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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Contingency Tables

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Introduction to Contingency Tables

Contingency Tables also called cross tables or two-way frequency tables describe the relationship between several categorical (qualitative) variables. A bivariate relationship is defined by the joint distribution of the two associated random variables.

Contingency Tables

Let $X$ and $Y$ be two categorical response variables. Let variable $X$ have $I$ levels and variable $Y$ have $J$. The possible combinations of classifications for both variables are $I\times J$. The response $(X, Y)$ of a subject randomly chosen from some population has a probability distribution, which can be shown in a rectangular table having $I$ rows (for categories of $X$) and $J$ columns (for categories of $Y$).

The cells of this rectangular table represent the $IJ$ possible outcomes. Their probability (say $\pi_{ij}$) denotes the probability that ($X, Y$) falls in the cell in row $i$ and column $j$. When these cells contain frequency counts of outcomes, the table is called a contingency or cross-classification table and it is referred to as a $I$ by $J$ ($I \times J$) table.

Joint and Marginal Distribution

The probability distribution {$\pi_{ij}$} is the joint distribution of $X$ and $Y$. The marginal distributions are the rows and columns totals obtained by summing the joint probabilities. For the row variable ($X$) the marginal probability is denoted by $\pi_{i+}$ and for column variable ($Y$) it is denoted by $\pi_{+j}$, where the subscript “+” denotes the sum over the index it replaces; that is, $\pi_{i+}=\sum_j \pi_{ij}$ and $\pi_{+j}=\sum_i \pi_{ij}$ satisfying

$l\sum_{i} \pi_{i+} =\sum_{j} \pi_{+j} = \sum_i \sum_j \pi_{ij}=1$

Note that the marginal distributions are single-variable information, and do not pertain to association linkages between the variables.

Contingency Tables, Cross Tabulation

In (many) contingency tables, one variable (say, $Y$) is a response, and the other $X$) is an explanatory variable. When $X$ is fixed rather than random, the notation of a joint distribution for $X$ and $Y$ is no longer meaningful. However, for a fixed level of $X$, the variable $Y$ has a probability distribution. It is germane to study how this probability distribution of $Y$ changes as the level of $X$ changes.

Contingency Table Uses

  • Identify relationships between categorical variables.
  • See if one variable is independent of the other (i.e. if the frequency of one category is the same regardless of the other variable’s category).
  • Calculate probabilities of specific combinations occurring.
  • Often used as a stepping stone for further statistical analysis, like chi-square tests, to determine if the observed relationship between the variables is statistically significant.

Read More about Contingency Tables

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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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