A scatter plot is a graph that relates numerical data with two parameters in a coordinate plane. The data points are plotted as ordered pairs. For instance, the number of ice cream cones sold daily at a kiosk can be plotted against the temperature that day.
When there is a statistical connection between two parameters of data, such that a change in one is associated with a change in the other, they are said to be correlated. For instance, up until approximately age there is a correlation between age and height: older people are generally taller, and taller people are generally older.
Causation is a relationship between two correlated quantities where one directly affects the other.
If two quantities correlate in such a way that an increase of one quantity is associated with an increase in the other, they are said to be positively correlated. Likewise, an increase in one quantity associated with a decrease in the other, is called a negative correlation.
The more the data points appear to follow a specific trend, the more correlated they are. If they are situated almost exactly on a line, the quantities are said to be strongly correlated, while if they are more spread out, the quantities are weakly correlated.
How strongly two quantities correlate can be described using the correlation coefficient, It can take on values between and Values near means that the correlation is strong and negative, while a strong, positive correlation leads to a value close to If there is no correlation, it has a value of
Data that has been drawn in a scatter plot and shows a moderate or strong correlation can be modeled using a line of fit. This is a line drawn on the scatter plot that is as close to as many data points as possible. For instance, a line of fit can be drawn for the weight of kittens plotted against their age.
The music-and-animal amusement park "Hiphop-opotamus Park" has hippos that they regularly measure. The length and height of each hippo at the time of the last measurement has been plotted below.
Draw a line of fit to the data. Then, comment on the apparent association in the data. Lastly, estimate the equation of the line and interpret its slope and -intercept in context.
A line of fit should be drawn so that as many points as possible are close to the line, with roughly half above and half below.
Looking at the graph, we can see that an increase in height is associated with an increase in length. Thus, the line of fit has a positive slope. Therefore, there is a positive correlation between a hippo's height and length. Note that height does not directly cause length, nor does length directly cause height. Thus, we can argue that there is no causation between these quantities, only a correlation.
Now that a line of fit has been drawn, we can approximate its equation in point-slope form. To find the slope of the line, we'll use two points that are on the line — not necessarily points from the data set.
The calculations above approximate the following about the line of fit.
Since represents the height of a hippo and represents the length, the slope tells us that for every foot in height the hippo has an additional feet in length.
Similarly, the -intercept means that a hippo whose height measures feet will be feet long. Notice that this last statement doesn't make sense. This is because the point lies outside the given data set. The relationship established for the given data set only applies to values that fall within it. In other words, extrapolation is not reliable while interpolation is.
When a line of fit has been drawn on a scatter plot, it is possible to determine how well the line models the data. This can be done by analyzing the residuals. A residual is the difference between a data point and the line of fit.
Most graphing calculators have a function called linear regression which can be used to find a precise line of fit, using strict rules. This line of fit is then called the line of best fit or the regression line.
The first step is to enter the data points in the calculator. On a TI calculator, this is done by first pressing the STAT button, and then selecting the option Edit....
This gives a number of columns, marked L, L, L etc.
With the help of the arrow keys, choose where in the lists to fill in the data values. Enter the data points' -values in list L and the corresponding -values in L Values are entered into the fields using the number buttons followed by ENTER.
After entering the values, press the STAT button, then select the menu item CALC to the right.
The option LinReg(ax+b) gives a line of best fit, expressed as a linear function in slope-intercept form.