Basic Statistics - Statistics for Data Analyst

Quantitative Qualitative Variables: Statistical Data (2021)

Jul 23, 2024Apr 28, 2021 by Muhammad Imdad Ullah

This article is about Quantitative Qualitative Variables. First, we need to understand the concept of data and variables. Let us start with some basics.

The word “data” is frequently used in many contexts and ordinary conversations. Data is Latin for “those that are given” (the singular form is “datum”). Data may therefore be thought of as the results of observation. In this post, we will learn about quantitative qualitative variables with examples.

Data are collected in many aspects of everyday life.

Statements given to a police officer, physician, or psychologist during an interview are data.
So are the correct and incorrect answers given by a student on a final examination.
Almost any athletic event produces data.
The time required by a runner to complete a marathon,
The number of spelling errors a computer operator commits in typing a letter.

Data are also obtained in the course of scientific inquiry:

the positions of artifacts and fossils in an archaeological site,
The number of interactions between two members of an animal colony during a period of observation,
The spectral composition of light emitted by a star.

Data comprise variables. Variables are something that changes from time to time, place to place, and/or person to person. Variables may be classified into quantitative and qualitative according to the form of the characters they may have.

Quantitative Qualitative Variables

Let us understand the major concept of Quantitative Qualitative variables by defining these types of variables and their related examples. The examples are self-explanatory and all of the examples are from real-life problems.

Qualitative Variables

A variable is called a quantitative variable when a characteristic can be expressed numerically such as age, weight, income, or several children, that is, the variables that can be quantified or measured from some measurement device/ scales (such as weighing machine, thermometer, and liquid measurement standardized container).

On the other hand, if the characteristic is non-numerical such as education, sex, eye color, quality, intelligence, poverty, satisfaction, etc. the variable is referred to as a qualitative variable. A qualitative characteristic is also called an attribute. An individual or an object with such a characteristic can be counted or enumerated after having been assigned to one of the several mutually exclusive classes or categories (or groups).

Quantitative Variables

Mathematically, a quantitative variable may be classified as discrete or continuous. A discrete variable can take only a discrete set of integers or whole numbers, which are the values taken by jumps or breaks. A discrete variable represents count data such as the number of persons in a family, the number of rooms in a house, the number of deaths in an accident, the income of an individual, etc.

A variable is called a continuous variable if it can take on any value- fractional or integral––within a given interval, that is, its domain is an interval with all possible values without gaps. A continuous variable represents measurement data such as the age of a person, the height of a plant, the weight of a commodity, the temperature at a place, etc.

A variable whether countable or measurable is generally denoted by some symbol such as $X$ or $Y$ and $X_i$ or $X_j$ represents the $i$th or $j$th value of the variable. The subscript $i$ or $j$ is replaced by a number such as $1,2,3, \cdots, n$ when referred to a particular value.

Examples of Statistical Data

Note that statistical data can be found everywhere, few examples are:

Any financial/ economics data
Transactional data (from stores, or banks)
The survey, or census (of unemployment, houses, population, roads, etc)
Medical history
Price of product
Production, and yields of a crop
My history, your history is also statistical data

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Secrets of Skewness and Measures of Skewness (2021)

Oct 12, 2024Mar 10, 2021 by Muhammad Imdad Ullah

If the curve is symmetrical, a deviation below the mean exactly equals the corresponding deviation above the mean. This is called symmetry. Here, we will discuss Skewness and Measures of Skewness.

Skewness is the degree of asymmetry or departure from the symmetry of a distribution. Positive Skewness means when the tail on the right side of the distribution is longer or fatter. The mean and median will be greater than the mode. Negative Skewness is when the tail of the left side of the distribution is longer or fatter than the tail on the right side.

Measures of Skewness

Karl Pearson Measures of Relative Skewness

In a symmetrical distribution, the mean, median, and mode coincide. In skewed distributions, these values are pulled apart; the mean tends to be on the same side of the mode as the longer tail. Thus, a measure of the asymmetry is supplied by the difference ($mean – mode$). This can be made dimensionless by dividing by a measure of dispersion (such as SD).

The Karl Pearson measure of relative skewness is
$$\text{SK} = \frac{\text{Mean}-\text{mode}}{SD} =\frac{\overline{x}-\text{mode}}{s}$$
The value of skewness may be either positive or negative.

The empirical formula for skewness (called the second coefficient of skewness) is

$$\text{SK} = \frac{3(\text{mean}-\text{median})}{SD}=\frac{3(\tilde{X}-\text{median})}{s}$$

Bowley Measures of Skewness

In a symmetrical distribution, the quartiles are equidistant from the median ($Q_2-Q_1 = Q_3-Q_2$). If the distribution is not symmetrical, the quartiles will not be equidistant from the median (unless the entire asymmetry is located in the extreme quarters of the data). The Bowley suggested measure of skewness is

$$\text{Quartile Coefficient of SK} = \frac{Q_(2-Q_2)-(Q_2-Q_1)}{Q_3-Q_1}=\frac{Q_2-2Q_2+Q_1}{Q_3-Q_1}$$

This measure is always zero when the quartiles are equidistant from the median and is positive when the upper quartile is farther from the median than the lower quartile. This measure of skewness varies between $+1$ and $-1$.

Moment Coefficient of Skewness

In any symmetrical curve, the sum of odd powers of deviations from the mean will be equal to zero. That is, $m_3=m_5=m_7=\cdots=0$. However, it is not true for asymmetrical distributions. For this reason, a measure of skewness is devised based on $m_3$. That is

\begin{align}
\text{Moment of Coefficient of SK}&= a_3=\frac{m_3}{s^3}=\frac{m_3}{\sqrt{m_2^3}}\\
&=b_1=\frac{m_3^2}{m_2^3}
\end{align}

For perfectly symmetrical curves (normal curves), $a_3$ and $b_1$ are zero.

FAQs about SKewness

What is skewness?
If a curve is symmetrical then what is the behavior of deviation below and above the mean?
What is Bowley’s Measure of Skewness?
What is Karl Person’s Measure of Relative Skewness?
What is the moment coefficient of skewness?
What is the positive and negative skewness?

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Quantiles or Fractiles Uncovered (2020)

Dec 21, 2024Nov 30, 2020 by Muhammad Imdad Ullah

When the number of observations is sufficiently large, the principle by which a distribution is divided into two equal parts may be extended to divide the distribution into four, five, eight, ten, or hundred equal parts. The median, quartiles, deciles, and percentiles values are collectively called quantiles or fractiles. Let us start learning about Quantiles or Fractiles.

Quantiles or Fractiles

Quartiles

These are the values that divide a distribution into four equal parts. There are three quartiles denoted by $Q_1, Q_2$, and $Q_3$. If $x_1,x_2,\cdots,x_n$ are $n$ observations on a variable $X$, and $x_{(1)}, x_{(2)}, \cdots, x_{(n)}$ is their array then $r$th quartile $Q_r$ is the values of $X$, such that $\frac{r}{4}$ of the observations is less than that value of $X$ and $\frac{4-r}{4}$ of the observations is greater.

The $Q_1$ is the value of $X$ such that $\frac{1}{4}$ of the observations is less than the value of $X$ and $\frac{4-1}{4}$ of the observations is greater, the $Q_3$ is the value of $X$, such that $\frac{3}{4}$ of the observations is less than that of $X$ and $\frac{4-3}{4}$ of the observations is greater.

Deciles

These are the values that divide a distribution into ten equal parts. There are 9 deciles $D_1, D_2, \cdots, D_9$.

Percentiles

These are the values that divide a distribution into a hundred equal parts. There are 99 percentiles denoted as $P_1,P_2,\cdots, P_{99}$.

The median, quartiles, deciles, percentiles, and other partition values are collectively called quantiles or fractiles. All quantiles are percentages. For example, $P_{50}, Q_2$, and $D_5$ are also median.

\begin{align*}
Q_2 &= D_5 = P_{50}\\
Q_1 &= P_{25} = D_{2.5}\\
Q_3 &= P_{75}=D_{7.5}
\end{align*}
The $r$th quantile, $k$th decile, and $j$th percentile are located in the array by the following relation:

For ungrouped Date
\begin{align}
Q_r &=\frac{r(n+1)}{4}\text{th value in the distribution and } r=1,2,3\\
D_k &=\frac{k(n+1)}{10}\text{th value in the distribution and } k=1,2,\cdots, 9\\
P_j &=\frac{j(n+1)}{100}\text{th value in the distribution and } k=1,2,\cdots, 99
\end{align}

For grouped Data
\begin{align}
Q_r&= l+\frac{h}{f}\left(\frac{rn}{4}-c\right)\\
D_k&= l+\frac{h}{f}\left(\frac{kn}{10}-c\right)\\
P_j&= l+\frac{h}{f}\left(\frac{jn}{100}-c\right)
\end{align}

Procedure for obtaining Percentile

A procedure for obtaining percentile (quartiles, deciles) of a data set of size $n$ is as follows:

Step 1: Arrange the data in ascending/ descending order.
Step 2: Compute an index $i$ as follows: $i=\frac{p}{100} (n+1)$th (in case of odd observation).

If $i$ is an integer, the $p$th percentile is the average of the $i$th and $(i+1)$th data values.
if $i$ is not an integer then round $i$ up to the nearest integer and take the value at that position or use some mathematics to locate the value of percentile between $i$th and $(i+1)$th value.

Percentile Example

Consider the following (sorted) data values: 380, 600, 690, 890, 1050, 1100, 1200, 1900, 890000.

For the $p=10$th percentile, $i=\frac{p}{100} (n+1) =\frac{10}{100} (9+1)= 1$. So the 10th percentile is the first sorted value or 380.

For the $p=75$ percentile, $i=\frac{p}{100} (n+1)= \frac{75}{100}(9+1) = 7.5$

To get the actual value we need to compute 7th value + (8th value – 7th value) $\times 0.5$. That is, $1200 + (1900-1200)\times 0.5 = 1200+350 = 1550$.

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Frequently Asked Questions Fractiles

What is meant by quartile, deciles, and percentiles?
Describe the procedure of obtaining percentiles (quartiles, and deciles).
What is the interquartile range?
Why do we need to sort the data first when computing quartiles, deciles, and percentiles?

Quantitative Qualitative Variables: Statistical Data (2021)