13. Statistics

10 topics3 h

Chapter 13 of NCERT Mathematics for Class 11: Statistics. Three sections — Why a single central value is not enough, Averaging the distances from a centre and Squaring instead of discarding the sign. Ten videos, 3 h in all.

Worked answers to the 28 exercise questions in this chapter: Exercise 13.1 · Exercise 13.2 · Miscellaneous Exercise

0 of 10 done — sign in to track this chapter

Part 1

Why a single central value is not enough

18 minTwo records with the same average that describe very different playersTwo players, ten games each, and every summary of the middle comes out at 53 for both of them - while one player's scores run from 0 to 117 and the other's never leave the forties and fifties. The explanation builds a whole family of 47 records with the average, the middle value and the commonest score all frozen at 53 and the strewing running free, which is why a second number is needed.18 min · 17 parts15 minThe cheapest measure of spread, and everything it fails to noticeTwo batsmen average the same fifty-three runs and share the same middle score - one subtraction, largest innings less smallest, finally tells them apart: 117 against 14.15 min · 14 parts

Part 2

Averaging the distances from a centre

16 minSigned deviations cancel to nothing, so the sign has to be discardedSubtract the mean from every reading and add up the results, and the answer is nothing - for this data set, and for every one ever written down. It is an identity, not misfortune.16 min · 16 parts14 minWorking it out about the mean and about the median, for a plain listSlide a reference point along a data set, and the total distance to every reading falls only while more readings lie ahead of it than behind - which is why nothing beats the median, not even the mean.14 min · 16 parts21 minThe same procedure once the data arrive already groupedA table of bands, unlike a table of values, loses the readings themselves. Every mean deviation computed from it afterwards assumes each reading sat at its band's exact midpoint.21 min · 21 parts17 minWhere this measure breaks down, and why another was neededThe mean deviation is retired within a page of being defined, and the usual complaints barely matter. The real reason is that a modulus cannot be expanded into further algebra.17 min · 17 parts

Part 3

Squaring instead of discarding the sign

13 minSquaring removes the sign and punishes the far-out valuesSix readings spread from 5 to 55, and thirty-one packed from 15 to 45 - and the plain total of their squared deviations ranks the packed set as more scattered.13 min · 17 parts13 minTaking the root to get the spread back into the units of the dataA batting record averaging fifty-three runs, best score a hundred and seventeen, carries a variance of 1300.6 - bigger than the record's entire span.13 min · 15 parts16 minCarrying the frequencies through, for discrete and for grouped dataA deviation column cannot begin until the mean has already been written out in full, and nothing guarantees that mean will be a tidy number.16 min · 18 parts19 minShifting and scaling the observations to make the arithmetic smallSlide every observation along by the same amount and not one gap between any two of them changes, so the variance never notices a shift at all.19 min · 18 parts

What you will be able to do

  • Compute the mean of each of the chapter's two ten-innings records and show that both come to 53
  • List the four measures of dispersion §13.2 names, and state which one this chapter sets aside
  • Prove that the deviations of a data set about its own mean add to zero
  • Carry out the four steps of §13.4.1 for the mean deviation about the mean of a given list
  • Compute the mean deviation about the mean of a discrete frequency distribution, using a worked table with the products written out
  • State the three objections §13.4.3 raises against the mean deviation
  • State why squaring removes the sign problem without introducing a modulus
  • State why the variance cannot be compared with the mean or plotted on the observations' own axis
  • Write, in terms of the frequencies, the variance of a discrete frequency distribution and its standard deviation
  • Write the step-deviation substitution and its inverse

What this chapter assumes you already know

  • The arithmetic mean of a list, and the sigma notation the chapter writes it in
  • The median rule for a list of odd length and for a list of even length
  • Arranging a list in ascending order
  • Plotting numbers as points on a number line
  • Reading the largest and smallest entries out of an unsorted list
  • Subtraction of integers, and the idea of a difference as a distance on a line
  • The word deviation as the difference between an observation and a fixed value
  • The definition of the mean as total divided by count, in sigma notation

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Equal averages mean equal performance."
  • "The median would have caught what the mean missed."
  • "The median is always one of the observations."
  • "Scatter just means the biggest value is big."
  • "You could just look at the dots and skip the number."
  • "A mean tells you what a typical innings looked like."
  • "The range is a bad measure."
  • "A bigger range always means more scatter."