CMA Foundation · Fundamentals of Business Mathematics and Statistics

Statistical Representation of Data for CMA Foundation

Statistical representation of data means organising raw figures into classes, tables and frequency distributions, then showing them through diagrams, histograms, frequency polygons and ogives. To solve questions, find the class width, class limits and boundaries, build the table with care, and read values off the correct graph.

What this chapter covers

This chapter teaches you how to turn a pile of raw numbers into something you can read at a glance. You start by classifying data and putting it in tables. Then you group it into a frequency distribution with class intervals. Finally you draw it as bar diagrams, pie charts, histograms, frequency polygons and ogives.

Most questions in Paper 3 are MCQs, so the chapter is tested through definitions, small calculations and graph reading. Expect to be asked for a class mark, a class width, a cumulative frequency, the correct graph for a given data type, or which curve gives the median.

The chapter is the base for the statistics half of the paper. Measures of central tendency, dispersion and correlation all start from a frequency table. If you can build and read one quickly, those later chapters become much easier.

The chapter is mostly concept and short calculation, with very little heavy maths, so it is a good place to score reliably. The same ideas, such as class boundaries, cumulative frequency and class mark, are reused in mean, median, mode and quartile questions. Since there is no negative marking and each MCQ carries 2 marks, solid clarity here gives you quick marks and saves time for the harder numerical chapters.

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

1Classification and Tabulation of DataStart here because you must understand types of data, classification bases and the parts of a table before you group numbers.
2Frequency Distribution and Class IntervalsThis is the core skill: class limits, boundaries, width, class mark and cumulative frequency feed every later topic.
3Diagrammatic Representation of DataBar diagrams and pie charts are simple and use only the table ideas you already know, so they come before graphs on a scale.
4Graphical Representation: Histogram and Frequency PolygonThese need correct class boundaries and widths from the frequency distribution, so study them after it.
5Ogives and Locating Median and Quartiles GraphicallyOgives need cumulative frequencies and graph skills, and they link directly to median and quartiles, so keep them last.

How to prepare Statistical Representation of Data

Aim to build one skill at a time and practise each on small, clean numbers. Do not just read; draw and calculate.

Learn the definitions first: primary and secondary data, classification bases, and the parts of a table. Write each in one line.
Take a small raw data set of about 20 values and make a frequency table by tally marks. Do this a few times until it is fast.
Practise class limits versus class boundaries. Convert inclusive classes such as 10-19, 20-29 into exact boundaries by adjusting half the gap.
Compute class width, class mark, relative frequency and both cumulative frequencies for several tables.
Sketch a histogram, a frequency polygon and both ogives by hand. Note which axis holds what and which points you plot.
Practise reading the median and quartiles from the intersection of the two ogives or from n/2, n/4 and 3n/4 on the cumulative scale.
Finish with timed MCQs on concepts such as which graph suits which data. Review every wrong answer and note the reason.

Common mistakes in Statistical Representation of Data

Using class limits instead of class boundaries when drawing a histogram or ogive.
Fix: Always convert to boundaries first (9.5-19.5, 19.5-29.5) before finding width, plotting or locating the median.
Calculating class width from limits of inclusive classes, giving 9 instead of 10.
Fix: Subtract the boundaries, or subtract consecutive lower limits, 20 − 10 = 10.
Plotting ogive points at class mid-points.
Fix: Remember that ogives use class boundaries (upper for less-than, lower for more-than) and polygons use mid-points.
Making cumulative frequency errors by skipping a class or adding the wrong way.
Fix: Check that the last cumulative frequency equals the total frequency. For more-than, the first value must equal the total.
Drawing a histogram with unequal class widths using raw frequencies as heights.
Fix: Check the widths. If they differ, use frequency density (or adjusted frequency) as the height.
Choosing the wrong diagram for the data in MCQs.
Fix: Link them: pie chart for parts of a whole, histogram for continuous frequency data, bar diagram for discrete or categorical comparison, ogive for cumulative values.

Last-day revision: Statistical Representation of Data

Class width = upper boundary − lower boundary of the class.
Class mark = (lower limit + upper limit) ÷ 2.
For inclusive classes, boundary adjustment = (next lower limit − previous upper limit) ÷ 2.
Exclusive classes already have boundaries equal to their limits.
Relative frequency = class frequency ÷ total frequency.
A histogram uses class boundaries on the x-axis, with bars touching each other.
With unequal class widths, a histogram needs frequency density = frequency ÷ width.
A frequency polygon joins the mid-points of the tops of histogram bars.
A less-than ogive plots upper boundaries against less-than cumulative frequencies.
A more-than ogive plots lower boundaries against more-than cumulative frequencies.
The two ogives cross at a point whose x-value is the median.
In a pie chart, angle of a sector = (value ÷ total) × 360°.

Statistical Representation of Data practice questions

Statistical Representation of Data in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Statistical Representation of Data: frequently asked questions

What is the difference between class limits and class boundaries?

Class limits are the stated smallest and largest values of a class, such as 10 and 19. Class boundaries are the exact dividing points between classes, such as 9.5 and 19.5, which remove the gap between inclusive classes. Use boundaries for graphs and for finding width.

How do I find the median from an ogive?

Draw the less-than and more-than ogives on the same axes. The x-value at their intersection is the median. You can also take n/2 on the cumulative frequency axis of the less-than ogive and read across to the curve, then down to the x-axis.

Is a histogram the same as a bar diagram?

No. A histogram shows continuous frequency data with bars that touch, and the area relates to frequency. A bar diagram compares separate categories, with gaps between bars, and only the height matters.

Do I need to draw graphs in the exam?

The exam is all MCQs on an OMR sheet, so you will not draw anything. You will need to know how graphs are built so you can answer questions on plotting points, shapes and values read from a graph.