Data Presentation: Visualisation and Graphical Presentation: formula sheet

Key formulas

Purpose of presentation

Raw data → organised table → chart / text → interpretation

Use this flow to explain why presentation is needed. There is no calculation in this topic.

Parts of a table

Table number + Title + Caption + Stub + Body + Headnote (unit) + Footnote + Source note

Learn the list in this order. Questions often ask you to name or identify parts.

Textual vs tabular

Textual = sentences, few figures; Tabular = rows and columns, many figures

Use this one-line contrast in difference questions.

Rules of a good table

Clear title, logical order, units stated, totals shown, ruled neatly, source given, not overcrowded

Write each rule with a short reason when asked to explain.

Check of totals

Sum of row totals = Sum of column totals = Grand total

Use this to verify a table you have built.

Chart choice rule: comparison

Compare categories → bar / column chart

Use horizontal bars when category names are long.

Chart choice rule: trend

Change over time → line graph

Time goes on the x-axis.

Chart choice rule: composition

Parts of a whole → pie chart (few categories) or stacked bar

Slices must add up to 100%. Avoid pies with many slices.

Chart choice rule: distribution

Continuous data in class intervals → histogram

Bars touch because the classes are continuous.

Chart choice rule: relationship

Two numeric variables → scatter plot

Shows direction and strength of correlation.

Pie chart angle

Angle = (Category value ÷ Total) × 360°

All angles must add up to 360°.

Angle of a sector in a pie chart

Angle = (Component value ÷ Total) × 360°

All angles must add up to 360°. Use this to check your work.

Percentage share of a component

Percentage = (Component value ÷ Total) × 100

Used for percentage bar charts and to label pie sectors. Shares add up to 100%.

Angle from percentage

Angle = Percentage × 3.6°

Since 360° ÷ 100 = 3.6°. Quick shortcut when percentages are given.

Bar length from a scale

Bar length = Value ÷ Value represented by 1 cm

Choose a scale so the largest bar fits the page, for example 1 cm = ₹10,000.

Choosing the chart

Compare categories: bar. Show shares of one total: pie. Show trend over time: line.

A rule of thumb for choosing; use multiple or component bars when more than one series is given.

Class mark (mid-point)

Class mark = (Lower limit + Upper limit) ÷ 2

Used as the X-value for each point of a frequency polygon.

Adjustment for inclusive classes

Adjustment factor = (Lower limit of next class − Upper limit of this class) ÷ 2

Subtract it from every lower limit and add it to every upper limit to get true (exclusive) limits. For classes 10–19, 20–29 the factor is 0.5.

Class width

Class width = Upper limit − Lower limit (true limits)

Check this before drawing a histogram. If widths are unequal, adjust the heights.

Frequency density (unequal widths)

Frequency density = Class frequency ÷ Class width

Plot this as the bar height when class widths are unequal, so that area stays proportional to frequency.

Cumulative frequency

Less than: running total from the first class. More than: running total from the last class, upwards.

The last less than total and the first more than total both equal N, the total frequency.

Median from an ogive

Median = X-value at which cumulative frequency = N ÷ 2

Mark N ÷ 2 on the Y-axis, go across to the curve, then drop down to the X-axis. The same value comes from the intersection of the two ogives.

Median by interpolation (check)

Median = L + [(N ÷ 2 − c) ÷ f] × h

L = lower limit of median class, c = cumulative frequency before it, f = its frequency, h = its width. Use it to verify your graph reading.

Interquartile range (IQR)

IQR = Q3 − Q1

Length of the box in a box plot. It holds the middle 50% of the data.

Outlier fences (common 1.5 × IQR rule)

Lower fence = Q1 − 1.5 × IQR; Upper fence = Q3 + 1.5 × IQR

Values outside the fences are usually plotted as outliers. This is a convention used by many tools, not a law.

Treemap area

Area of a rectangle ∝ category value ÷ total of all values

Bigger area means bigger share.

Reading a scatter plot

Upward trend = positive; downward trend = negative; no pattern = no linear relationship

Association does not prove causation.

KPI variance against target

Variance = Actual − Target

A positive value is favourable for revenue or profit KPIs. For cost KPIs, a positive value is adverse.

KPI achievement percentage

Achievement % = (Actual ÷ Target) × 100

Used for gauges and traffic-light signals. Check whether higher or lower is better for that KPI.

Growth rate KPI

Growth % = (Current period − Previous period) ÷ Previous period × 100

Divide by the earlier period, not the later one.

Chart choice rule

Trend over time → line; comparison of categories → bar/column; share of a whole → pie (few parts); relationship of two variables → scatter

A rule of thumb for matching display to purpose, not a law.

Quick revision

• A table has a title, headings, body and source note; use text when only a few figures need to be stated.
• Bar chart: compares categories; bars share a common baseline and equal width.
• Pie chart: shows parts of a whole; each slice angle = (value ÷ total) × 360°.
• Line graph: best for trends over time, with time on the horizontal axis.
• Histogram: for continuous frequency distributions; bars touch, and area reflects frequency.
• Frequency polygon: plot frequency against class midpoints and join the points with straight lines, closing the ends on the axis (or join the midpoints of the tops of the histogram bars).
• Ogive: plots cumulative frequency against class limits; there are less-than and more-than types.
• The two ogives intersect at the median.
• Scatter plot: shows the relationship between two variables, with each point being one pair of values.
• Heat map: uses colour intensity to show values across a grid.
• A dashboard brings key measures onto one screen for monitoring and decisions.
• A good chart has a title, labelled axes, units and a sensible scale.

Common mistakes

• Leaving out the title or unit of measurement. Fix: Write the title and the unit (for example ₹ in lakh) before filling any number.
• Mixing up stub and caption. Fix: Remember: stub is the row heading on the left; caption is the column heading on top.
• Using a pie chart for many categories or for data that does not form a whole. Fix: Use pie only for a few parts of one total. Otherwise use a bar chart.
• Using a bar chart instead of a line graph for trends over time, or the reverse for categories. Fix: Ask first: is it comparison or change over time?
• Pie chart angles do not add up to 360°. Fix: Add the components to get the total before you start, and add the angles at the end. Adjust rounding by a degree if needed.
• Confusing a multiple bar chart with a component bar chart. Fix: In a multiple bar chart, bars stand side by side and each is a separate value. In a component bar chart, parts are stacked in one bar and the bar height is the total.
• Leaving gaps between bars in a histogram or using inclusive limits as they are. Fix: Convert inclusive classes to true limits first (9.5–19.5, 19.5–29.5). Draw the bars touching each other.
• Plotting a less than ogive against class mark or lower limits. Fix: Less than uses the upper limit. More than uses the lower limit. Remember that a cumulative total is complete only at the end of the class.
• Saying a scatter plot proves that X causes Y. Fix: Say the variables are associated. Cause needs separate evidence.
• Using a scatter plot for a category against a number. Fix: Use a bar chart or box plot when one variable is a category.

Exam tips

• For "parts of a table" questions, list every part with a one-line meaning. A bare list earns fewer marks.
• For difference questions, give at least four distinct points in paired form.
• In a construct-a-table question, the title, unit and totals usually carry marks even if one figure is wrong.
• In MCQs, watch the words stub (rows) and caption (columns). Examiners often swap them in the wrong options.
• If asked to comment, add one sentence of interpretation using the totals or the largest and smallest values.
• In MCQs, match the verb in the question (compare, trend, share, distribution, relationship) to the chart.
• In written answers, give the chart, the reason and the labelling features. Each earns step marks.
• For pie chart numbers, always show the angle formula and the 360° check.