Regression Line

Finance allah shmoop what is a regression line You know

that one friend in your group who never has a

beef with anyone who always remembers little details about everyone's

lives who always has a kind word and even volunteers

that help you move All right A regression line is

kind of like that for a set of linear ish

data always trying to stay as close to every data

point as possible no matter how far away some data

points try to get In fact it's the one line

of best fit that minimizes the distances between the data

points and the line That's the regression line Before we

get too carried away with the ins and outs of

finding these elusive regression lines it's important to remember that

we only find regression lines for data that is probably

linear We safe probably because there's no way to be

sure that a data set must be linear only a

bunch of circumstantial evidence that it might be linear There

are several bits that all work together to help us

feel okay with saying data are linear ish but probably

the most important and the only one we really need

to worry about Here in finance land is a linear

pattern in the scatter plot Okay Example Here we go

Polly the ceo of the world renowned plant distributor polly's

pretty plants sells vegetable plants She has grown in her

vast network of greenhouses Okay you got us She works

out of her mom's basement and she's thirteen Still polly

is a maven for experimentation in statistical data gathering So

much so that she has gathered the following data on

the different amounts of her special fertilizer water mixture given

to several plants from the same packet of tomato plant

seeds all planted in soil from the same spot in

her mom's backyard Well polly would like to be able

to predict the amount of growth a seed will experience

based on how much fertilizers puts in the water The

whole point of the regression line is to allow her

to do this at least to a certain degree of

accuracy Right Like that's we're getting it but first she

needs to know if the data points are linear okay

so polly quickly whips up a scatter plot that's what

thirteen year old girls d'oh isn't it and see what

appears to be a roughly believing your pattern right Well

now we could draw lines on that scatter plot until

the cows come home But only one is the line

of best fit or the line that gets a smallest

bunch of distances from each data point possible All right

well what would this regression line look like Okay well

as an eyeball on lee approach a line of best

fit does not have to hit any of the data

points but it should follow the slant or slope of

the data and tries to split the points so that

the distance is straight up from the line to The

points are balanced out by distances straight down from the

line to the point Unless you start with line any

line and it goes up from left to right All

right this first line does nothing right We usually try

to fix the slow first and what we need to

make the slope less steep In other words decrease the

slow period little by little until we have a pretty

good match to the slope of the data That last

line looks like it's pretty close to the slant of

the data And again this is just an eyeball approach

so yeah it won't be perfect We've got the slow

part pretty locked in But we don't have the split

The data points with equal distances thing going for us

yet we need to move the whole line straight up

until it kind of splits The data points a little

by little until we can get a good split There

don't have to be equal numbers of points above and

below It's more about equal distances So we have three

points all about the same medium ish distance above with

a very close point a medium close point and a

kind of farpoint below Yeah down there s o The

total distance above is about equal to the total distance

below What We could find the equation of the eyeball

line but it definitely isn't the perfect best fit line

It's just close So how do we find the equation

by hand Well the formula for the slope of the

regression line is m equals The correlation coefficient are times

the standard deviation of the wide data s sub y

divided by the standard deviation of the ecs data s

sub x right We typically don't find the correlation coefficient

or the standard deviations by hand especially since they're super

duper easy to get via technology like a graphing calculator

spreadsheet our website But well you can check out some

of our other videos of you really want to gold

if we pop polly's data from before into a t

I graphing calculator can run a lillian wreg when your

aggression to get pr value Then run one of our

stats on both the x and y data to get

standard deviations But we get the correlation Coefficient are to

be zero point eight six five four in santa deviation

of the ecs data s sub x to be one

point eight seven Oh wait right there and stare a

deviation of the white data s Why to be two

point seven Oh five three Easy Okay that in turn

gives a slope of our regression line of point eight

six five Four times two point seven oh five three

divided by one point eight seven away which is approximately

one point two five one four So we did all

the math there for you What We're still missing the

y intercept So how do we find that for polly

Wealth The formula for the y intercept of the regression

line b is found by taking the mean of the

wide data Why bar Minus the product of the slope

em And we just calculated in the mean of the

ecs data x bar again we usually get the two

means x bar And why bar using tech This is

the twenty first century after all People using the same

one var stats from before on each data set using

our tricked out diamond plate covered voice activated t i

graphing count Well that gives us exper equal to four

point five and wiebe are equaled a fourteen point two

six six seven to go along with our slope of

one point two five one four Well r y intercept

then is fourteen point two six six seven minds one

point two five one four times four point five which

is approximately eight point six three five for a lot

of numbers at you don't We've thrown them Yeah but

what's our regression equation than wealth We jam the slope

And why intercept into slope intercept form that y equals

mx plus b thing and we get y equals one

point two five one four Acts plussed And that's b

8 point six whatever well plotted on the original scatter

plot This is how that line looks right there Finding

the slope and white intercept using the formulas or find

again twenty first century people Come on ask your parents

No one except maybe old man hostetler who still thinks

calculators or a tool of the devil Does any of

this by hand And what any of you be really

mad if we told you that we already had the

equation on a screen way back at the beginning of

the by hand calculations Yeah we knew you'd be be

cool about it Seems that when we found the r

value we also had the slope And why intercepted the

regression line staring us right in the face just above

the are there see that value in the a equals

ro that's the slope of the regression line See the

b equals ro Yeah that's the why intercept the fact

that the values are a wee bit different than ours

Like poor decimal places in That's just a rounding here

so polly's got a regression line What can she do

with it Well she could use it to predict likely

growth of another of those same variety of seeds based

on how much fertilizer she adds And let's say probably

uses a five point five cubic centimeters of fertilizer How

tall might her plant be What we just plug five

point five in for acts in the regression equation giving

us white schools one point two five one four times

five point five plus eight point six three five four

which is fifteen point five and change centimeters She could

expect the plan to be about fifteen point five plus

centimeters tall after the same time period passes and probably

won't be exactly that value But it should at least

be in the neighborhood Regression lines give predictions not guarantees

So to recap regression lines are the best fit lines

to a set of data with linear pattern In this

case the phrase best If it means the line reduces

the vertical distance between the points and the best fit

line to as small as possible we can find the

slope and intercept of the regression line using the formula

or we could just use tact to do everything for

us And we can use the regression equation to help

us predict either x or y values with the expectation

that the real result will probably be close to the

value predicted by the regression equation Now we just need

some regression to help us figure out if we were 00:07:39.223 --> [endTime] albert einstein or fred flintstone in a previous life