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Weighted mean · 4 marks

Observe the following infographic.

  1. (i) Identify the correct inference(s) from the statements given.
    (a) More schools have a playground in Haryana compared to Uttarakhand.
    (b) Approximately every 2 out of 3 schools in Arunachal Pradesh have a playground.
    (c) Punjab has the highest number of schools with a playground.
    (d) Suppose it is given that Maharashtra has more schools than Telangana. Then the number of schools having a playground is more in Maharashtra.
  2. (ii) Using the information given, can we find the nation-wide percentage of schools with a playground? If not, what additional information is needed?
What percentage of schoolshave a playground?(the map’s values, listed from highest to lowest)DNH & Daman & Diu99.8%Punjab98%Lakshadweep97%Delhi96%Andaman & Nicobar96%Maharashtra93%Puducherry92%Haryana90%Goa89%Gujarat88%Kerala88%Himachal Pradesh87%Manipur87%Tamil Nadu87%Sikkim86%Madhya Pradesh81%Karnataka81%Rajasthan80%Mizoram80%Chhattisgarh79%Andhra Pradesh79%Uttarakhand78%Assam78%Tripura78%Ladakh76%Uttar Pradesh76%Telangana74%Nagaland73%Jharkhand71%Arunachal Pradesh68%West Bengal66%Odisha64%Jammu & Kashmir62%Bihar50%Meghalaya44%DNH = Dadra & Nagar Haveli. Source: UDISE 2021–22
Answer: (i) Correct inferences: (b) and (d). (a) and (c) compare numbers of schools, which percentages alone cannot do. (ii) No; we also need the number of schools in each state (or the number with playgrounds), to take a weighted mean.

Step-by-step solution

Idea: A percentage tells us the share within a state, not the number of schools. Comparing actual numbers across states needs the totals; a nation-wide percentage is the weighted mean of the state percentages, weighted by the number of schools.

(i) Identify the correct inference(s) from the statements given.
(a) More schools have a playground in Haryana compared to Uttarakhand.
(b) Approximately every 2 out of 3 schools in Arunachal Pradesh have a playground.
(c) Punjab has the highest number of schools with a playground.
(d) Suppose it is given that Maharashtra has more schools than Telangana. Then the number of schools having a playground is more in Maharashtra.

  1. (a) Haryana 90% vs Uttarakhand 78% compares shares; if Uttarakhand had many more schools it could still have more with playgrounds. Cannot be inferred.½ mark
  2. (b) Arunachal Pradesh 68% ≈ 23 (66.7%). Correct.½ mark
  3. (c) Punjab has a high percentage (98%), but ‘highest number’ needs the number of schools; also DNH & Daman & Diu has a higher percentage (99.8%). Cannot be inferred.½ mark
  4. (d) Maharashtra has more schools and a higher percentage (93% > 74%), so 0.93 × (more schools) > 0.74 × (fewer schools). Correct.½ mark
(b) and (d)

(ii) Using the information given, can we find the nation-wide percentage of schools with a playground? If not, what additional information is needed?

  1. No. The nation-wide percentage is not the simple average of the state percentages, because states have very different numbers of schools.1 mark
  2. We need the number of schools in each state. Then nation-wide % = Σ(schools in state × % in state)Σ(schools in state), a weighted mean (or simply total schools with playgrounds ÷ total schools × 100).1 mark
No; the number of schools in each state is needed
(i) Only (b) and (d) are correct inferences. (ii) No. The nation-wide percentage is a weighted mean of the state percentages, so we need the number of schools in each state (weights).

Answer to write in the exam

(i)

(a) only percentages known, not numbers ⇒ cannot infer

(b) 68% ≈ 23 ⇒ correct

(c) number of schools not known ⇒ cannot infer

(d) M > T and 93% > 74% ⇒ 0.93M > 0.74T ⇒ correct

∴ Correct inferences: (b) and (d)

(ii)

States have different numbers of schools

Nation-wide % = Σ nipiΣ ni (weighted mean)

∴ No; we need the number of schools in each state

Common mistakes that cost marks

  • Accepting (a) because 90% > 78%. A higher percentage does not mean more schools.
  • Reading ‘highest percentage’ as ‘highest number’ in (c).
  • Averaging all the state percentages to get a national figure.

How this can come in the exam

MCQ (1 mark)

State P has 1000 schools, 80% with libraries; State Q has 3000 schools, 60% with libraries. The percentage for P and Q together is

  1. 70%
  2. 65%
  3. 75%
  4. 68%
Show answer

(B) 65%
800 + 18004000 × 100 = 65%.

Try one yourself

District A has 200 schools (90% with toilets for girls) and District B has 600 schools (70%). Which district has more such schools, and what is the combined percentage?

Show answer

A: 180, B: 420, so B has more. Combined: 600800 × 100 = 75%.

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