411 REFERENCE · Wikibooks
Linear Algebra/Topic: Line of Best Fit
Reference material by Wikibooks contributors. This retains its source identity and is not 411 news reporting.
Linear Algebra Topic: Line of Best Fit Projection Onto a Subspace Topic: Geometry of Linear Maps
Scientists are often presented with a system that
has no solution and they must find an answer anyway.
That is, they must
find a value that is as close as possible to being an answer.
For instance,
suppose that we have a coin to use in flipping.
This coin has some proportion of heads
to total flips, determined by how it is
physically constructed, and we want to know if is near .
We can get experimental data by flipping it
many times.
This is the result a penny experiment, including some
intermediate numbers.
<TR>
<TD style="border-right:2px black solid;border-bottom:2px black solid;"> number of flips
<TD style="border-bottom:2px black solid;"> 30
<TD style="border-bottom:2px black solid;"> 60
<TD style="border-bottom:2px black solid;"> 90
<TR>
<TD style="border-right:2px black solid;">number of heads
<TD> 16
<TD> 34
<TD> 51
Because of randomness, we do not find the exact proportion
with this sample — there is no solution to this system.
That is, the vector of experimental data is not in the subspace
of solutions.
However, as described above, we want to find the that most nearly works.
An orthogonal projection of the data vector into the line subspace
gives our best guess.
The estimate () is a bit high but not much,
so probably the penny is fair enough.
best fitThe line with the slope
is called the line of best fit for this data.
x200px
least squaresMinimizing the distance
between the given vector and the vector used as the right-hand side
minimizes the total of these vertical lengths,
and consequently
we say that the line has been
obtained through fitting by least-squares
x200px
(the vertical scale here has been exaggerated ten times
to make the lengths visible).
We arranged the equation above so that the line
must pass through
because we take it to be (our best guess at)
the line whose slope is this coin's true proportion
of heads to flips.
We can also handle cases where the line need not
pass through the origin.
For example, the different denominations of U.S. money have different average
times in circulation
(the $2 bill is left off as a special case).
How long should we expect a $25 bill to last?
<TR>
<TD style="border-right:2px black solid;border-bottom:2px black solid;">denomination
<TD style="border-bottom:2px black solid;"> 1
<TD style="border-bottom:2px black solid;"> 5
<TD style="border-bottom:2px black solid;"> 10
<TD style="border-bottom:2px black solid;"> 20
<TD style="border-bottom:2px black solid;"> 50
<TD style="border-bottom:2px black solid;"> 100
<TR>
<TD style="border-right:2px black solid;"> average life (years)
<TD> 1.5
<TD> 2
<TD> 3
<TD> 5
<TD> 9
<TD> 20
The plot (see below) looks roughly linear.
It isn't a perfect line, i.e.,
the linear system with equations , ..., has no
solution, but we can again use orthogonal projection
to find a best approximation.
Consider the matrix of coefficients of that linear system and also its
vector of constants, the experimentally-determined values.
The ending result in the subsection on Projection into a Subspace says
that coefficients and so that the linear combination
of the columns of is as close as possible to the vector
are the entries of .
Some calculation
gives an intercept of and a slope
of .
x200px
Plugging into the equation of the line shows that such a
bill should last between five and six years.
We close by considering
the times for the men's mile race Oakley Baker 1977.
These are the world records that were in force on January first
of the given years.
We want to project when a 3:40 mile
will be run.
<TR>
<TD style="border-right:2px black solid;border-bottom: 2px black solid;">year
<TD style="border-bottom:2px black solid;"> 1870
<TD style="border-bottom:2px black solid;"> 1880
<TD style="border-bottom:2px black solid;"> 1890
<TD style="border-bottom:2px black solid;"> 1900
<TD style="border-bottom:2px black solid;"> 1910
<TD style="border-bottom:2px black solid;"> 1920
<TD style="border-bottom:2px black solid;"> 1930
<TR>
<TD style="border-right:2px black solid"> seconds
<TD> 268.8
<TD> 264.5
<TD> 258.4
<TD> 255.6
<TD> 255.6
<TD> 252.6
<TD> 250.4
<TR style="height:20px">
<TR>
<TD style="border-right:2px black solid;border-bottom: 2px black solid;">year
<TD style="border-bottom:2px black solid;"> 1940
<TD style="border-bottom:2px black solid;"> 1950
<TD style="border-bottom:2px black solid;"> 1960
<TD style="border-bottom:2px black solid;"> 1970
<TD style="border-bottom:2px black solid;"> 1980
<TD style="border-bottom:2px black solid;"> 1990
<TD style="border-bottom:2px black solid;"> 2000
<TR>
<TD style="border-right:2px black solid"> seconds
<TD> 246.4
<TD> 241.4
<TD> 234.5
<TD> 231.1
<TD> 229.0
<TD> 226.3
<TD> 223.1
We can see below that the data is surprisingly linear.
With this input
the Python program at this Topic's end gives
and
(rounded to two places; the original data
is good to only about a quarter of a second
since much of it was hand-timed).
x200px
When will a second mile be run?
Solving the equation of the line of best fit
gives an estimate of the year .
This example is amusing, but serves as a caution — obviously the
linearity of the data will break down someday (as indeed it does
prior to 1860).
Exercises
The calculations here are best done on a computer. In addition, some of the problems require more data, available in your library, on the net, in the answers to the exercises, or in the section following the exercises.
Problem 1
Use least-squares to judge if the coin in this experiment is fair.
<TR>
<TD style="border-right:2px black solid; border-bottom:2px black solid;">flips
<TD style="border-bottom:2px black solid;"> 8
<TD style="border-bottom:2px black solid;"> 16
<TD style="border-bottom:2px black solid;"> 24
<TD style="border-bottom:2px black solid;"> 32
<TD style="border-bottom:2px black solid;"> 40
<TR>
<TD style="border-right:2px black solid;"> heads
<TD> 4
<TD> 9
<TD> 13
<TD> 17
<TD> 20
Problem 2
For the men's mile record, rather than give each of the many
records and its exact date, we've "smoothed" the data somewhat by
taking a periodic sample.
Do the longer calculation and compare the conclusions.
Problem 3
Find the line of best fit for the men's meter run.
How does the slope compare with that for the men's mile?
(The distances are close; a mile is about meters.)
Problem 4 Find the line of best fit for the records for women's mile.
Problem 5
Do the lines of best fit for the men's and women's miles cross?
Problem 6
When the space shuttle Challenger exploded in 1986, one of the
criticisms made of NASA's decision to launch was in the way the
analysis of number of O-ring failures versus temperature was made
(of course, O-ring failure caused the explosion).
Four O-ring failures will cause the rocket to explode.
NASA had data from 24 previous flights.
<TR>
<TD style="border-right:2px black solid; border-bottom:2px black solid;">temp °F
<TD style="border-bottom:2px black solid;"> 53
<TD style="border-bottom:2px black solid;"> 75
<TD style="border-bottom:2px black solid;"> 57
<TD style="border-bottom:2px black solid;"> 58
<TD style="border-bottom:2px black solid;"> 63
<TD style="border-bottom:2px black solid;"> 70
<TD style="border-bottom:2px black solid;"> 70
<TD style="border-bottom:2px black solid;"> 66
<TD style="border-bottom:2px black solid;"> 67
<TD style="border-bottom:2px black solid;"> 67
<TD style="border-bottom:2px black solid;"> 67
<TR>
<TD style="border-right:2px black solid;">failures
<TD> 3
<TD> 2
<TD> 1
<TD> 1
<TD> 1
<TD> 1
<TD> 1
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TR style="height:20px">
<TR>
<TD style="border-right:2px black solid; border-bottom:2px black solid;">temp °F
<TD style="border-bottom:2px black solid;"> 68
<TD style="border-bottom:2px black solid;"> 69
<TD style="border-bottom:2px black solid;"> 70
<TD style="border-bottom:2px black solid;"> 70
<TD style="border-bottom:2px black solid;"> 72
<TD style="border-bottom:2px black solid;"> 73
<TD style="border-bottom:2px black solid;"> 75
<TD style="border-bottom:2px black solid;"> 76
<TD style="border-bottom:2px black solid;"> 76
<TD style="border-bottom:2px black solid;"> 78
<TD style="border-bottom:2px black solid;"> 79
<TD style="border-bottom:2px black solid;"> 80
<TD style="border-bottom:2px black solid;"> 81
<TR>
<TD style="border-right:2px black solid;">failures
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
<TD> 0
The temperature that day was forecast to be .
<li> NASA based the decision to launch partially on a chart showing
only the flights that had at least one O-ring failure.
Find the line that best fits these seven flights.
On the basis of this data,
predict the number of O-ring failures when the temperature is , and
when the number of failures will exceed four.
<li> Find the line that best fits all 24 flights.
On the basis of this extra data,
predict the number of O-ring failures when the temperature is , and
when the number of failures will exceed four.
Which do you think is the more accurate method of predicting?
(An excellent discussion appears in Dalal Folkes Hoadley 1989.)
Problem 7
This table lists the average distance from the sun to
each of the first seven planets, using earth's average as a unit.
<TR>
<TD style="border-bottom:2px black solid">Mercury
<TD style="border-bottom:2px black solid"> Venus
<TD style="border-bottom:2px black solid"> Earth
<TD style="border-bottom:2px black solid"> Mars
<TD style="border-bottom:2px black solid"> Jupiter
<TD style="border-bottom:2px black solid"> Saturn
<TD style="border-bottom:2px black solid"> Uranus
<TR>
<TD> 0.39
<TD> 0.72
<TD> 1.00
<TD> 1.52
<TD> 5.20
<TD> 9.54
<TD> 19.2
<li> Plot the number of the planet
(Mercury is , etc.) versus the distance.
Note that it does not look like a line, and so finding the
line of best fit is not fruitful.
<li> It does, however look like an exponential curve.
Therefore, plot the number of the planet versus the
logarithm of the distance.
Does this look like a line?
<li> The asteroid belt between Mars and
Jupiter is thought
to be what is left of a planet that broke apart.
Renumber so that Jupiter is , Saturn is , and Uranus is
, and plot against the log again.
Does this look better?
<li> Use least squares on that data to predict the
location of Neptune.
<li> Repeat to predict where Pluto is.
<li> Is the formula accurate for Neptune and Pluto?
This method was used to help discover Neptune (although the second item is
misleading about the history; actually, the discovery of Neptune
in position
prompted people to look for the "missing planet" in position ).
See Gardner 1970
Problem 8
William Bennett has proposed an Index of Leading Cultural Indicators
for the US Bennett 1993.
Among the statistics cited are the average daily hours spent watching TV,
and the average combined SAT scores.
<TR>
<TD style="border-right:2px black solid;border-bottom:2px black solid;">
<TD style="border-bottom:2px black solid;"> 1960
<TD style="border-bottom:2px black solid;"> 1965
<TD style="border-bottom:2px black solid;"> 1970
<TD style="border-bottom:2px black solid;"> 1975
<TD style="border-bottom:2px black solid;"> 1980
<TD style="border-bottom:2px black solid;"> 1985
<TD style="border-bottom:2px black solid;"> 1990
<TD style="border-bottom:2px black solid;"> 1992
<TR>
<TD style="border-right:2px black solid;">TV
<TD> 5:06
<TD> 5:29
<TD> 5:56
<TD> 6:07
<TD> 6:36
<TD> 7:07
<TD> 6:55
<TD> 7:04
<TR>
<TD style="border-right:2px black solid;">SAT
<TD> 975
<TD> 969
<TD> 948
<TD> 910
<TD> 890
<TD> 906
<TD> 900
<TD> 899
Suppose that a cause and effect relationship is proposed between
the time spent watching TV and the decline in SAT scores (in this article,
Mr. Bennett does not argue that there is a direct connection).
<li> Find the line of best fit relating the independent variable
of average daily TV hours to the dependent variable of SAT scores.
<li> Find the most recent estimate of the average daily TV hours
(Bennett's cites Neilsen Media Research as the source of these
estimates).
Estimate the associated SAT score.
How close is your estimate to the actual average?
(Warning: a change has been made recently in the SAT,
so you should investigate whether some adjustment needs to be made to
the reported average to make a valid comparison.)
/Solutions/
Computer Code
#!/usr/bin/python
# least_squares.py calculate the line of best fit for a data set
# data file format: each line is two numbers, x and y
n = 0
sum_x = 0
sum_y = 0
sum_x_squared = 0
sum_xy = 0
fn = raw_input("Name of the data file? ")
datafile = open(fn,"r")
while 1:
ln = datafile.readline()
if ln:
data = ln.split()
x = float(data[0])
y = float(data[1])
n += 1
sum_x += x
sum_y += y
sum_x_squared += x*x
sum_xy += x*y
else:
break
datafile.close()
slope = (n*sum_xy - sum_x*sum_y) / (n*sum_x_squared - sum_x**2)
intercept = (sum_y - slope*sum_x)/n
print "line of best fit: slope= %f intercept= %f" % (slope, intercept)
Additional Data
Data on the progression of the world's records (taken from the Runner's World web site) is below.
<TR>
<TD colspan="3" style="border-left:2px black solid;border-right:2px black solid;border-top:2px black solid; border-bottom:2px black solid; text-align:center;">
Progression of Men's Mile Record
<TR>
<TD style="border-left:2px black solid;border-bottom:2px black solid;"> time <TD style="border-bottom:2px black solid;"> name <TD style="border-right:2px black solid; border-bottom:2px black solid;"> date
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:52.0 <TD> Cadet Marshall (GBR) <TD style="border-right:2px black solid;"> 02Sep52
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:45.0 <TD> Thomas Finch (GBR) <TD style="border-right:2px black solid;"> 03Nov58
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:40.0 <TD> Gerald Surman (GBR) <TD style="border-right:2px black solid;"> 24Nov59
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:33.0 <TD> George Farran (IRL) <TD style="border-right:2px black solid;"> 23May62
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:29 3/5 <TD> Walter Chinnery (GBR) <TD style="border-right:2px black solid;"> 10Mar68
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:28 4/5 <TD> William Gibbs (GBR) <TD style="border-right:2px black solid;"> 03Apr68
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:28 3/5 <TD> Charles Gunton (GBR) <TD style="border-right:2px black solid;"> 31Mar73
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:26.0 <TD> Walter Slade (GBR) <TD style="border-right:2px black solid;"> 30May74
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:24 1/2 <TD> Walter Slade (GBR) <TD style="border-right:2px black solid;"> 19Jun75
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:23 1/5 <TD> Walter George (GBR) <TD style="border-right:2px black solid;"> 16Aug80
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:19 2/5 <TD> Walter George (GBR) <TD style="border-right:2px black solid;"> 03Jun82
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:18 2/5 <TD> Walter George (GBR) <TD style="border-right:2px black solid;"> 21Jun84
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:17 4/5 <TD> Thomas Conneff (USA) <TD style="border-right:2px black solid;"> 26Aug93
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:17.0 <TD> Fred Bacon (GBR) <TD style="border-right:2px black solid;"> 06Jul95
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:15 3/5 <TD> Thomas Conneff (USA) <TD style="border-right:2px black solid;"> 28Aug95
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:15 2/5 <TD> John Paul Jones (USA) <TD style="border-right:2px black solid;"> 27May11
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:14.4 <TD> John Paul Jones (USA) <TD style="border-right:2px black solid;"> 31May13
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:12.6 <TD> Norman Taber (USA) <TD style="border-right:2px black solid;"> 16Jul15
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:10.4 <TD> Paavo Nurmi (FIN) <TD style="border-right:2px black solid;"> 23Aug23
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:09 1/5 <TD> Jules Ladoumegue (FRA) <TD style="border-right:2px black solid;"> 04Oct31
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:07.6 <TD> Jack Lovelock (NZL) <TD style="border-right:2px black solid;"> 15Jul33
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:06.8 <TD> Glenn Cunningham (USA) <TD style="border-right:2px black solid;"> 16Jun34
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:06.4 <TD> Sydney Wooderson (GBR) <TD style="border-right:2px black solid;"> 28Aug37
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:06.2 <TD> Gunder Hagg (SWE) <TD style="border-right:2px black solid;"> 01Jul42
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:04.6 <TD> Gunder Hagg (SWE) <TD style="border-right:2px black solid;"> 04Sep42
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:02.6 <TD> Arne Andersson (SWE) <TD style="border-right:2px black solid;"> 01Jul43
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:01.6 <TD> Arne Andersson (SWE) <TD style="border-right:2px black solid;"> 18Jul44
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:01.4 <TD> Gunder Hagg (SWE) <TD style="border-right:2px black solid;"> 17Jul45
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:59.4 <TD> Roger Bannister (GBR) <TD style="border-right:2px black solid;"> 06May54
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:58.0 <TD> John Landy (AUS) <TD style="border-right:2px black solid;"> 21Jun54
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:57.2 <TD> Derek Ibbotson (GBR) <TD style="border-right:2px black solid;"> 19Jul57
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:54.5 <TD> Herb Elliott (AUS) <TD style="border-right:2px black solid;"> 06Aug58
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:54.4 <TD> Peter Snell (NZL) <TD style="border-right:2px black solid;"> 27Jan62
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:54.1 <TD> Peter Snell (NZL) <TD style="border-right:2px black solid;"> 17Nov64
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:53.6 <TD> Michel Jazy (FRA) <TD style="border-right:2px black solid;"> 09Jun65
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:51.3 <TD> Jim Ryun (USA) <TD style="border-right:2px black solid;"> 17Jul66
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:51.1 <TD> Jim Ryun (USA) <TD style="border-right:2px black solid;"> 23Jun67
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:51.0 <TD> Filbert Bayi (TAN) <TD style="border-right:2px black solid;"> 17May75
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:49.4 <TD> John Walker (NZL) <TD style="border-right:2px black solid;"> 12Aug75
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:49.0 <TD> Sebastian Coe (GBR) <TD style="border-right:2px black solid;"> 17Jul79
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:48.8 <TD> Steve Ovett (GBR) <TD style="border-right:2px black solid;"> 01Jul80
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:48.53 <TD> Sebastian Coe (GBR) <TD style="border-right:2px black solid;"> 19Aug81
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:48.40 <TD> Steve Ovett (GBR) <TD style="border-right:2px black solid;"> 26Aug81
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:47.33 <TD> Sebastian Coe (GBR) <TD style="border-right:2px black solid;"> 28Aug81
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:46.32 <TD> Steve Cram (GBR) <TD style="border-right:2px black solid;"> 27Jul85
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:44.39 <TD> Noureddine Morceli (ALG) <TD style="border-right:2px black solid;"> 05Sep93
<TR>
<TD style="border-left:2px black solid; border-bottom:2px black solid; border-right:2px black solid;"> 3:43.13 <TD style="border-bottom:2px black solid;"> Hicham el Guerrouj (MOR) <TD style="border-bottom:2px black solid; border-right:2px solid black;"> 07Jul99
<TR>
<TD colspan="3" style="border-left:2px black solid;border-right:2px black solid;border-top:2px black solid; border-bottom:2px black solid; text-align:center;">
Progression of Men's 1500 Meter Record
<TR>
<TD style="border-left:2px black solid;border-bottom:2px black solid;"> time <TD style="border-bottom:2px black solid;"> name <TD style="border-right:2px black solid; border-bottom:2px black solid;"> date
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:09.0 <TD> John Bray (USA) <TD style="border-right:2px black solid;"> 30May00
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:06.2 <TD> Charles Bennett (GBR) <TD style="border-right:2px black solid;"> 15Jul00
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:05.4 <TD> James Lightbody (USA) <TD style="border-right:2px black solid;"> 03Sep04
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:59.8 <TD> Harold Wilson (GBR) <TD style="border-right:2px black solid;"> 30May08
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:59.2 <TD> Abel Kiviat (USA) <TD style="border-right:2px black solid;"> 26May12
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:56.8 <TD> Abel Kiviat (USA) <TD style="border-right:2px black solid;"> 02Jun12
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:55.8 <TD> Abel Kiviat (USA) <TD style="border-right:2px black solid;"> 08Jun12
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:55.0 <TD> Norman Taber (USA) <TD style="border-right:2px black solid;"> 16Jul15
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:54.7 <TD> John Zander (SWE) <TD style="border-right:2px black solid;"> 05Aug17
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:53.0 <TD> Paavo Nurmi (FIN) <TD style="border-right:2px black solid;"> 23Aug23
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:52.6 <TD> Paavo Nurmi (FIN) <TD style="border-right:2px black solid;"> 19Jun24
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:51.0 <TD> Otto Peltzer (GER) <TD style="border-right:2px black solid;"> 11Sep26
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:49.2 <TD> Jules Ladoumegue (FRA) <TD style="border-right:2px black solid;"> 05Oct30
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<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:49.0 <TD> Luigi Beccali (ITA) <TD style="border-right:2px black solid;"> 17Sep33
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<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:48.8 <TD> William Bonthron (USA) <TD style="border-right:2px black solid;"> 30Jun34
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<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:47.8 <TD> Jack Lovelock (NZL) <TD style="border-right:2px black solid;"> 06Aug36
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<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:47.6 <TD> Gunder Hagg (SWE) <TD style="border-right:2px black solid;"> 10Aug41
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<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:45.8 <TD> Gunder Hagg (SWE) <TD style="border-right:2px black solid;"> 17Jul42
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<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:45.0 <TD> Arne Andersson (SWE) <TD style="border-right:2px black solid;"> 17Aug43
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<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:43.0 <TD> Gunder Hagg (SWE) <TD style="border-right:2px black solid;"> 07Jul44
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:42.8 <TD> Wes Santee (USA) <TD style="border-right:2px black solid;"> 04Jun54
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:41.8 <TD> John Landy (AUS) <TD style="border-right:2px black solid;"> 21Jun54
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<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:40.8 <TD> Sandor Iharos (HUN) <TD style="border-right:2px black solid;"> 28Jul55
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:40.6 <TD> Istvan Rozsavolgyi (HUN) <TD style="border-right:2px black solid;"> 03Aug56
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:40.2 <TD> Olavi Salsola (FIN) <TD style="border-right:2px black solid;"> 11Jul57
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:38.1 <TD> Stanislav Jungwirth (CZE) <TD style="border-right:2px black solid;"> 12Jul57
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:36.0 <TD> Herb Elliott (AUS) <TD style="border-right:2px black solid;"> 28Aug58
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:35.6 <TD> Herb Elliott (AUS) <TD style="border-right:2px black solid;"> 06Sep60
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:33.1 <TD> Jim Ryun (USA) <TD style="border-right:2px black solid;"> 08Jul67
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:32.2 <TD> Filbert Bayi (TAN) <TD style="border-right:2px black solid;"> 02Feb74
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:32.1 <TD> Sebastian Coe (GBR) <TD style="border-right:2px black solid;"> 15Aug79
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:31.36 <TD> Steve Ovett (GBR) <TD style="border-right:2px black solid;"> 27Aug80
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:31.24 <TD> Sydney Maree (usa) <TD style="border-right:2px black solid;"> 28Aug83
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:30.77 <TD> Steve Ovett (GBR) <TD style="border-right:2px black solid;"> 04Sep83
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:29.67 <TD> Steve Cram (GBR) <TD style="border-right:2px black solid;"> 16Jul85
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:29.46 <TD> Said Aouita (MOR) <TD style="border-right:2px black solid;"> 23Aug85
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:28.86 <TD> Noureddine Morceli (ALG) <TD style="border-right:2px black solid;"> 06Sep92
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 3:27.37 <TD> Noureddine Morceli (ALG) <TD style="border-right:2px black solid;"> 12Jul95
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid; border-bottom:2px black solid;"> 3:26.00 <TD style="border-bottom:2px black solid;"> Hicham el Guerrouj (MOR) <TD style="border-right:2px black solid; border-bottom:2px black solid;"> 14Jul98
<TD colspan="3" style="border-left:2px black solid;border-right:2px black solid;border-top:2px black solid; border-bottom:2px black solid; text-align:center;">
Progression of Women's Mile Record
<TR>
<TD style="border-left:2px black solid;border-bottom:2px black solid;"> time <TD style="border-bottom:2px black solid;"> name <TD style="border-right:2px black solid; border-bottom:2px black solid;"> date
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 6:13.2 <TD> Elizabeth Atkinson (GBR) <TD style="border-right:2px black solid;"> 24Jun21
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:27.5 <TD> Ruth Christmas (GBR) <TD style="border-right:2px black solid;"> 20Aug32
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:24.0 <TD> Gladys Lunn (GBR) <TD style="border-right:2px black solid;"> 01Jun36
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:23.0 <TD> Gladys Lunn (GBR) <TD style="border-right:2px black solid;"> 18Jul36
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:20.8 <TD> Gladys Lunn (GBR) <TD style="border-right:2px black solid;"> 08May37
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:17.0 <TD> Gladys Lunn (GBR) <TD style="border-right:2px black solid;"> 07Aug37
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:15.3 <TD> Evelyne Forster (GBR) <TD style="border-right:2px black solid;"> 22Jul39
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:11.0 <TD> Anne Oliver (GBR) <TD style="border-right:2px black solid;"> 14Jun52
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:09.8 <TD> Enid Harding (GBR) <TD style="border-right:2px black solid;"> 04Jul53
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:08.0 <TD> Anne Oliver (GBR) <TD style="border-right:2px black solid;"> 12Sep53
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:02.6 <TD> Diane Leather (GBR) <TD style="border-right:2px black solid;"> 30Sep53
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:00.3 <TD> Edith Treybal (ROM) <TD style="border-right:2px black solid;"> 01Nov53
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 5:00.2 <TD> Diane Leather (GBR) <TD style="border-right:2px black solid;"> 26May54
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:59.6 <TD> Diane Leather (GBR) <TD style="border-right:2px black solid;"> 29May54
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:50.8 <TD> Diane Leather (GBR) <TD style="border-right:2px black solid;"> 24May55
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:45.0 <TD> Diane Leather (GBR) <TD style="border-right:2px black solid;"> 21Sep55
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:41.4 <TD> Marise Chamberlain (NZL) <TD style="border-right:2px black solid;"> 08Dec62
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:39.2 <TD> Anne Smith (GBR) <TD style="border-right:2px black solid;"> 13May67
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:37.0 <TD> Anne Smith (GBR) <TD style="border-right:2px black solid;"> 03Jun67
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:36.8 <TD> Maria Gommers (HOL) <TD style="border-right:2px black solid;"> 14Jun69
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:35.3 <TD> Ellen Tittel (FRG) <TD style="border-right:2px black solid;"> 20Aug71
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:34.9 <TD> Glenda Reiser (CAN) <TD style="border-right:2px black solid;"> 07Jul73
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:29.5 <TD> Paola Pigni-Cacchi (ITA) <TD style="border-right:2px black solid;"> 08Aug73
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:23.8 <TD> Natalia Marasescu (ROM) <TD style="border-right:2px black solid;"> 21May77
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:22.1 <TD> Natalia Marasescu (ROM) <TD style="border-right:2px black solid;"> 27Jan79
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:21.7 <TD> Mary Decker (USA) <TD style="border-right:2px black solid;"> 26Jan80
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:20.89 <TD> Lyudmila Veselkova (SOV) <TD style="border-right:2px black solid;"> 12Sep81
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:18.08 <TD> Mary Decker-Tabb (USA) <TD style="border-right:2px black solid;"> 09Jul82
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:17.44 <TD> Maricica Puica (ROM) <TD style="border-right:2px black solid;"> 16Sep82
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:15.8 <TD> Natalya Artyomova (SOV) <TD style="border-right:2px black solid;"> 05Aug84
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:16.71 <TD> Mary Decker-Slaney (USA) <TD style="border-right:2px black solid;"> 21Aug85
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid;"> 4:15.61 <TD> Paula Ivan (ROM) <TD style="border-right:2px black solid;"> 10Jul89
<TR>
<TD style="border-left:2px black solid; border-right:2px black solid; border-bottom:2px black solid;"> 4:12.56 <TD style=" border-bottom:2px black solid;"> Svetlana Masterkova (RUS) <TD style="border-right:2px black solid; border-bottom:2px black solid;"> 14Aug96
References
Bennett William Quantifying America's Decline Wall Street Journal March 15, 1993 1993
Dalal Siddhartha Folkes Edward Hoadley Bruce Lessons Learned from Challenger: A Statistical Perspective Stats: the Magazine for Students of Statistics Fall 1989 1989 14-18
Gardner Martin Mathematical Games, Some mathematical curiosities embedded in the solar system Scientific American April 1970 1970 108-112
Oakley Cletus Baker Justine Least Squares and the 3:40 Mile Mathematics Teacher April 1977 1977
Linear Algebra Topic: Line of Best Fit Projection Onto a Subspace Topic: Geometry of Linear Maps