## Levels of Measurement (2021): A Comprehensive Tutorial

### 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.

### 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).

### 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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## Quantitative Qualitative Variables: Statistical Data (2021)

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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## Statistical Data: Introduction and Real Life Examples (2020)

By statistical Data we mean, the piece of information collected for descriptive or inferential statistical analysis of the data. Data is everywhere. Therefore, everything that has past and/ or features is called statistical data.

#### One can find the Statistical data

• 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

#### Data

Data is the plural of datum — it is a piece of information. The value of the variable (understudy) associated with one element of a population or sample is called a datum (or data in a singular sense or data point). For example, Mr. Asif entered college at the age of 18 years, his hair is black, has a height of 5 feet 7 inches, and he weighs about 140 pounds. The set of values collected for the variable from each of the elements belonging to the sample is called data (or data in a plural sense). For example, a set of 25 weights was collected from the 25 students.

#### Types of Data

The data can be classified into two general categories: quantitative data and qualitative data. The quantitative data can further be classified as numerical data that can be either discrete or continuous. The qualitative data can be further subdivided into nominal, ordinal, and binary data.

Qualitative data represent information that can be classified by some quality, characteristics, or criterion—for example, the color of a car, religion, blood type, and marital status.

When the characteristic being studied is non-numeric it is called a qualitative variable or an attribute. A qualitative variable is also known as a categorical variable. A categorical variable is not comparable to taking numerical measurements. Observations falling in each category (group, class) can only be counted for examples, gender (either male or female), general knowledge (poor, moderate, or good), religious affiliation, type of automobile owned, city of birth, eye color (red, green, blue, etc), etc. Qualitative variables are often summarized in charts graphs etc. Other examples are what percent of the total number of cars sold last month were Suzuki, what percent of the population has blue eyes?

Quantitative data result from a process that quantifies, such as how much or how many. These quantities are measured on a numerical scale. For example, weight, height, length, and volume.

When the variables studied can be reported numerically, the variable is called a quantitative variable. e.g. the age of the company president, the life of an automobile battery, the number of children in a family, etc. Quantitative variables are either discrete or continuous.

Note that some data can be classified as either qualitative or quantitative, depending on how it is used. If a numerical is used as a label for identification, then it is qualitative; otherwise, it is quantitative. For example, if a serial number on a car is used to identify the number of cars manufactured up to that point then it is a quantitative measure. However, if this number is used only for identification purposes then it is qualitative data.

#### Binary Data

The binary data has only two possible values/states; such as, defected or non-defective, yes or no, and true or false, etc. If both of the values are equally important then it is binary symmetric data (for example, gender). However, if both of the values are not equally important then it can be called binary asymmetric data (for example, result: pass or fail, cancer detected: yes or no).

For quantitative data, a count will always give discrete data, for example, the number of leaves on a tree. On the other hand, a measure of a quantity will usually be continuous, for example, weigh 160 pounds, to the nearest pound. This weight could be any value in the interval say 159.5 to 160.5.

The following are some examples of Qualitative Data. Note that the outcomes of all examples of Qualitative Variables are non-numeric.

• The type of payment (cheque, cash, or credit) used by customers in a store
• The color of your new cell phone
• The make of the types on your car

The following are some examples of Quantitative Data. Note that the outcomes of all examples of Quantitative Variables are numeric.

• The age of the customer in a stock
• The length of telephone calls recorded at a switchboard
• The cost of your new refrigerator
• The weight of your watch
• The air pressure in a tire
• the weight of a shipment of tomatoes
• The duration of a flight from place A to B

Learn about the Measures of Central Tendency

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## Data Transformation (Variable Transformation)

The data transformation is a rescaling of the data using a function or some mathematical operation on each observation. When data are very strongly skewed (negative or positive), we sometimes transform the data so that they are easier to model. In another way, if variable(s) does not fit a normal distribution then one should try a DatavTransformation to fit the assumption of using a parametric statistical test.

The most common data transformation is log (or natural log) transformation, which is often applied when most of the data values cluster around zero relative to the larger values in the data set and all of the observations are positive.

## Data Transformation Techniques

Variable transformation can also be applied to one or more variables in scatter plot, correlation, and regression analysis to make the relationship between the variables more linear; hence it is easier to model with a simple method. Other transformations than log are square root, reciprocal, etc.

### Reciprocal Transformation

The reciprocal transformation $x$ to $\frac{1}{x}$ or $(-\frac{1}{x})$ is a very strong transformation with a drastic effect on the shape of the distribution. Note that this transformation cannot be applied to zero values, but can be applied to negative values. Reciprocal transformation is not useful unless all of the values are positive and reverses the order among values of the same sign i.e. largest becomes smallest etc.

### Logarithmic Transformation

The logarithm $x$ to log (base 10) (or natural log, or log base 2) is another strong transformation that affects the shape of the distribution. Logarithmic transformation is commonly used for reducing right skewness, but cannot be applied to negative or zero values.

### Square Root Transformation

The square root x to $x^{\frac{1}{2}}=\sqrt(x)$ transformation has a moderate effect on the distribution shape and is weaker than the logarithm. Square root transformation can be applied to zero values but not negative values.

The purpose of data transformation are:

• Convert data from one format or structure to another (like changing a messy spreadsheet into a table).
• Clean and prepare data for analysis (fixing errors, inconsistencies, and missing values).
• Standardize data for easier integration and comparison (making sure all your data uses the same units and formats).

### Goals of transformation

The goals of transformation may be

• one might want to see the data structure differently
• one might want to reduce the skew that assists in modeling
• one might want to straighten a nonlinear (curvilinear) relationship in a scatter plot. In other words, a transformation may be used to have approximately equal dispersion, making data easier to handle and interpret

There are many techniques used in data transformation, these techniques are:

• Cleaning and Filtering: Identifying and removing errors, missing values, and duplicates.
• Data Normalization: Ensuring data consistency across different fields.
• Aggregation: Summarizing data by combining similar values.

The Benefits of data tranformation and data clean are:

• Improved data quality: Less errors and inconsistencies lead to more reliable results.
• Easier analysis: Structured data is easier to work with for data analysts and scientists.
• Better decision-making: Accurate insights from clean data lead to better choices.

Data transformation is a crucial step in the data pipeline, especially in tasks like data warehousing, data integration, and data wrangling.

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