Chapter 13 ยท Introduction

Chapter 13 โ€” Statistics

Two batsmen can have the same average and be completely different players. Measure how far data spread around their centre โ€” with the range, the mean deviation, the variance and the standard deviation.

A selector has to pick one opening batsman for a big final. Over their last ten innings both candidates average exactly $50$ runs. Is it a coin toss? Look at the scores themselves. The first batsman made $48, 52, 45, 55, 50, 47, 53, 49, 51, 50$ โ€” never far from fifty. The second made $0, 112, 8, 95, 30, 74, 2, 120, 15, 44$ โ€” two centuries, but also a duck and several cheap dismissals. The averages are identical; the players are not. One gives the team a steady start almost every time, the other wins some matches on his own and loses others. A single "centre" number โ€” mean, median or mode โ€” cannot tell them apart. We need a second number that measures how spread out the data are.

The same question appears everywhere data are collected. Two cities can have the same average annual rainfall while one gets it evenly and the other in a few violent monsoon weeks; two machines can fill bottles with the same average volume while one wastes far more through over- and under-filling; two classes can have the same mean mark while one has a long tail of struggling students. In Class 10 you learnt to find the mean, median and mode of grouped data. In this chapter you will learn the measures of dispersion: the range (the simplest), the mean deviation (the average distance from the centre), and the variance and standard deviation (the measures used throughout science, finance and quality control). You will compute each of them for a plain list of numbers, for a discrete frequency table and for a table with class intervals, and you will learn the shortcut method that keeps the arithmetic small.

A centre is not enoughโ†’Range: largest โˆ’ smallestโ†’Mean deviation: average of $|x_i-a|$โ†’Variance $\sigma^2$ and standard deviation $\sigma$
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • explain why a measure of central tendency must be accompanied by a measure of dispersion, and find the range of a data set;
  • find the mean deviation about the mean and about the median for a list of observations, and explain why absolute values are needed;
  • find the mean deviation for a discrete frequency distribution and for a continuous (class-interval) frequency distribution, using cumulative frequencies to locate the median;
  • find the variance and standard deviation of ungrouped data, of a discrete frequency distribution and of a continuous frequency distribution, by the direct formula and by the shortcut (step-deviation) method;
  • use the formula "variance = mean of the squares โˆ’ square of the mean" to work backwards from given summaries, correct wrongly copied observations, and interpret a standard deviation in context.
Teaser โ€” Same Average, Different Spread

Two data sets with the same mean. Press Play: the observations arrive one at a time, each dropping onto its row, and a thin line joins it to the mean (dashed). The shaded band covers everything from the smallest to the largest value โ€” the range. Watch the readouts: the means agree, but the typical distance from the mean is completely different. Switch the pair to see the same story in another setting.

mean, top rowโ€”
mean, bottom rowโ€”
range (top ยท bottom)โ€”
average distance from meanโ€”

Sections in this chapter

13.1

Measures of Dispersion: Range and Mean Deviation

Why spread matters, the range, deviations from a central value, why their plain sum is useless, and the mean deviation about the mean and about the median for ungrouped data.

$\text{M.D.}(a)=\dfrac{1}{n}\displaystyle\sum|x_i-a|$
Open section โ†’13.2

Mean Deviation for Grouped Data

Mean deviation about the mean and the median for discrete frequency tables and for continuous class-interval tables, with cumulative frequencies and the step-deviation method for the mean.

$\text{M.D.}(\bar x)=\dfrac{1}{N}\displaystyle\sum f_i|x_i-\bar x|$
Open section โ†’13.3

Variance and Standard Deviation

Squaring the deviations, the variance and standard deviation for ungrouped, discrete and continuous data, the "mean of squares minus square of mean" formula and the shortcut method.

$\sigma^2=\dfrac{1}{n}\displaystyle\sum x_i^2-\bar x^{\,2}$
Open section โ†’
โœ… Before you start
From Class 10 Mathematics, Chapter 13 (Statistics), you need: the mean of a list $\bar x=\tfrac1n\sum x_i$ and of a frequency table $\bar x=\tfrac{\sum f_ix_i}{\sum f_i}$; class marks (mid-points of class intervals); the assumed-mean and step-deviation methods for the mean; cumulative frequency; and the median of a list and of grouped data, $M=l+\dfrac{\frac N2-\text{c.f.}}{f}\times h$. Section 13.2 recalls each of these when it is first used. From this course you need the modulus $|x|$ โ€” the distance of $x$ from $0$ โ€” from Section 2.3, and the sigma notation $\Sigma$ from Section 8.1. A calculator that finds square roots is useful for standard deviations.
โ† 12.4 Algebra of Derivatives13.1 Measures of Dispersion: Range and Mean Deviation โ†’