating teaching methods, curriculum, and student outcomes.
  • Testing and Assessment: Developing and analyzing standardized tests.
  • Student Data Analysis: Identifying trends and addressing educational disparities.

Science and Technology

  • Research: Designing experiments, collecting data, and analyzing results.
  • Data Analysis: Discovering patterns, relationships, and insights in large datasets.
  • Machine Learning: Developing algorithms that can learn from data and make predictions.

Sports

  • Player Performance Analysis: Evaluating athlete performance and identifying areas for improvement.
  • Team Strategy: Developing game plans and making tactical decisions.
  • Sports Betting: Analyzing data to predict game outcomes.

For learning about the Basics of Statistics Follow the link Basic Statistics

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