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## Statistics and probability

### Course: Statistics and probability > Unit 5

Lesson 4: Least-squares regression equations- Introduction to residuals and least squares regression
- Introduction to residuals
- Calculating residual example
- Calculating and interpreting residuals
- Calculating the equation of a regression line
- Calculating the equation of the least-squares line
- Interpreting slope of regression line
- Interpreting y-intercept in regression model
- Interpreting a trend line
- Interpreting slope and y-intercept for linear models

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# Interpreting a trend line

The graph shows how studying affects test scores. The line slope of 15 means that for each extra hour of studying, there is usually a 15-point increase in test score. But it's not a guarantee. Some students may do better or worse than the trend. Created by Sal Khan.

## Want to join the conversation?

- Can someone please explain what bivariate data is ?(21 votes)
- Same as above, but maybe a better way to understand vocab:

bi = 2

variate = variables(6 votes)

- According to that line, if someone studied about 7 hours, then their score should be ~125 - which is off the chart. The maximum score appears to be 100, so what's going on? How can the line show that 7 hours is a score of ~125?(10 votes)
- The line doesn't go on infinitely (I guess it is technically a line segment). If you plotted more and more points and the hours went up and up the line would just level off. Remember that this is data taken from the real world. It doesn't have the precision most math has. If you have a line that is plotting the amount of money you pay for flowers and one flower is 2 dollars you can have an exact, perfect line. This type of data is not like that. If you have a student who studies for 10 hours he'll probably get in the 90s but it's not definite. The line is just an estimate.(10 votes)

- Hi moderators,

i noticed that the content in this video is repeated in another video in the same module "interpreting slope of a line". the content is same in both the videos.(7 votes) - I am not sure if there is an easier way to go about this but there has to be some sort of formula. It is hard eyeballing the line and then it is even more difficult trying to measure a line on a computer screen........(4 votes)
- There is another way - the least-squares method, but it is advanced. I would suggest just zooming in your screen using CTRL and +.

https://www.khanacademy.org/math/ap-statistics/bivariate-data-ap/least-squares-regression/v/regression-residual-intro(2 votes)

- At1:58, what does Sal mean?(3 votes)
- if the first statement were true, at x=0 (did not study), you would have seen a score of 15 --> (0,15) -> but there is no datapoint to indicate this, so this first statement is false/incorrect.(3 votes)

- Does line of best fit have to be exact? The line of best fit can also be used to find slope, so if you don't place the line of best fit perfectly, the actual slope maybe a bit off. How can I fix this kind of problem?(3 votes)
- Why is it important that for a best-fit line be drawn with an equal number of data points above and below the line?(2 votes)
- It isn't actually too important. It usually just turns out that way because it's the average line of the data points make.(4 votes)

- This isn't exactly a question about the video, but is it possible to determine the line of best fit without a graph? If you have several points, and no graph, how would you determine the equation of the line of best fit?

Thanks.(2 votes)- If you had no graph, then you wouldn't be able to calculate the slope. So, I'd say your best bet is to eyeball it, and try to keep the line in the center of the data trend. If you're answering a question on paper, then definitely use a ruler. If you really wanted to calculate slope, then you could make a graph and plot the points on the graph. I find that calculating the slope is really helpful when determining the line.(3 votes)

- In the video, Sal mentions that the slope of the line on the graph, which is 15, means that for every hour a student studies, there is a 15 point improvement on their test.

Does this work every time? Is it always going to be increasing on the x-axis by one? Or does it just depend on the graph you are working with?

Also, to find the slope and analyze what it is saying (just like the question Sal is solving in the video) do you just start at a "whole number point" on the line and then travel vertically until you reach another "whole number" in order to see the change? For example, Sal first begins at point (0.5, 45) and travels horizontally to point (1.5, 60) to find the meaning in the change of slope.

Well, that was a mouthful! Thanks for anyone who helps, and I hope this helped other viewers too! :)(3 votes)

## Video transcript

Shira's math test
included a survey question asking how many
hours students had spent studying for the test. The graph below shows
the relationship between how many hours
students spent studying and their score on the test. Shira drew the line below to
show the trend in the data. Assuming the line is correct,
what does the line slope of 15 mean? So let's see. The horizontal axis is
time studying in hours. The vertical axis is
scores on the test. And each of these blue
dots represent the time and the score for
a given student. So this student right over
here spent-- I don't know, it looks like they spend
about 0.6 hours studying. And they didn't do
too well on the exam. They look like they got
below a 45, looks like a 43 or a 44 on the exam. This student over here spent
almost 4 and 1/2 hours studying and got, looks like, a 94,
close to a 95 on the exam. And what Shira
did is try to draw a line that tries
to fit this data. And it seems like it
does a pretty good job of at least showing
the trend in the data. Now, slope of 15 means
that if I'm on the line-- so let's say I'm here--
and if I increase in the horizontal
direction by 1-- so there, I increase the horizontal
direction by 1-- I should be increasing in
the vertical direction by 15. And you see that. If we increase by one hour here,
we increase by 15% on the test. Now, what that means is
that the trend it shows is that, in general,
along this trend, if someone studies
an extra hour, then if we're going
with that trend, then, hey, it seems
reasonable that they might expect to see a
15% gain on their test. Now, let's see which of
these are consistent. In general, students
who didn't study at all got scores of about
15 on the test. Well, let's see. This is neither true--
these are the people who didn't study at all, and they
didn't get a 15 on the test. And that's definitely
not what this 15 implies. This doesn't say what the people
who didn't study at all get. So this one is not true. That one is not true. Let's try this one. If one student studied for one
hour more than another student, the student who studied
more got exactly 15 more points on the test. Well, this is getting
closer to the spirit of what the slope means. But this word "exactly" is
what, at least in my mind, messes this choice up. Because this isn't saying that
it's a guarantee that if you study an hour extra that you'll
get 15% more on the test. This is just saying that
this is the general trend that this line is seeing. So it's not guaranteed. For example, we could
find this student here who studied exactly two hours. And if we look at the students
who studied for three hours, well, there's no one
exactly at three hours. But some of them-- so
this was, let's see, the student who
was at two hours. You go to three hours,
there's no one exactly there. But there's going to be
students who got better than what would be
expected and students who might get a
little bit worse. Notice, there's points
above the trend line, and there's points
below the trend line. So this "exactly,"
you can't say it's guaranteed an hour
more turns into 15%. Let's try this choice. In general, studying
for one extra hour was associated with a 15-point
improvement in test score. That feels about right. In general, studying
for 15 extra hours was associated with a 1-point
improvement in test score. Well, no, that would get the
slope the other way around. So that's definitely
not the case. So let's check our answer. And we got it right.