A continuous variable can take any numerical value, including integral and fractional values, and its values can be broken down into infinite gradations. Height is cited as an example.

To restore continuity, the difference (gap) between the upper limit of the first class and the lower limit of the second class is found and divided by two. This value is then subtracted from all lower limits and added to all upper limits.

While both inclusive and exclusive methods can be used for discrete variables, for continuous variables, exclusive class intervals (or adjusted inclusive intervals) are necessary to maintain continuity, though the source notes that inclusive class intervals are used “very often” for continuous variables (referring to methods requiring adjustment like those in Table 3.4/3.5). Given the options, exclusive tends to be the standard method taught for pure continuous data without gaps, but since the source uses the ‘inclusive’ method followed by adjustment for continuous data (Table 3.4/3.5), we choose the type that is defined to include both limits. However, since exclusive intervals (like 10-20, 20-30) inherently manage continuity better, let’s prioritize the need for continuity in continuous variables. The inclusive method with adjustment restores continuity. Standard practice is to use methods that cover all points. The source says, “In the case of continuous variables, inclusive class intervals are used very often”. Let’s stick to the direct statement in the source.

In the Inclusive Method, values equal to the lower and upper limits of a class are included in the frequency of that same class.

The class intervals like 30 Kg – 39.999… Kg are understood as “30 Kg and above and under 40 Kg”.

After collecting data (learnt in the previous chapter), the next step is to organise and present them in a classified form. Statistical analysis happens after classification.

Characteristics like height, weight, age, income, and marks are quantitative in nature. When collected data of such characteristics are grouped into classes, it becomes a Quantitative Classification.

A Bivariate Frequency Distribution is defined as the frequency distribution of two variables.

The number of classes is usually between six and fifteen.

In the Exclusive Method, an item equal to either the upper or the lower class limit is excluded from the frequency of that class. Typically, in classes like 10-20, 20-30, etc., the upper limit (20) is excluded from the first class and included in the subsequent class (20-30) to maintain continuity.

Unequal sized intervals are used when the range is very high (e.g., income) because equal, moderate intervals would result in too many classes, and large intervals would suppress information on small or very high levels.

Once data is grouped into classes, an individual observation has no significance. All values in a class are assumed to be equal to the class mark, leading to a loss of information regarding the actual values of the observations.

The counting of class frequency is done by tally marks against the particular class. Tallies are counted as groups of five, where four are put as //// and the fifth is placed across them as $begin{smallmatrix}text{////}\text{/}end{smallmatrix}$.

Once data is grouped into classes, individual observations are not used in further calculations. Instead, the class mark (middle value) is used to represent the class.

The purpose of classifying raw data is to bring order in them so that they can be subjected to further statistical analysis easily.

Raw data is unclassified and often very large and cumbersome to handle, making it difficult to draw meaningful conclusions or yield to statistical methods easily.

Class limits should be definite and clearly stated. Generally, open-ended classes such as “70 and over” or “less than 10” are not desirable.

Chronological Classification is achieved when data is grouped according to time, such as years, quarters, months, weeks, etc..

For a discrete variable, the classification of its data is known as a Frequency Array, where frequencies correspond to each of its integral values.

A discrete variable changes only by finite “jumps” and cannot take any intermediate value between them. The number of students in a class is an example, as “half of a student” is absurd.

Characteristics like nationality, literacy, religion, and gender are Qualities or Attributes that cannot be measured quantitatively. Classification based on these is called Qualitative Classification.

Class Interval or Class Width is defined as the difference between the upper class limit and the lower class limit.

To obtain the frequency curve, the class marks are plotted on the X-axis and frequency is plotted on the Y-axis.

Range is the difference between the largest and the smallest values of the variable.

Frequency Curve is a graphic representation of a frequency distribution.

Class Frequency means the number of values in a particular class.

Spatial Classification is when data is classified with reference to geographical locations such as countries, states, cities, districts, etc..

In raw data tables (like Table 3.1), the numbers are not arranged in any order, necessitating arranging them (e.g., ascending or descending order) first, which is a tedious task, especially if the number of observations is large (100 or 1,000 students).

Frequency expressed as proportion or percentage of total frequency is known as relative frequency.

The Class Mid-Point or Class Mark is the middle value of a class and is ascertained by the formula: (Upper Class Limit + Lower Class Limit)/2.