Statistical Description of Data: CA Foundation Quantitative Aptitude

What this chapter covers

Statistical Description of Data: topics in the order to study them

1. 1Collection and Types of DataStart here because every later idea depends on knowing what data is and where it comes from.
2. 2Scales of Measurement and Types of VariablesIt builds on data types and gives you the vocabulary (nominal, ordinal, interval, ratio; discrete, continuous) used in all later topics.
3. 3Classification and Frequency DistributionThis is the core skill of the chapter: grouping data into classes and counting frequencies.
4. 4Cumulative and Relative Frequency DistributionsThese are built directly from a frequency table, so learn them right after it.
5. 5Tabular Presentation of DataOnce you can make a frequency table, learn the parts and rules of a good table.
6. 6Diagrammatic Presentation of DataBar, pie and similar diagrams are easier once the table is clear, and they are mostly for categorical data.
7. 7Graphical Presentation: Histogram, Polygon and OgiveStudy it last because it needs frequency, cumulative frequency and class marks together.

How to prepare Statistical Description of Data

1. Read the first two topics and make a one-page list of terms with a one-line example each, such as primary data, discrete variable and ordinal scale.
2. Practise sorting examples into types. For each item, ask: is it qualitative or quantitative, and which scale does it use?
3. Build frequency tables from raw data by hand. Use tally marks, find the range, choose class widths, and check that the frequencies add up to the total count.
4. For each table, also write the class marks, cumulative frequencies (less than and more than) and relative frequencies. Check that relative frequencies add up to 1 or 100%.
5. Learn the parts of a table and which diagram suits which data. Then sketch a histogram, polygon and ogive from one dataset so you remember how each is built.
6. Solve MCQs in timed sets. Mark questions where you were unsure, and review the definition behind each one.
7. In the last days, revise your term list and formula notes only, then re-attempt your marked questions.

Common mistakes in Statistical Description of Data

• Mixing up discrete and continuous variables, for example calling the number of students continuous.

  Fix: Ask: can it take values like 2.5? If counts must be whole numbers, it is discrete. Measures such as height or weight are continuous.

• Treating all ordered categories as interval or ratio scale data.

  Fix: If only the order matters and gaps are not equal or meaningful, it is ordinal. Look for a true zero before choosing ratio.

• Using wrong class limits or class marks, especially with inclusive classes.

  Fix: Check the class type first. For inclusive classes, convert to boundaries before drawing a histogram or ogive, and find class width from the boundaries. Then compute class marks from the correct limits.

• Errors in cumulative frequency, such as adding a class twice or mixing less than and more than.

  Fix: Write the running total in a separate column. The last less than value must equal the total frequency. Match the cumulative type to the question.

• Choosing the wrong chart for the data.

  Fix: Link each chart to a data type: bars and pie charts for categories, histogram and polygon for grouped continuous data, ogive for cumulative frequency.

• Guessing definition-based MCQs without recall, then losing marks to negative marking.

  Fix: Revise your term list often. If you can eliminate at least two of the four options, guess between the remaining two. If you cannot narrow the options down, skip the question.

Last-day revision: Statistical Description of Data

• ✓Primary data is collected first-hand by you; secondary data is taken from existing sources.
• ✓A qualitative variable describes a quality or category; a quantitative variable is measured in numbers.
• ✓A discrete variable takes separate values (often counts); a continuous variable can take any value in a range.
• ✓The four scales, from weakest to strongest, are nominal, ordinal, interval and ratio. Only the ratio scale has a true zero.
• ✓Range = largest value − smallest value.
• ✓Class width = upper limit − lower limit (for exclusive classes). For inclusive classes, class width = upper boundary − lower boundary, not the difference between the stated limits, so convert to boundaries first. Class mark = (lower limit + upper limit) ÷ 2, which gives the same value for inclusive classes whether you use limits or boundaries.
• ✓Frequency is the count in a class; the frequencies add up to the total number of observations.
• ✓Relative frequency = class frequency ÷ total frequency; these add up to 1.
• ✓A less than cumulative frequency adds frequencies from the lowest class up; a more than one adds from the highest class down.
• ✓A histogram shows a continuous frequency distribution as touching bars; a frequency polygon joins the class-mark points.
• ✓An ogive plots cumulative frequency against class boundaries; the two ogives (less than and more than) cross at the median.
• ✓Bar diagrams and pie charts suit categorical data; histograms and ogives suit grouped continuous data.

Statistical Description of Data practice questions

• In the layout of a statistical table, the descriptive headings given to the rows are collectively known as:
• For a firm's 50 employees, the 'more than' cumulative frequencies of age (in years) are: more than 10: 50; more than 20: 42; more than 30: 3…
• The monthly budget of Mr. Iyer's household is ₹1,80,000, of which ₹54,000 is spent on education. In a pie chart of the budget, the angle at …
• A frequency table uses inclusive class intervals 10–19, 20–29, 30–39 and so on. What are the true class boundaries of the class 20–29?
• The mean of a sample of 8 measurements is 24. If one measurement is removed, the mean of the remaining 7 measurements becomes 22. What was t…
• A frequency distribution of daily wages (in ₹) of workers in a factory is given as: Wages (₹): 200–300, 300–400, 400–500, 500–600 Frequency…
• The Sharma family has a monthly income of ₹60,000 and spends ₹24,000 on food. If the budget is shown in a pie chart, what central angle repr…
• A dataset of monthly sales (in ₹'000) for a retail store shows the following values: 45, 52, 48, 55, 50. What is the range of this sales dat…