A scatter plot is a graph that shows each observation of a bivariate data set as an ordered pair in a coordinate plane. Consider the following example in which a scatter plot shows the results obtained at a local ice cream parlor of a study that recorded the number of ice creams sold and the corresponding air temperature.
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 18, 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, r. It can take on values between -1 and 1. Values near -1 means that the correlation is strong and negative, while a strong, positive correlation leads to a value close to 1. If there is no correlation, it has a value of 0.
When data sets have a positive or negative correlation, the trend of the data can be modeled using a line of fit, also called a trend line. This line is drawn on a scatter plot near most of the data points, which appear evenly distributed above and below the line.
The scatter plot above shows the mean weights of kittens from the same litter in relation to their age. In this case, a line of fit could be drawn quite seamlessly. When drawing such a line of fit, the following characteristics should be considered.
The music-and-animal amusement park "Hiphop-opotamus Park" has 20 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 y-intercept in context.
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.
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. For example, consider the following data set.
The line of best fit can be calculated following these 3 steps.
On a graphing calculator, begin by entering the data points. To do so, press the button and select the option
This gives a number of columns, labeled L1, L2, L3, and so on.
Use the arrow keys to choose where in the lists to fill in the data values. Enter the x-values of the data points in L1 and press after each value. The same can be done for the the corresponding y-values in column L2.
After entering the values, press the button and select the menu item
In this case, the line of best fit is described by the function y=-7.354x+39.506. The correlation coefficient r is less than -0.99, indicating a strong negative correlation. If r does not appear, press 0, then select
DiagnosticOn and enable it by pressing twice. Once more, find the line of best fit.
A graphing calculator can also be used to graph the line of best fit. After selecting the option
LinReg(ax+b), choose the option
Press and move to the Y-VARS menu. Then, select the option
FUNCTION and press
Press the button until the parameters are given. To graph the scatter plot, first push the buttons and Then, choose one of the plots in the list. Select the option
ON, choose the type to be a scatter plot, and assign L1 and L2 as
Then, the plot can be made by pressing the button It is possible that after drawing the plot the window-size is not large enough to see all of the information.
To fix this, press and select the option
ZoomStat. After doing that, the window will resize to show the important information.