Organisation of Data

• (i) The class midpoint is equal to:
  (a) The average of the upper class limit and the lower class limit.
  Reason: Midpoint = $\frac{L_1 + L_2}{2}$
• (ii) The frequency distribution of two variables is known as:
  (b) Bivariate Distribution
  Reason: ‘Bi’ means two; univariate involves one variable.
• (iii) Statistical calculations in classified data are based on:
  (d) the class midpoints
  Reason: In grouped data, the exact values are lost, so the midpoint is assumed to represent all values in that class.
• (iv) Range is the:
  (a) difference between the largest and the smallest observations
  Reason: Range = Maximum Value – Minimum Value.

Yes, classification condenses raw data, facilitates comparison, and helps in studying relationships.
Example: In a kitchen, spices, pulses, and utensils are kept in separate, classified containers/shelves. This organization saves time and effort when cooking, compared to a situation where everything is mixed in one big pile (Raw Data).

A variable is a characteristic or phenomenon which is capable of being measured and changes its value over time (e.g., Height, Weight).

Discrete Variable	Continuous Variable
Takes only specific/exact values (jumps).	Can take any value within a range (fractions).
No intermediate values are possible.	All intermediate values are possible.
Example: Number of students (1, 2, 3… not 1.5).	Example: Height (5.2, 5.21, 5.25 ft).

• Exclusive Method: The upper limit of one class is the lower limit of the next class (e.g., 10-20, 20-30). The upper limit is excluded from that class (20 goes to the 20-30 group). Used for continuous variables.
• Inclusive Method: The upper limit of one class does not equal the lower limit of the next (e.g., 10-19, 20-29). The upper limit is included in that class. Used for discrete variables.

When raw data is grouped into classes, the individual identity of the observations is lost. For statistical calculations, we assume all values in a class are equal to the class midpoint. The actual values (e.g., whether a value was 11 or 19 in the 10-20 class) are no longer known. This disappearance of exact details is called ‘loss of information’.

Yes, classified data is generally better because:

• Simplification: It condenses a huge mass of unorganized data into a manageable form.
• Comparability: It makes comparison between different sets of data easier.
• Pattern Recognition: It highlights characteristics (like clustering of values) that raw data hides.

• Univariate Distribution: A frequency distribution based on a single variable (e.g., Marks of students).
• Bivariate Distribution: A frequency distribution based on two variables simultaneously (e.g., Marks in Maths and Marks in Physics for the same students). It typically uses a two-way table.

(Data from NCERT Table 3.2: Max=5090, Min=1007, N=50)

(i) Range:
Range = Largest Value – Smallest Value = $5090 – 1007 = 4083$.

(ii) Frequency Distribution (Interval = 500):

Expenditure (₹)	No. of Households (f)
1000 – 1500	20
1500 – 2000	13
2000 – 2500	6
2500 – 3000	5
3000 – 3500	2
3500 – 4000	1
4000 – 4500	2
4500 – 5000	0
5000 – 5500	1
Total	50

(iii) Find the number of households:
(a) Less than ₹ 2000: (1000-1500) + (1500-2000) = $20 + 13 = 33$.
(b) More than ₹ 3000: Sum of all classes > 3000 = $2 + 1 + 2 + 0 + 1 = 6$.
(c) Between ₹ 1500 and ₹ 2500: (1500-2000) + (2000-2500) = $13 + 6 = 19$.

No. of Cell Phones (X)	Tally Marks	Frequency (f)
0	|	1
1	|||| ||	7
2	|||| |||| ||||	15
3	|||| |||| ||	12
4	||||	5
5	||	2
6	||	2
7	|	1
Total		45

(Range: Min=1, Max=37. N=60)

Class Interval (Inclusive)	Tally Marks	Frequency (f)
1 – 7	|||| |||| ||||	15
8 – 14	|||| |||| ||	12
15 – 21	|||| |||| |||| |	16
22 – 28	|||| |||| |	11
29 – 35	||||	4
36 – 42	||	2
Total		60

Word Breakdown:
The (3), quick (5), brown (5), fox (3), jumps (5), over (4), the (3), lazy (4), dog (3).

Frequency Array:

No. of Letters (X)	Tally Marks	Frequency (f)
3	||||	4
4	||	2
5	|||	3
Total		9