Class | Subject | Book | Chapter | Session | Questions covered |
|---|---|---|---|---|---|
XI | Economics | Statistics for Economics (NCERT) | 3 – Organisation of Data | 2026-27 | 11 of 11 |
Chapter at a glance
Topic | Key facts |
|---|---|
Classification | Arranging raw data into groups: chronological, spatial, qualitative, quantitative |
Variable | Characteristic that varies; discrete (whole values, e.g. number of children) or continuous (any value in a range, e.g. height) |
Frequency distribution | Classes with their frequencies; class limits, class interval, class mark = (upper limit + lower limit) ÷ 2 |
Exclusive method | Upper limit of one class = lower limit of next; upper limit excluded (e.g. 0–10, 10–20) |
Inclusive method | Both limits included (e.g. 0–9, 10–19) |
Loss of information | Classified data treat all values in a class as equal to the class mark |
Bivariate distribution | Frequency distribution of two variables together |
Exercise Questions
Which of the following alternatives is true?
Answer
- The class midpoint is equal to:
- The average of the upper class limit and the lower class limit.
- The product of upper class limit and the lower class limit.
- The ratio of the upper class limit and the lower class limit.
- None of the above.
- The frequency distribution of two variables is known as
- Univariate Distribution (b) Bivariate Distribution (c) Multivariate Distribution (d) None of the above
- Statistical calculations in classified data are based on
- the actual values of observations (b) the upper class limits (c) the lower class limits (d) the class midpoints
- Range is the
- difference between the largest and the smallest observations (b) difference between the smallest and the largest observations (c) average of the largest and the smallest observations (d) ratio of the largest to the smallest observation
Answer
- (a) The class midpoint is the average of the upper and lower class limits.
- (b) Bivariate distribution.
- (d) The class midpoints represent each class in calculations.
- (a) Range = largest observation − smallest observation.
- a (ii) b (iii) d (iv) a
Can there be any advantage in classifying things? Explain with an example from your daily life.
Answer
Answer
Yes. Classification groups similar things together, which makes it easy to find, compare and analyse them, and saves time.
Example: in a library, books are arranged by subject (Science, History, Literature), and within subjects by author. A student can quickly find the needed book. Similarly, at home we keep clothes, utensils and books in separate places; a grocery store arranges items such as grains, spices and soaps in separate sections.
In statistics, classifying raw data in the same way brings out its main features and makes it suitable for further analysis.
Yes; classification makes things easy to locate and compare, e.g. books arranged subject-wise in a library.
What is a variable? Distinguish between a discrete and a continuous variable.
Answer
Answer
A variable is a characteristic that can be measured and that takes different values for different units (e.g. height, income, marks, number of children).
Discrete variable | Continuous variable |
|---|---|
Takes only specific (whole-number) values, increasing in jumps | Can take any value (integer or fraction) within a range |
No values between two consecutive values | Infinitely many values between any two values |
Example: number of children in a family (1, 2, 3), number of students in a class | Example: height (152.5 cm), weight, temperature, income |
A variable is a measurable characteristic that varies; a discrete variable takes only certain values (number of children), a continuous one takes any value in a range (height).
Explain the ‘exclusive’ and ‘inclusive’ methods used in classification of data.
Answer
Answer
Exclusive method. The upper class limit of one class is the lower class limit of the next class. An observation equal to the upper limit is excluded from that class and included in the next one. Example: 0–10, 10–20, 20–30. A value of 10 goes into 10–20. This method suits continuous variables.
Inclusive method. Both the lower and the upper class limits are included in the class. Example: 0–9, 10–19, 20–29. A value of 10 goes into 10–19. There is a gap between classes, so it suits discrete data. For continuous data the classes can be adjusted by subtracting half the gap from lower limits and adding it to upper limits (e.g. 9.5–19.5).
Exclusive: upper limit of a class excluded and equals the next lower limit (0–10, 10–20). Inclusive: both limits included (0–9, 10–19).
Use the data in Table 3.2 that relate to monthly household expenditure (in Rs) on food of 50 households and
Answer
- Obtain the range of monthly household expenditure on food.
- Divide the range into appropriate number of class intervals and obtain the frequency distribution of expenditure.
- Find the number of households whose monthly expenditure on food is (a) less than Rs 2000 (b) more than Rs 3000 (c) between Rs 1500 and Rs 2500
Answer
(i) Range.
Largest value = ₹5090; smallest value = ₹1007.
Formula: Range = largest value − smallest value
(ii) Frequency distribution (exclusive classes of width ₹500, starting at ₹1000; 9 classes):
Expenditure (₹) | Frequency |
|---|---|
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)
- Less than ₹2000: 20 + 13 = 33 households.
- More than ₹3000: 2 + 1 + 2 + 0 + 1 = 6 households.
- Between ₹1500 and ₹2500: 13 + 6 = 19 households.
- Range = ₹4083. (ii) Nine classes of ₹500 from ₹1000 (frequencies 20, 13, 6, 5, 2, 1, 2, 0, 1). (iii) (a) 33 (b) 6 (c) 19 households.
In a city 45 families were surveyed for the number of Cell phones they used. Prepare a frequency array based on their replies as recorded below.
Answer
1 3 2 2 2 2 1 2 1 2 2 3 3 3 3 3 3 2 3 2 2 6 1 6 2 1 5 1 5 3 2 4 2 7 4 2 4 3 4 2 0 3 1 4 3
Answer
A frequency array is used for a discrete variable: each value is listed with the number of times it occurs.
Number of cell phones | Number of families |
|---|---|
0 | 1 |
1 | 7 |
2 | 15 |
3 | 12 |
4 | 5 |
5 | 2 |
6 | 2 |
7 | 1 |
Total | 45 |
Frequencies for 0–7 phones: 1, 7, 15, 12, 5, 2, 2, 1 (total 45).
What is ‘loss of information’ in classified data?
Answer
Answer
When raw data are grouped into a frequency distribution, the individual observations disappear: we only know how many values lie in each class, not what they are. In further calculations every value in a class is taken to be equal to the class mark (mid-point). For example, all values in the class 10–20 are treated as 15. The details of the actual values are thus lost. This is called loss of information, and it is the price paid for summarising data.
In classified data the actual individual values are lost; all values in a class are represented by its class mark.
Do you agree that classified data is better than raw data? Why?
Answer
Answer
Yes, generally.
- Raw data are large and unorganised; it is hard to see any pattern in them.
- Classified data are concise: they summarise the data and bring out their main features.
- They make comparison and further statistical analysis (averages, dispersion, graphs) easy.
- They save time and effort.
The only drawback is a small loss of information, because individual values are not shown, but the benefits are much greater.
Yes; classified data are concise, reveal patterns and are easy to analyse, though some detail is lost.
Distinguish between univariate and bivariate frequency distribution.
Answer
Answer
Univariate frequency distribution | Bivariate frequency distribution |
|---|---|
Frequency distribution of a single variable | Frequency distribution of two variables simultaneously |
Example: marks of students | Example: sales and advertisement expenditure of companies |
Shown in a simple table with classes and frequencies | Shown in a two-way table: classes of one variable as rows, of the other as columns, frequencies in the cells |
Univariate deals with one variable; bivariate classifies data by two variables together in a two-way table.
Prepare a frequency distribution by inclusive method taking class interval of 7 from the following data.
Answer
28 17 15 22 29 21 23 27 18 12 7 2 9 4 1 8 3 10 5 20 16 12 8 4 33 27 21 15 3 36 27 18 9 2 4 6 32 31 29 18 14 13 15 11 9 7 1 5 37 32 28 26 24 20 19 25 19 20 6 9
Answer
There are 60 observations; smallest = 1, largest = 37. With inclusive classes of width 7 starting from 1:
Class (inclusive) | Frequency |
|---|---|
1–7 | 15 |
8–14 | 12 |
15–21 | 15 |
22–28 | 10 |
29–35 | 6 |
36–42 | 2 |
Total | 60 |
Classes 1–7, 8–14, 15–21, 22–28, 29–35, 36–42 with frequencies 15, 12, 15, 10, 6, 2 (total 60).
“The quick brown fox jumps over the lazy dog”. Examine the above sentence carefully and note the numbers of letters in each word. Treating the number of letters as a variable, prepare a frequency array for this data.
Answer
Answer
Letters in each word: The (3), quick (5), brown (5), fox (3), jumps (5), over (4), the (3), lazy (4), dog (3).
Number of letters | Number of words |
|---|---|
3 | 4 |
4 | 2 |
5 | 3 |
Total | 9 |
3 letters: 4 words; 4 letters: 2 words; 5 letters: 3 words.
Topic checklist
- Raw and classified data; types of classification
- Discrete and continuous variables
- Frequency distribution: class limits, interval, class mark
- Exclusive and inclusive methods
- Loss of information
- Frequency array and bivariate distribution