Statistics 2: Spread, Quartiles and Distribution Shape

SSC · RRB · State exams — method by method

5 Part0 worked examples0 practice questions

If the marks of two classes are 40, 40, 40, 40 and 10, 20, 60, 70, both have mean 40, yet they are very different: in the first all are alike, in the second they are widely scattered. A single number for the centre does not show this difference. That is why we need measures of spread (dispersion): they tell how close together or far apart the values are around the centre.

Question	Measures	In this chapter
How spread out?	range, mean deviation, variance, standard deviation, coefficient of variation	group B
How many are below a point?	quartiles, deciles, percentiles, the five-number summary	group C
What is the shape?	skewness, the normal curve	group C
Where do slips happen?	sample vs whole, inclusive classes, open ends, the sum of deviations	group D

Here: group A (the first) repeats the measures of the centre from the previous chapter in one table, because every formula of spread uses the mean or the median. Group B has spread, group C position and shape, group D the traps, and the end has the summary, mixed practice and revision.

Previous chapterThis is the second part of statistics. The first part, “Statistics 1: Data, Mean, Median and Mode”, teaches arranging data, frequency tables, class intervals and the three measures of the centre; this chapter will feel easier after it.

Before you start: the centre

Mean · median · mode

The three measures of the centre from Statistics 1 in one table.

Spread is always measured against some centre, so first revise the three measures of the centre at a glance. The full methods are in Statistics 1.

The measures of the centre at a glance

Formulas that will come again and again

The formulas of the mean, the median and the mode for raw and grouped data.

Measure	Raw data / table	Grouped data
Mean $\overset{x}{ˉ}$	$\frac{\sum x}{n}$ , $\frac{\sum f x}{\sum f}$	$\frac{\sum f x}{\sum f}$ , $a + h \frac{\sum f u}{\sum f}$
Median $M$	after sorting, the $\frac{n + 1}{2}$ th value	$L + \frac{N /2 - c f}{f} \times h$
Mode $Mo$	the value that occurs most often	$L + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h$

Here: $n$ = number of values, $N = \sum f$ , $x$ = a value (the class mark for grouped data), $a$ = assumed mean, $u = \frac{x - a}{h}$ , $h$ = class width, $L$ = lower boundary of the median/modal class, $c f$ = cumulative frequency up to the class before it, $f$ = frequency of that class, $f_{0}, f_{1}, f_{2}$ = frequencies of the class before the modal class, the modal class and the class after it.

How far the values are from the mean — this is the base of the mean deviation, variance and SD; so getting the mean right is the first condition.
The median and the quartiles are one family — quartiles come from the same formula as the median with only the position changed.
In inclusive classes (20–29, 30–39) first make the true boundaries (19.5, 29.5 …) and take the width as the difference of two consecutive lower limits.

Statistics 1If any of these formulas feels unfamiliar, first open the previous chapter, “Statistics 1: Data, Mean, Median and Mode”, read its method, and then return here.

Spread (dispersion)

Range · deviation · variance · SD

Range, mean deviation (raw and grouped), variance and standard deviation, variance with an assumed mean, the effect of changes, correcting a wrong value, combined SD, the coefficient of variation and choosing the right measure.

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Range and mean deviation (raw data)

Largest − smallest · average distance

The range, the coefficient of range, and the mean deviation about the mean or the median (list and frequency table).

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Mean deviation: grouped data

\frac{\sum f ∣ x - A ∣}{\sum f}

· the working columns

The mean deviation about the mean or the median from the class marks, its coefficient, and the table of working columns.

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Variance and standard deviation

σ^{2} = \frac{\sum ( x - x ˉ ) ^{2}}{n}

σ = σ^{2}

Variance and standard deviation of raw data and a frequency table: the definition and the short formula.

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Variance with an assumed mean (big numbers)

σ^{2} = \frac{\sum d ^{2}}{n} - (\frac{\sum d}{n})^{2}

Variance and standard deviation of raw data and of a frequency table with $d = x - A$ : the fast way for big values.

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Standard deviation: grouped data

The step-deviation method

Variance and standard deviation from class marks and $u = \frac{x - a}{h}$ .

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Effect of changes, and SD of a series

Unchanged by adding · changed by multiplying

The effect of adding or multiplying every value on SD, variance and CV; the SD of the first $n$ natural numbers and of an equally spaced series.

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Correcting a wrong value, and combined SD

Correct both

\sum x

and

\sum x^{2}

The correct mean and SD after a wrongly noted value, and the combined mean and SD of two groups.

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Coefficient of variation and consistency

CV = \frac{σ}{x ˉ} \times 100

Comparing the consistency of two data sets with a relative measure (CV).

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Choosing the right measure of spread

Extreme values · different means · a quick idea

Which of range, quartile deviation, standard deviation and coefficient of variation to use when, and the effect of one extreme value on each.

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Quartiles, percentiles and shape

Position · skewness · the normal curve

Values that cut the data into 4, 10 or 100 equal parts, the five-number summary, skewness and the normal curve.

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Quartiles, deciles, percentiles: raw data

Position

\frac{k ( n + 1 )}{4}

· IQR · QD

$Q_{1}$ , $Q_{3}$ , deciles, percentiles, the inter-quartile range (IQR), quartile deviation and percentile rank.

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Quartiles, deciles, percentiles: grouped data

The median formula with a different position

The formula for $Q_{1}$ , $Q_{3}$ , $D_{k}$ , $P_{k}$ and the quartile deviation.

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Five-number summary and the box plot

Min ·

Q_{1}

· median ·

Q_{3}

· max

The five-number summary, reading a box plot, and outliers by the 1.5 × IQR rule.

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Skewness: shape, Pearson and Bowley coefficients

Which way the mean is pulled

Symmetric and skewed shapes, the order of mean-median-mode, and the Pearson and Bowley coefficients of skewness.

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The normal curve and the z-score

68 – 95 – 99.7

The bell-shaped normal distribution, the 68-95-99.7 rule and the z-score.

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Traps in spread questions

Where marks are lost

The traps of sample vs whole data, the SD with inclusive classes, open ends and the sum of deviations.

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Traps: $n - 1$ , inclusive classes, open ends, the sum of deviations

A slip behind every option

The sample SD ( $n - 1$ ), the SD with inclusive classes, the quartile deviation of a table with open ends, and the deviations from the mean adding up to zero.

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Summary, mixed practice, revision

All formulas in one place

One table of all the formulas of this chapter, over 40 exam-style mixed questions and revision cards.

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Summary of all formulas

The whole chapter on one page

All the formulas of spread, position and shape, each with one thing to remember.

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Mixed practice — set 1

Basic to standard

Questions from all methods mixed; spot the method first, then solve.

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Mixed practice — set 2

Standard to advanced

Mixed questions from the hard end of the exam.

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Revision cards and exam tips

Flip, then recall

Flip cards of the most important facts, last tips, and the way back to the previous chapter.

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Statistics 2: Spread, Quartiles and Distribution Shape

Before you start: the centre

The measures of the centre at a glance

Spread (dispersion)

Range and mean deviation (raw data)

Mean deviation: grouped data

Variance and standard deviation

Variance with an assumed mean (big numbers)

Standard deviation: grouped data

Effect of changes, and SD of a series

Correcting a wrong value, and combined SD

Coefficient of variation and consistency

Choosing the right measure of spread

Quartiles, percentiles and shape

Quartiles, deciles, percentiles: raw data

Quartiles, deciles, percentiles: grouped data

Five-number summary and the box plot

Skewness: shape, Pearson and Bowley coefficients

The normal curve and the z-score

Traps in spread questions

Traps: $n - 1$ , inclusive classes, open ends, the sum of deviations

Summary, mixed practice, revision

Summary of all formulas

Mixed practice — set 1

Mixed practice — set 2

Revision cards and exam tips

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