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Pearson's Correlation, Clearly Explained!!! - YouTube
Channel: StatQuest with Josh Starmer
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correlation it's the sensation across
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the nation stack quests hello I'm Josh
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starburns welcome to stack quest today
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is part two in our series on covariance
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and correlation this time we're going to
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talk about correlation however before we
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dive deep into correlation I want to
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talk about relationships not the fun
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and/or confusing kind we sometimes find
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ourselves in will you hold my hand um
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you don't have a hand
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you're just a stick figure dang instead
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I want to talk about the relationships
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between data on the x-axis and data on
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the y-axis
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in this example we're looking at mRNA
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transcripts from gene X in five
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different cells on the x-axis and from
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gene Y in the same five different cells
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on the y-axis
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however if mRNA transcripts doesn't mean
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anything to you
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imagine we went into five different
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grocery stores and put the number of
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green apples on the x-axis in the number
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of red apples on the y-axis
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each pair of measurements were taken
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from a single cell or grocery store and
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can be represented by a blue dot
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we can see that in general relatively
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low values for gene X are paired with
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relatively low values for gene Y and
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relatively high values for gene X are
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paired with relatively high values for
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gene y
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we can use a straight line with a
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positive slope to represent this trend
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and if someone told us that they
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collected a new measurement for gene X
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20 then we can use the line to predict
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that when gene X equals 20 then the
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value for gene Y should be somewhere
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around 27
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alternatively if someone gave us a value
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for gene Y we could use the trend to
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predict a range of values for gene X
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both cases we made guesses based on the
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trend we observed in the data
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if the data were closer to the trendline
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then given a gene X value
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we might guess that the value for gene Y
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falls in a smaller range
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in this case the closer the data are to
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the line the more gene X can tell us
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about gene y alternatively we could say
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that the relationship between gene X and
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gene Y is relatively strong
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the data were further from the trendline
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then we might guess that the value for
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Jean Y falls in a larger range
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in this case we could say that the
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values for gene X tell us less about the
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values for gene y alternatively we could
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say that the relationship between gene X
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and gene Y is relatively weak
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note just to be clear all we are saying
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is that we observed that low values for
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gene X tend to be paired with low values
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for gene Y and that high values for gene
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X tend to be paired with relatively high
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values for gene Y and that this
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observation suggests a trend that we can
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use to make predictions and inferences
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aka
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educated guesses we are not saying that
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a low value for gene X causes gene Y to
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have a low value or that a high value
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for gene Y causes gene X to have a high
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value in other words we are not ruling
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out the possibility that something else
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causes the trend that we observe small
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bam so far we have looked at a
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relatively weak relationship and a
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relatively strong relationship we can
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quantify the strength of a relationship
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with correlation
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in other words these data with a
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relatively weak relationship have a
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small correlation value these data with
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a moderate relationship have a moderate
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correlation value
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and these data with a strong
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relationship have a relatively large
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correlation value the maximum value for
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correlation is 1
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correlation equals one when a straight
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line with a positive slope can go
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through the center of every data point
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this means that if someone gave us a
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value for gene X
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then we could guess that jean y had a
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value in a very very narrow range
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note correlation does not depend on the
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scale of the data
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in fact I intentionally omitted putting
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numbers on the axes because they do not
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affect correlation at all
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in other words regardless of the scale
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of the data correlation equals one when
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a straight line with a positive slope
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can go through all of the data
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that means that correlation can equal
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one when the slope is large and when the
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slope is small
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note when a straight line with a
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positive slope goes through the data
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correlation equals one regardless of how
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much data we have
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for example if we only had two data
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points then we can draw a straight line
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with a positive slope by just connecting
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the two dots
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and then correlation equals one and that
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makes the relationship appear strong
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but we should not have any confidence in
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predictions made with this line because
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we have so little data
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to understand why we should have low
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confidence in correlations made with
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small datasets let's start with an empty
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graph and draw two random points on it
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then just like before we could draw a
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straight line that goes through the
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center of each point just by connecting
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the dots and that means correlation
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equals one for these two randomly drawn
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dots
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in fact we can always draw a straight
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line between any two random dots
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now let's go back to the original data
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and imagine that instead of two pairs of
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measurements we had three pairs of
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measurements
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now just like before since we can draw a
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straight line through all three points
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correlation equals one however now we
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can have more confidence in the
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predictions we make with this line
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this is because if we started with an
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empty graph and drew three random points
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on it then even though it's easy to draw
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a straight line to connect any two
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points there is a very small chance that
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we will be able to draw a straight line
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through all three points ultimately the
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probability that we can connect three
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randomly drawn points with a straight
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line is very small and thus we can have
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more confidence that the observed
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correlation isn't just the result of
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random chance
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in general the more data we have the
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more confidence we have in the
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predictions we make with the line
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because the probability that we can draw
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a straight line through the same number
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of randomly placed points gets smaller
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and smaller with each additional point
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note we could draw a squiggly line that
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connects all of the dots
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but when we're talking about correlation
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were only talking about using straight
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lines
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oh no it's the dreaded terminology alert
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for correlation a p-value tells us the
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probability that randomly drawn dots
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will result in a similarly strong
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relationship or stronger
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thus the smaller the p-value the more
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confidence we have in the predictions we
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make with the line in this case the
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p-value is crazy small 2.2 times 10 to
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the negative 16 which means that the
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probability of random data creating a
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similarly strong or stronger
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relationship is crazy small
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to summarize what we've talked about so
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far the maximum value for correlation
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one occurs whenever you can draw a
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straight line with a positive slope that
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goes through all of the data and our
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confidence in how useful the
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relationship is depends on how much data
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we have
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of these three examples we should have
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the least confidence in this
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relationship since it is supported by
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the least amount of data and we should
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have the most confidence in this
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relationship since it is supported by
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the most data and has the smallest
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p-value BAM
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when a straight line with a negative
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slope can go through the center of every
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data point then the correlation equals
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negative one
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since a straight line can go through all
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of the data points correlation equals
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negative one implies that there is a
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strong relationship in the data and if
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someone gives us a value for gene X then
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we can guess a value for gene Y within a
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very narrow range
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just like before our confidence in that
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guess which we quantify with a p-value
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depends on how much data we have if we
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had a lot of data we could have a lot of
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confidence in the guess because the
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p-value would be super small and the
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less data we have the less confidence we
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have in the guess because the p-value
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gets larger
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like before as long as a straight line
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goes through all of the data and the
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slope of the line is negative
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correlation equals negative one when the
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slope is large and when the slope is
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small BAM so far we've seen that when
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the slope of the line is negative the
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strongest relationship has correlation
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equal to negative one and when the slope
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of the line is positive the strongest
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relationship has correlation equal to
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one
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in both cases if a straight line cannot
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go through all of the data then we will
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get correlation values closer to zero
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and the worse the fit the closer the
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correlation gets to zero
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and when there is no relationship that
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we can represent with a straight line
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correlation equals zero when correlation
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equals zero a value on the x-axis
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doesn't tell us anything about what to
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expect on the y-axis because there is no
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reason to choose one value over another
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BAM as long as the correlation value is
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not zero we can still use the line to
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make inferences
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but our guesses become more refined the
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closer the correlation values get to
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negative one or one
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and just like before our confidence in
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our inferences depends on the amount of
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data we have collected and the p-value
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in the left graph we have very little
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confidence in the trim because we have
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very little data and the p value equals
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0.8
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in the middle we have moderate
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confidence in the trend because we have
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more data and the p value equals 0.08 on
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the right we have a lot of confidence in
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the trend because we have even more data
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in the p value equals zero point zero
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zero eight note the correlation equals
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zero point three in all three examples
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in this case increasing the sample size
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did not increase correlation and that
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means adding data did not refine our
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guests
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all it did was increase our confidence
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in the guess
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thus our guesses will probably be pretty
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bad in all three cases however we'll
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have the most confidence in the bad
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guest that came from this data
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in other words just because you have a
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lot of data and you have a lot of
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confidence in your guests
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if the correlation value is small your
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guests will still be bad double bam if
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you know how to calculate variance and
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covariance calculating correlation is a
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snap note if you're not already familiar
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with the concepts of variance and
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covariance check out the quests the
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links are in the description below
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if this were the data then the
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correlation equals the covariance of
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gene X and gene Y divided by the square
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root of the variance for gene X times
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the square root of the variance for gene
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y
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as we saw in the stat quest on
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covariance the numerator can be any
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value between positive and negative
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infinity depending on whether the slope
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of the line that represents the
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relationship is positive or negative how
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far the data are spread out around the
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means and the scale of the data
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thus when we calculate correlation the
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denominator squeezes the covariance to
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be a number from negative 1 to 1 in
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other words the denominator ensures that
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the scale of the data does not affect
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the correlation value and this makes
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correlations much easier to interpret
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when the data all fall on a straight
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line with a positive or negative slope
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then the covariance and the product of
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the square root of the variance terms
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are the same and division gives us 1 or
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negative 1 depending on the slope
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when the data do not fall on a straight
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line with a positive or negative slope
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then the covariance accounts for less of
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the variance in the data and the
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correlation is closer to zero
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as we saw in the stat quest on
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covariance the covariance value for this
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data is 116 so the denominator will
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squeeze 116 down to a value from
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negative 1 to 1
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the variants in the gene X data is 100
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1.8 and the variants in the gene Y data
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is 160 point 3 and when we do the math
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we get 0.9
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like I mentioned earlier we can quantify
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our confidence in this relationship with
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a p-value
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the smaller the p-value the more
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confidence we can have in the guesses we
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make in this case the p-value is 0.03
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that means that there is a 3% chance
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that random data could produce a
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similarly strong relationship or
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stronger
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triple bam
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before we go there's one last important
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thing I want to mention about
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correlation even though correlation
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values are way easier to interpret then
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covariance values they are still not
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super easy to interpret for example it's
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not super obvious that this relationship
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where correlation equals zero point nine
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is twice as good as making predictions
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as this relationship where correlation
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equals zero point six four
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the good news is that R squared which is
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related to correlation solves this
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problem the better news is that if you
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want to learn more about r-squared you
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can check out these quests the links are
[1038]
in the description below
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pS another awesome thing about R squared
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is that it can quantify relationships
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that are more complicated than simple
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straight lines
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in summary correlation quantifies the
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strengths of relationships if you have a
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weak relationship then you will have a
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small correlation value if you have a
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moderate relationship then you'll have a
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moderate correlation value and if you
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have a strong relationship then you will
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have a large correlation value
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correlation values go from negative one
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which is the strongest linear
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relationship with a negative slope to
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one which is the strongest linear
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relationship with a positive slope in
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both cases if a straight line cannot go
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through all of the data then we will get
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correlation values closer to zero and
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the worse the fit the closer the
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correlation values get to zero and when
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there is no relationship that we can
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represent with a straight line
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correlation equals zero lastly our
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confidence in the inferences depends on
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the amount of data we have collected and
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the p-value
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the more data we have the smaller the
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p-value and the more confidence we have
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in our inferences BAM hooray
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we made it to the end of another
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exciting stat quest if you like this
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stack quest and want to see more please
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subscribe and if you want to support
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stack quest consider buying one or two
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of my original songs or a t-shirt or a
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hoodie or just donate the links are in
[1144]
the description below alright until next
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time quest on
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