Basic Statistics

Introduction to statistics

Levels of Measurement (2021): A Comprehensive Tutorial

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Levels of Measurement (Scale of Measure)

The levels of measurement (scale of measures) have been classified into four categories. It is important to understand these measurement levels since they play an important part in determining the arithmetic and different possible statistical tests carried on the data. The scale of measure is a classification that describes the nature of the information within the number assigned to a variable. In simple words, the level of measurement determines how data should be summarized and presented.

It also indicates the type of statistical analysis that can be performed. The four-level of measurements are described below:

Nominal Level of Measurement (Nominal Scale)

At the nominal level of measurement, the numbers are used to classify the data (unordered group) into mutually exclusive categories. In other words, for the nominal level of measurement, observations of a qualitative variable are measured and recorded as labels or names.

Ordinal Level of Measurement (Ordinal Scale)

In the ordinal level of measurement, the numbers are used to classify the data (ordered group) into mutually exclusive categories. However, it does not allow for a relative degree of difference between them. In other words, for the ordinal level of measurement, observations of a qualitative variable are either ranked or rated on a relative scale and recorded as labels or names.

Interval Level of Measurement (Interval Scale)

For data recorded at the interval level of measurement, the interval or the distance between values is meaningful. The interval scale is based on a scale with a known unit of measurement.

Ratio Level of Measurement (Ratio Scale)

Data recorded at the ratio level of measurement are based on a scale with a known unit of measurement and a meaningful interpretation of zero on the scale. Almost all quantitative variables are recorded on the ratio level of measurement.

Levels of Measurement

Examples of levels of measurement

Examples of Nominal Level of Measurement

  • Religion (Muslim, Hindu, Christian, Buddhist)
  • Race (Hispanic, African, Asian)
  • Language (Urdu, English, French, Punjabi, Arabic)
  • Gender (Male, Female)
  • Marital Status (Married, Single, Divorced)
  • Number plates on Cars/ Models of Cars (Toyota, Mehran)
  • Parts of Speech (Noun, Verb, Article, Pronoun)

Examples of Ordinal Level of Measurement

  • Rankings (1st, 2nd, 3rd)
  • Marks Grades (A, B, C, D)
  • Evaluations such as High, Medium, and Low
  • Educational level (Elementary School, High School, College, University)
  • Movie Ratings (1 star, 2 stars, 3 stars, 4 stars, 5 stars)
  • Pain Ratings (more, less, no)
  • Cancer Stages (Stage 1, Stage 2, Stage 3)
  • Hypertension Categories (Mild, Moderate, Severe)

Examples of Interval Levels of Measurement

  • Temperature with Celsius scale/ Fahrenheit scale
  • Level of happiness rated from 1 to 10
  • Education (in years)
  • Standardized tests of psychological, sociological, and educational discipline use interval scales.
  • SAT scores

Examples of Ratio Level of Measurement

  • Height
  • Weight
  • Age
  • Length
  • Volume
  • Number of home computers
  • Salary

In essence, levels of measurement act like a roadmap for statistical analysis. They guide us in selecting the most appropriate methods to extract valuable insights from the data under study. The level of measures is very important because they help us in

  • Choosing the right statistical tools: Different levels of measurement are used for different statistical methods. For example, One can compute a measure of central tendency (such as mean and median) for data on income (which is interval level), but a measure of central tendency (such as mean and median) cannot be computed for data on favorite color (which is nominal level, the mode can be computed regarding the measure of central tendency).
  • Drawing valid conclusions: If the statistical test is misused because of a misunderstanding of the measurement level of the data, the conclusions might be misleading or even nonsensical. Therefore, measurement levels help us ensure that analysis reflects the actual characteristics of the data.
  • Making meaningful comparisons: Levels of measurement also allow us to compare data points appropriately. For instance, one can say someone is 2 years older than another person (ordinal data), but one cannot say that their preference for chocolate ice cream is twice as strong (nominal data).
Levels of Measurement

FAQS About Levels of Measurements

  1. What do you mean by measurement levels?
  2. What is the role of Levels of Measurement in Statistics?
  3. Compare, nominal, ordinal, ratio, and interval scale.
  4. What measures of central tendency can be performed on which measurement level?
  5. What is the importance of measurement levels?
  6. Give at least five, five examples of each measurement level.

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Errors in Measurement

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Errors in Measurement: It is a fact and from experience, it is observed that a continuous variable can not be measured with perfect (true) value because of certain habits and practices, measurement methods (techniques), instruments (or devices) used, etc. It means that the measurements are thus always recorded correctly to the nearest units and hence are of limited accuracy. The actual values are, however, assumed to exist.

Errors in Measurement Example

For example, if the weight of a student is recorded as 60 kg (correct to the nearest kilogram), his/her true (actual) weight, may lie between 59.5 kg and 60.5 kg. The weight recorded as 60.00 kg for that student means the true weight is known to lie between 59.995 and 60.005 kg.

Thus, there is a difference, however, it is small which may be between the measured value and the true value. This sort of departure from the true value is technically known as errors in measurement. In other words, if the observed value and the true value of a variable are denoted by $x$ and $x + \varepsilon$, respectively, then the difference $(x + \varepsilon) – x=\varepsilon$, is the error. This error involves the unit of measurement of $x$ and is, therefore, called an absolute error.

An absolute error divided by the true value is called the relative error. Thus the relative error can be measured as $\frac{\varepsilon}{x+\varepsilon}$. Multiplying this relative error by 100 gives the percentage error. These errors are independent of the units of measurement of $x$. It ought to be noted that an error has both magnitude and direction and that the word error in statistics does not mean a mistake which is a chance inaccuracy.

Errors in Measurement

An error is said to be biased when the observed value is higher or lower than the true value. Biased errors arise from the personal limitations of the observer, the imperfection in the instruments used, or some other conditions that control the measurements. These errors are not revealed by repeating the measurements. They are cumulative, that is, the greater the number of measurements, the greater would be the magnitude of the error. They are thus more troublesome. These errors are also called cumulative or systematic errors.

An error, on the other hand, is said to be unbiased when the deviations from the true value tend to occur equally often. Unbiased errors tend to cancel out in the long run. These errors are therefore compensating and are also known as random errors or accidental errors.

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We can reduce errors in measurement by

  • Double-checking all measurements for accuracy
  • Double-checking the formulas are correct
  • Making sure observers and measurement takers are well-trained
  • Measuring with the instrument has the highest precision
  • Take the measurements under controlled conditions
  • Pilot test your measuring instruments
  • Use multiple measures for the same construct

Types of Errors: Errors can be classified into two main categories:

  • Random Errors: These are variations in the reading/recording due to limitations of the instrument being used, the environment, or even the person taking the measurement. These errors are random by nature and fluctuate slightly up or down from the true value with each measurement.
  • Systematic Errors: Systematic Errors are consistent errors that cause your measurements to deviate from the true value predictably. For example, a ruler with a slightly inaccurate scale would introduce a systematic error in every measurement you make with it.
Types of Errors

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Quantitative Qualitative Variables: Statistical Data (2021)

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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.

Quantitative Qualitative Variables

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)

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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.

Skewness and Measures of Skewness

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.

Skewness ad Measure of Skewness

FAQs about SKewness

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

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