Chapter 4 — Presentation Of Data

Overview

You exceeded your current quota, please check your plan and billing details. For more information on this error, read the docs: https://platform.openai.com/docs/guides/error-codes/api-errors.

Topics in this chapter

1 topics · tap a topic title to jump straight to it.

🔌1

You exceeded your current quota

Fig 1 — Educational Diagram: You exceeded your current quota

⚡ PHYSICAL LAW / FORMULA

You exceeded your current quota

Key Point: Relative frequency (proportion) of exceedance = count_exceed / N

Meaning: In everyday language the message "You exceeded your current quota" means that an observed value (usage, imports, catch, applications, etc.) has gone beyond a preset limit (quota). In the context of data presentation (Class 11 Economics: Presentation of Data) this phrase is treated as a situation to be described, measured and shown graphically so that the extent and frequency of exceedance are clear.

How to present and analyse such cases:

• Identify the quota (fixed limit) and the observed values. Separate observations into two groups: ≤ quota and > quota (exceeding).
• Use a frequency table to show how many observations exceed the quota and how many do not. Compute relative frequencies and percentages to show proportions.
• If values are continuous (e.g., GB used), use a grouped frequency distribution and show the portion of the distribution that lies above the quota. For censored or truncated data (values recorded only up to the quota), note the censoring and, where necessary, treat them separately in analysis.
• Use cumulative frequency (ogive) to find the number or percentage of observations up to a value; by comparing cumulative frequency at the quota with total N you get the share exceeding it.
• When presenting graphs, mark the quota as a clear horizontal (or vertical) reference line and highlight the exceedance portion for quick visual interpretation.

Practical presentation tips: label axes and quota value, show counts and percentages on bars or pie slices, annotate exact values of exceedance and percent-over-quota, and if required, compute summary measures (mean, median) with and without the exceedances to show their effect.

📌 Examples

• Internet data plan: Quota = 100 GB/month. Users' monthly usages: 70, 85, 120, 95, 135, 60. Frequency table: ≤100 GB: 4 users; >100 GB: 2 users. Percentage exceeding = (2/6)*100 = 33.33%. Percentage over quota for a user with 135 GB = ((135−100)/100)*100 = 35%. Graph: bar chart of individual usages with a horizontal line at 100 GB.
• Import quota: Country quota for a commodity = 1,000 tonnes. Imports over a year (monthly): 90, 120, 85, 110, 125, 95, 105, 130, 140, 80, 115, 105 (in tonnes ×10). Build a grouped frequency distribution; compute cumulative frequency. If cumulative imports by month exceed quota in month 9, mark this on the ogive to show when and by how much the quota was exceeded.
• Fishing quota: Annual quota = 5,000 fish. Boats' catches (counts): many boats reported their catches; 12 boats exceeded the quota individually. Proportion exceeding = number of boats exceeding / total boats. Present as a pie chart: 'Within quota' vs 'Exceeded quota' slices.
• School admissions: Seat quota for a course = 50. Applications received = 130. Present a simple table: Seats = 50, Applicants = 130, Excess applications = 80, Ratio applicants:seats = 130:50 = 2.6:1. Graph: bar diagram comparing seats vs applicants and annotate the exceeded amount (80).

🧮 Formulas

1. \[Relative frequency (proportion) of exceedance = count_exceed / N\]
2. \[Percentage exceeding quota = (count_exceed / N) × 100\]
3. \[Percent over quota for an observation = ((observed − quota) / quota) × 100\]
4. \[Relative frequency of a class = f_i / N (where f_i is class frequency\]

   \[N total observations)\]
5. \[Cumulative frequency (up to class k) = Σ f_i (i = 1 to k)\]
6. \[Mean for grouped data = (Σ f_i × x_i) / N (x_i = class midpoint)\]

Key Concepts

Data

Facts, figures or observations collected for analysis or presentation.

Primary data

Data collected firsthand by the investigator for a specific purpose.

Secondary data

Data already collected by someone else and used by the investigator.

Qualitative data

Non-numeric data describing qualities or categories.

Quantitative data

Numeric data that can be measured or counted.

Discrete data

Quantitative data taking only distinct, separate values (often counts).

Continuous data

Quantitative data that can take any value within a range (measurable).

Classification

Arranging data into groups or classes based on common characteristics.

Tabulation

Organising data systematically in rows and columns to form a table.

Frequency

The number of times a value or class occurs in a data set.

Frequency distribution

A table that shows values/classes along with their frequencies.

Grouped frequency distribution

Frequency distribution where data are combined into class intervals.

Ungrouped frequency distribution

Frequency distribution listing each distinct value and its frequency.

Class interval

A range of values forming one group in a grouped distribution.

Class limits

The smallest (lower) and largest (upper) values included in a class interval.

Class boundaries

The true upper and lower limits of a class after removing gaps between adjacent classes.

Class mark (midpoint)

The central value of a class interval, calculated as (lower limit + upper limit) / 2.

Cumulative frequency

A running total of frequencies up to a given value or class.

Relative frequency

The proportion of total observations represented by a frequency (frequency / total).

Histogram

A graphical representation of a grouped frequency distribution using adjacent bars whose areas are proportional to class frequencies.

1. Distinguish between primary data and secondary data with one example of each. / प्राथमिक आंकड़ों और द्वितीयक आंकड़ों में अंतर बताइए तथा प्रत्येक का एक उदाहरण दीजिए।
   Show answer

   Primary data are collected firsthand by the investigator for a specific purpose (e.g., a survey conducted by a student on classmates' heights), whereas secondary data are already collected by someone else and reused (e.g., census figures taken from a government report). / प्राथमिक आंकड़े अन्वेषक द्वारा किसी विशेष उद्देश्य के लिए स्वयं एकत्रित किए जाते हैं (जैसे विद्यार्थी द्वारा सहपाठियों की ऊँचाई का सर्वेक्षण), जबकि द्वितीयक आंकड़े किसी अन्य द्वारा पहले से एकत्रित होते हैं और पुनः उपयोग किए जाते हैं (जैसे सरकारी रिपोर्ट से ली गई जनगणना संख्याएँ)।

2. Why are quantitative data classified into 'discrete' and 'continuous' types? Give one example of each. / मात्रात्मक आंकड़ों को 'विविक्त' और 'संतत' प्रकारों में क्यों वर्गीकृत किया जाता है? प्रत्येक का एक उदाहरण दीजिए।
   Show answer

   They are distinguished because discrete data take only distinct, separate values (often counts, e.g., number of children in a family), while continuous data can take any value within a range and are measurable (e.g., height or weight). / उन्हें इसलिए अलग किया जाता है क्योंकि विविक्त आंकड़े केवल पृथक, अलग मान लेते हैं (प्रायः गणना, जैसे परिवार में बच्चों की संख्या), जबकि संतत आंकड़े किसी परास के भीतर कोई भी मान ले सकते हैं और मापने योग्य होते हैं (जैसे ऊँचाई या भार)।

3. A data plan has a quota of 100 GB. Six users record usages 70, 85, 120, 95, 135, 60 GB. Find the percentage of users exceeding the quota and the percent-over-quota for the 135 GB user. / एक डेटा प्लान की सीमा 100 GB है। छह उपयोगकर्ता 70, 85, 120, 95, 135, 60 GB उपयोग दर्ज करते हैं। सीमा से अधिक उपयोग करने वाले उपयोगकर्ताओं का प्रतिशत तथा 135 GB उपयोगकर्ता के लिए सीमा-से-अधिक प्रतिशत ज्ञात कीजिए।
   Show answer

   Two users (120, 135) exceed the quota, so percentage exceeding = (2/6)×100 = 33.33%; percent over quota for the 135 GB user = ((135−100)/100)×100 = 35%. / दो उपयोगकर्ता (120, 135) सीमा पार करते हैं, अतः अधिक करने वालों का प्रतिशत = (2/6)×100 = 33.33%; 135 GB उपयोगकर्ता के लिए सीमा-से-अधिक प्रतिशत = ((135−100)/100)×100 = 35%।

4. Define class mark (midpoint) and compute it for the class interval 20–30. / वर्ग चिह्न (मध्य बिंदु) को परिभाषित कीजिए तथा वर्ग अंतराल 20–30 के लिए इसकी गणना कीजिए।
   Show answer

   The class mark is the central value of a class interval, calculated as (lower limit + upper limit)/2; for 20–30 it is (20+30)/2 = 25. / वर्ग चिह्न किसी वर्ग अंतराल का केंद्रीय मान है, जिसे (निम्न सीमा + उच्च सीमा)/2 से निकाला जाता है; 20–30 के लिए यह (20+30)/2 = 25 है।

5. Explain how an ogive (cumulative frequency curve) can be used to find how many observations exceed a given quota value. / समझाइए कि किसी दी गई सीमा मान से अधिक कितने प्रेक्षण हैं, यह ज्ञात करने के लिए तोरण (संचयी बारंबारता वक्र) का उपयोग कैसे किया जाता है।
   Show answer

   Plot cumulative frequency against the variable, draw a vertical line at the quota value and read off the cumulative frequency up to it; then subtract this from total N (N − cumulative) to get the number of observations exceeding the quota. / चर के सापेक्ष संचयी बारंबारता आलेखित कीजिए, सीमा मान पर एक ऊर्ध्वाधर रेखा खींचिए और उस तक की संचयी बारंबारता पढ़िए; फिर इसे कुल N से घटाइए (N − संचयी) ताकि सीमा से अधिक प्रेक्षणों की संख्या मिल जाए।

6. Why is a histogram preferred over a simple bar diagram for showing a grouped frequency distribution? / समूहीकृत बारंबारता बंटन दिखाने के लिए सरल दंड आरेख की तुलना में आयतचित्र को क्यों प्राथमिकता दी जाती है?
   Show answer

   A histogram uses adjacent bars whose areas are proportional to class frequencies, so it correctly represents continuous data and class widths, whereas a bar diagram with gaps suits discrete categories and does not show class intervals or area-based frequency. / आयतचित्र में संलग्न दंड होते हैं जिनके क्षेत्रफल वर्ग बारंबारताओं के समानुपाती होते हैं, अतः यह संतत आंकड़ों और वर्ग चौड़ाइयों को सही प्रकार दर्शाता है, जबकि अंतराल वाले दंड आरेख विविक्त श्रेणियों के लिए उपयुक्त हैं और वर्ग अंतराल या क्षेत्र-आधारित बारंबारता नहीं दिखाते।

7. A course has a seat quota of 50 but receives 130 applications. Compute the excess applications and the applicants-to-seats ratio. / एक पाठ्यक्रम में 50 सीटों की सीमा है परंतु 130 आवेदन प्राप्त होते हैं। अतिरिक्त आवेदन तथा आवेदक-से-सीट अनुपात की गणना कीजिए।
   Show answer

   Excess applications = 130 − 50 = 80, and the applicants-to-seats ratio = 130:50 = 2.6:1. / अतिरिक्त आवेदन = 130 − 50 = 80, तथा आवेदक-से-सीट अनुपात = 130:50 = 2.6:1।

8. What is relative frequency, and why is it useful when comparing classes of different total sizes? / सापेक्ष बारंबारता क्या है, और भिन्न कुल आकार वाले वर्गों की तुलना करते समय यह उपयोगी क्यों है?
   Show answer

   Relative frequency is the proportion of total observations in a class, calculated as f_i / N; it is useful because expressing frequencies as proportions or percentages allows fair comparison even when two distributions have different total numbers of observations. / सापेक्ष बारंबारता किसी वर्ग में कुल प्रेक्षणों का अनुपात है, जिसे f_i / N से निकाला जाता है; यह उपयोगी है क्योंकि बारंबारताओं को अनुपात या प्रतिशत के रूप में व्यक्त करने से दो बंटनों की कुल प्रेक्षण संख्या भिन्न होने पर भी निष्पक्ष तुलना संभव होती है।

Foundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.