Statistical Representation of Data: formula sheet

Key formulas

Bases of classification

Geographical (place) | Chronological (time) | Qualitative (attribute) | Quantitative (variable)

Decide the basis from the feature used to group the data. Time gives chronological. A measurable number gives quantitative.

Parts of a statistical table

Table number, Title, Caption, Stub, Body, Head note, Foot note, Source note

Caption is the column heading. Stub is the row heading. Body holds the figures.

Types of tables by number of characteristics

Simple (one characteristic) | Double (two) | Triple (three) | Manifold (more than three)

Count how many characteristics are shown together in the table.

Types of tables by purpose

General purpose (reference) table | Special purpose (summary) table

General purpose tables hold detailed data for reference. Special purpose tables show the findings of a particular study.

Class limits and width (for quantitative classes)

Class width = Upper limit − Lower limit (exclusive classes)

Use for grouped data. For example, in 10-20 the width is 20 − 10 = 10.

Class width (exclusive classes)

Class width = Upper limit − Lower limit

Use boundaries when classes are inclusive.

Mid-point (mid value)

Mid-point = (Lower limit + Upper limit) ÷ 2

For inclusive classes, using limits or boundaries gives the same answer.

Adjustment factor for inclusive classes

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

For integer data with gap 1, d = 0.5.

Class boundaries

Lower boundary = Lower limit − d; Upper boundary = Upper limit + d

This converts inclusive classes to exclusive classes.

Approximate number of classes

Number of classes ≈ Range ÷ Class width

Range = Largest value − Smallest value. Round up.

Less than cumulative frequency

Add frequencies from the first class onwards

Placed against the upper boundary of each class.

More than cumulative frequency

Add frequencies from the last class backwards

Placed against the lower boundary of each class.

Total frequency

N = Σf

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

Pie chart angle

Angle of a component = (Component value ÷ Total) × 360°

All angles must add up to 360°.

Angle from percentage

Angle = (Percentage ÷ 100) × 360° = percentage × 3.6°

10% = 36°, 25% = 90°, 50% = 180°.

Percentage bar height

Component percentage = (Component ÷ Bar total) × 100

Every bar in a percentage bar diagram has the same total height of 100%.

Scaling a two-dimensional diagram

Side of square ∝ √value; Radius of circle ∝ √value

Area is proportional to the value, so lengths scale by the square root.

Choosing the diagram

Compare values: simple or multiple bar. Parts of a total: component bar or pie. Share structure across bars: percentage bar.

Use this to answer 'which diagram is suitable' questions.

Class width

Class width = Upper limit − Lower limit (exclusive classes)

For inclusive classes, first convert to true limits.

Mid-value of a class

Mid-value = (Lower limit + Upper limit) ÷ 2

Used as the x-coordinate of each point on the frequency polygon.

Adjustment for inclusive classes

True lower limit = Lower limit − d ÷ 2; True upper limit = Upper limit + d ÷ 2

d is the gap between the upper limit of one class and the lower limit of the next. For 10-19, 20-29, d = 1.

Frequency density

Frequency density = Class frequency ÷ Class width

Use as the height of each rectangle when class widths are unequal. Some books first scale to a standard width; the shape is the same.

Less-than ogive points

Plot (upper limit, cumulative frequency of classes up to that limit)

Add a starting point (lower limit of first class, 0).

More-than ogive points

Plot (lower limit, frequency of that class and all later classes)

The last point is (upper limit of last class, 0).

Median from ogive

x-value where the two ogives intersect; also x at height N/2

N is the total frequency.

First quartile Q1

x-value at height N/4 on the less-than ogive

One quarter of items lie below it.

Third quartile Q3

x-value at height 3N/4 on the less-than ogive

Three quarters of items lie below it.

Quick revision

• 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°.

Common mistakes

• Swapping stub and caption Fix: Stub is the row heading on the left. Caption is the column heading on top. Think of a stub as the side.
• Treating classification and tabulation as the same thing Fix: Classification groups the data. Tabulation presents the grouped data in rows and columns. Classification comes first.
• Taking class width of 10-19 as 9. Fix: Convert to boundaries 9.5-19.5. Width is 10.
• Placing a value equal to the upper limit in the same class in an exclusive series. Fix: In exclusive classes, the upper limit is excluded. So 20 goes into 20-30, not 10-20.
• Mixing up a multiple bar diagram and a component bar diagram. Fix: In a multiple bar diagram, the bars stand side by side and each is a separate series. In a component bar, one bar is stacked into parts that add to the total.
• Using 100 instead of 360 in a pie chart angle. Fix: Always write angle = (value ÷ total) × 360°. If you have a percentage, multiply by 3.6.
• Leaving gaps between rectangles when the classes are inclusive (10-19, 20-29). Fix: Subtract d ÷ 2 from each lower limit and add d ÷ 2 to each upper limit first, so classes touch. The adjustment is 0.5 when the gap d is 1.
• Using frequency as height with unequal class widths. Fix: Compute frequency density = frequency ÷ class width and use it as the height.
• Plotting cumulative frequencies against class midpoints. Fix: For a less-than ogive use upper limits. For a more-than ogive use lower limits.
• Using the wrong limit for the more-than ogive. Fix: A more-than curve means 'this value or more', so use lower limits.

Exam tips

• Learn the eight parts of a table in order. Questions often ask which part holds the column headings or the units.
• Practise naming the basis of classification from a one-line example. These MCQs are quick marks.
• Remember that there is no negative marking. If unsure, eliminate and attempt every question.
• Do not spend time on long tables. Count characteristics, then answer.
• Revise this topic with frequency distributions, since class intervals build on classification.
• Always check the gap between classes first. It tells you the series type in two seconds.
• Class width, boundaries and mid-point questions are quick marks. Do them first.
• In cumulative frequency questions, confirm that the final total equals Σf. It catches addition slips.