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Basic Skills – Lesson 3 
Introduction to Standard 
Deviation and Error Estimation 
MMeeaann DDeevviiaattiioonn 
SSttaannddaarrdd DDeevviiaattiioonn ffoorr:: 
a.) AA ppooppuullaattiioonn b.) AA ssaammppllee ooff tthhee 
ppooppuullaattiioonn
When measurements are taken (and eexxppeerriimmeennttaall ddaattaa iiss 
oobbttaaiinneedd)) tthhee ddaattaa wwiillll uussuuaallllyy tteenndd ttoo vvaarryy ffrroomm tthhee mmeeaann oorr 
aavveerraaggee vvaalluuee.. {{IInnddiivviidduuaall vvaalluueess wwiillll tteenndd ttoo bbee both ggrreeaatteerr 
tthhaann aanndd lleessss tthhaann tthhee mmeeaann}}.. 
FFoorr eexxaammppllee,, iiff tthhee ddiiaammeetteerr ooff 110000 ssiimmiillaarr sstteeeell bbaallll bbeeaarriinnggss 
aarree mmeeaassuurreedd,, tthhee vvaalluueess wwoouulldd lliikkeellyy nnoott aallll bbee tthhee ssaammee iiff aa 
hhiigghh pprreecciissiioonn mmeeaassuurriinngg ddeevviiccee iiss uusseedd ((ssuucchh aass aa ddiiggiittaall 
ccaalliippeerr)) 
TThhee vvaarriiaattiioonn ((oorr ddiissppeerrssiioonn)) ooff tthhee ddaattaa ffrroomm tthhee mmeeaann vvaalluuee 
ccaann bbee tthhoouugghhtt ooff aass tthhee eerrrroorr tthhaatt iiss pprreesseenntt iinn tthhee 
mmaannuuffaaccttuurriinngg pprroocceessss uusseedd ttoo mmaakkee tthhee sstteeeell bbaallll bbeeaarriinnggss.. 
TThheerree aarree vvaarriioouuss wwaayyss ttoo qquuaannttiiffyy tthhiiss eerrrroorr,, tthhee mmoosstt 
ccoommmmoonn bbeeiinngg standard deviation.. 
IInn oorrddeerr ttoo uunnddeerrssttaanndd ssttaannddaarrdd ddeevviiaattiioonn,, wwee mmuusstt hhaavvee aa 
bbaassiicc uunnddeerrssttaannddiinngg ooff aa normal distribution.. TThhiiss ccaann bbee 
sseeeenn iinn tthhee ffoolllloowwiinngg ddiiaaggrraamm::
Normal Distribution ((IInnttrroo ttoo SSttaattiissttiiccss)) 
 TThhee nnoorrmmaall ddiissttrriibbuuttiioonn 
iiss sshhoowwnn ttoo bbee 
ssyymmmmeettrriiccaall aanndd 
rreesseemmbblleess aa bbeellll.. 
 WWiitthhiinn ±± 11 ssaammppllee ssttdd.. 
ddeevv.. ooff tthhee mmeeaann,, 6688%% 
ooff tthhee ddaattaa (( oorr 
mmeemmbbeerrss ooff tthhee ssaammppllee 
ppooppuullaattiioonn)) aarree ffoouunndd 
 WWiitthhiinn ±± 22 ssaammppllee ssttdd.. 
ddeevv.. 9955%% ooff tthhee ddaattaa iiss 
ffoouunndd.. 
9 5 % o f s a m p l e p o p u l a t i o n 
x 
+ 1 s 
+ 2 s 
- 1 s 
- 2 s 
Frequency of Occurrence 
N o r m a l D i s t r i b u t i o n 
( A B e l l C u r v e ) 
6 8 % o f s a m p l e 
p o p u l a t i o n
Sample Population and Population 
AA sample population iiss uusseedd wwhheenn iitt iiss impossible 
ttoo mmeeaassuurree tthhee eennttiirree ppooppuullaattiioonn.. FFoorr eexxaammppllee;; 
tthhee ddiiaammeetteerr ooff 220000 bbaallll bbeeaarriinnggss aarree rraannddoommllyy 
mmeeaassuurreedd ffrroomm aa ppooppuullaattiioonn ooff 110000 000000.. TThhee 
sample population hhaass aa ssiizzee ooff 220000.. WWhheenn tthhee 
ssttdd.. ddeevv.. ffoorr aa ssaammppllee iiss ccaallccuullaatteedd,, aa ssmmaallll “s” iiss 
uusseedd ttoo rreepprreesseenntt tthhiiss vvaalluuee.. IIff aallll tthhee mmeemmbbeerrss 
ooff tthhee ppooppuullaattiioonn ((110000 000000)) ccaann bbee mmeeaassuurreedd,, 
tthhee ssyymmbbooll ffoorr ssiiggmmaa ((ϭ )) iiss uusseedd ttoo rreepprreesseenntt 
tthhee vvaalluuee ffoorr ssttdd.. ddeevv.. 
 TThhee ccaallccuullaattiioonn ffoorr ssttdd.. ddeevv.. iiss slightly ddiiffffeerreenntt 
ffoorr ss aanndd ϭϭ ..
CCaallccuullaattiioonn ooff ss aanndd ϭϭ 
TToo ccaallccuullaattee tthhee ssttdd ddeevv ffoorr aa ssaammppllee ppooppuullaattiioonn 
ooff ““n”” mmeemmbbeerrss:: 
ss == ((((((xx11--xxMMEEAANN))22 ++ ((xx22--xxMMEEAANN))22 ++……++ ((xxnn--xxMMEEAANN))22 ))//(n-1) ))00..55 
TToo ccaallccuullaattee tthhee ssttdd ddeevv ffoorr aann eennttiirree ppooppuullaattiioonn,, 
ooff ““n”” mmeemmbbeerrss:: 
ϭϭ== ((((((xx11--xxMMEEAANN))22 ++ ((xx22--xxMMEEAANN))22 ++……++ ((xxnn--xxMMEEAANN))22 ))//((n)) ))00..55 
Your calculator will (likely) have these equations 
already preprogrammed, however, it may be a 
challenge to use them.
Mean Deviation 
Mean deviation (DD)) iiss aannootthheerr mmeetthhoodd ooff eexxpprreessssiinngg 
tthhee ddiissppeerrssiioonn ffrroomm tthhee mmeeaann.. TToo ccaallccuullaattee:: 
DD == ((||xx11--xxMMEEAANN|| ++ ||xx22--xxMMEEAANN|| ++……++ ||xxnn -- xxMMEEAANN||))//nn 
WWhheenn eexxpprreessssiinngg eexxppeerriimmeennttaall ddaattaa aass aa ssiinnggllee rreessuulltt 
((wwiitthh eerrrroorr)) wwee ccaann uussee tthhee ffoolllloowwiinngg:: 
XX == xxMMEEAANN ±± ΔΔxx,, wwhheerree ΔΔxx == ss,, oorr ΔΔxx == ϭϭ oorr ΔΔxx == DD 
YYoouurr hhoommeewwoorrkk qquueessttiioonnss wwiillll iinnddiiccaattee wwhhiicchh vvaalluuee iiss 
ttoo bbee ccaallccuullaatteedd..
Practice Questions 
 WWoorrkkbbooookk:: 
 PPaaggee 55,, QQ##44,, 55aa 
 Listed below are the diameters (in mm) of 
5 steel ball bearings taken from a 
population of 25: 
3.45, 3.62, 3.50, 3.44, 3.54 
Express the data a single value including 
error. Use standard deviation to calculate 
the error.. AAnnsswweerr ggiivveenn nneexxtt ccllaassss
 AAnnsswweerr:: 33..5511 ±± 00..0077 mmmm

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Grade 12 U0-L3-ErrorEstimation

  • 1. Basic Skills – Lesson 3 Introduction to Standard Deviation and Error Estimation MMeeaann DDeevviiaattiioonn SSttaannddaarrdd DDeevviiaattiioonn ffoorr:: a.) AA ppooppuullaattiioonn b.) AA ssaammppllee ooff tthhee ppooppuullaattiioonn
  • 2. When measurements are taken (and eexxppeerriimmeennttaall ddaattaa iiss oobbttaaiinneedd)) tthhee ddaattaa wwiillll uussuuaallllyy tteenndd ttoo vvaarryy ffrroomm tthhee mmeeaann oorr aavveerraaggee vvaalluuee.. {{IInnddiivviidduuaall vvaalluueess wwiillll tteenndd ttoo bbee both ggrreeaatteerr tthhaann aanndd lleessss tthhaann tthhee mmeeaann}}.. FFoorr eexxaammppllee,, iiff tthhee ddiiaammeetteerr ooff 110000 ssiimmiillaarr sstteeeell bbaallll bbeeaarriinnggss aarree mmeeaassuurreedd,, tthhee vvaalluueess wwoouulldd lliikkeellyy nnoott aallll bbee tthhee ssaammee iiff aa hhiigghh pprreecciissiioonn mmeeaassuurriinngg ddeevviiccee iiss uusseedd ((ssuucchh aass aa ddiiggiittaall ccaalliippeerr)) TThhee vvaarriiaattiioonn ((oorr ddiissppeerrssiioonn)) ooff tthhee ddaattaa ffrroomm tthhee mmeeaann vvaalluuee ccaann bbee tthhoouugghhtt ooff aass tthhee eerrrroorr tthhaatt iiss pprreesseenntt iinn tthhee mmaannuuffaaccttuurriinngg pprroocceessss uusseedd ttoo mmaakkee tthhee sstteeeell bbaallll bbeeaarriinnggss.. TThheerree aarree vvaarriioouuss wwaayyss ttoo qquuaannttiiffyy tthhiiss eerrrroorr,, tthhee mmoosstt ccoommmmoonn bbeeiinngg standard deviation.. IInn oorrddeerr ttoo uunnddeerrssttaanndd ssttaannddaarrdd ddeevviiaattiioonn,, wwee mmuusstt hhaavvee aa bbaassiicc uunnddeerrssttaannddiinngg ooff aa normal distribution.. TThhiiss ccaann bbee sseeeenn iinn tthhee ffoolllloowwiinngg ddiiaaggrraamm::
  • 3. Normal Distribution ((IInnttrroo ttoo SSttaattiissttiiccss))  TThhee nnoorrmmaall ddiissttrriibbuuttiioonn iiss sshhoowwnn ttoo bbee ssyymmmmeettrriiccaall aanndd rreesseemmbblleess aa bbeellll..  WWiitthhiinn ±± 11 ssaammppllee ssttdd.. ddeevv.. ooff tthhee mmeeaann,, 6688%% ooff tthhee ddaattaa (( oorr mmeemmbbeerrss ooff tthhee ssaammppllee ppooppuullaattiioonn)) aarree ffoouunndd  WWiitthhiinn ±± 22 ssaammppllee ssttdd.. ddeevv.. 9955%% ooff tthhee ddaattaa iiss ffoouunndd.. 9 5 % o f s a m p l e p o p u l a t i o n x + 1 s + 2 s - 1 s - 2 s Frequency of Occurrence N o r m a l D i s t r i b u t i o n ( A B e l l C u r v e ) 6 8 % o f s a m p l e p o p u l a t i o n
  • 4. Sample Population and Population AA sample population iiss uusseedd wwhheenn iitt iiss impossible ttoo mmeeaassuurree tthhee eennttiirree ppooppuullaattiioonn.. FFoorr eexxaammppllee;; tthhee ddiiaammeetteerr ooff 220000 bbaallll bbeeaarriinnggss aarree rraannddoommllyy mmeeaassuurreedd ffrroomm aa ppooppuullaattiioonn ooff 110000 000000.. TThhee sample population hhaass aa ssiizzee ooff 220000.. WWhheenn tthhee ssttdd.. ddeevv.. ffoorr aa ssaammppllee iiss ccaallccuullaatteedd,, aa ssmmaallll “s” iiss uusseedd ttoo rreepprreesseenntt tthhiiss vvaalluuee.. IIff aallll tthhee mmeemmbbeerrss ooff tthhee ppooppuullaattiioonn ((110000 000000)) ccaann bbee mmeeaassuurreedd,, tthhee ssyymmbbooll ffoorr ssiiggmmaa ((ϭ )) iiss uusseedd ttoo rreepprreesseenntt tthhee vvaalluuee ffoorr ssttdd.. ddeevv..  TThhee ccaallccuullaattiioonn ffoorr ssttdd.. ddeevv.. iiss slightly ddiiffffeerreenntt ffoorr ss aanndd ϭϭ ..
  • 5. CCaallccuullaattiioonn ooff ss aanndd ϭϭ TToo ccaallccuullaattee tthhee ssttdd ddeevv ffoorr aa ssaammppllee ppooppuullaattiioonn ooff ““n”” mmeemmbbeerrss:: ss == ((((((xx11--xxMMEEAANN))22 ++ ((xx22--xxMMEEAANN))22 ++……++ ((xxnn--xxMMEEAANN))22 ))//(n-1) ))00..55 TToo ccaallccuullaattee tthhee ssttdd ddeevv ffoorr aann eennttiirree ppooppuullaattiioonn,, ooff ““n”” mmeemmbbeerrss:: ϭϭ== ((((((xx11--xxMMEEAANN))22 ++ ((xx22--xxMMEEAANN))22 ++……++ ((xxnn--xxMMEEAANN))22 ))//((n)) ))00..55 Your calculator will (likely) have these equations already preprogrammed, however, it may be a challenge to use them.
  • 6. Mean Deviation Mean deviation (DD)) iiss aannootthheerr mmeetthhoodd ooff eexxpprreessssiinngg tthhee ddiissppeerrssiioonn ffrroomm tthhee mmeeaann.. TToo ccaallccuullaattee:: DD == ((||xx11--xxMMEEAANN|| ++ ||xx22--xxMMEEAANN|| ++……++ ||xxnn -- xxMMEEAANN||))//nn WWhheenn eexxpprreessssiinngg eexxppeerriimmeennttaall ddaattaa aass aa ssiinnggllee rreessuulltt ((wwiitthh eerrrroorr)) wwee ccaann uussee tthhee ffoolllloowwiinngg:: XX == xxMMEEAANN ±± ΔΔxx,, wwhheerree ΔΔxx == ss,, oorr ΔΔxx == ϭϭ oorr ΔΔxx == DD YYoouurr hhoommeewwoorrkk qquueessttiioonnss wwiillll iinnddiiccaattee wwhhiicchh vvaalluuee iiss ttoo bbee ccaallccuullaatteedd..
  • 7. Practice Questions  WWoorrkkbbooookk::  PPaaggee 55,, QQ##44,, 55aa  Listed below are the diameters (in mm) of 5 steel ball bearings taken from a population of 25: 3.45, 3.62, 3.50, 3.44, 3.54 Express the data a single value including error. Use standard deviation to calculate the error.. AAnnsswweerr ggiivveenn nneexxtt ccllaassss
  • 8.  AAnnsswweerr:: 33..5511 ±± 00..0077 mmmm