Scatter plots

Scatter plots exercise

• Summarize your knowledge about scatter plots.

  Hints

  We examine if the value of $x$ has an impact on the value of $y$.

  A line is given by the equation $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.

  A line with a positive slope is increasing. This one has a negative slope and is decreasing.

  Solution

  On a coordinate plane, scatter plots are used to show relationships between variables in order to recognize trends.

  Take the first scatter plot as an example: It shows the impact of the popularity rating of a DJ to the number of guest attending a party. A DJ with a 50% popularity rating has 200 guests in attendance and one with 80% leads to 350 guests.

  So we can assume a trend (correlation): The higher the DJ popularity rating the higher the number of guests.

  The points on the graph are grouped closely together. This indicates a high correlation. In this case the correlation is positive.

  So you can draw a trend line, also called the line of best fit.

  For the line of best fit, you can calculate the slope as well as the $y$-intercept using two given points on the line.

• Interpret the different scatter plots.

  Hints

  Is there a line that fits the given data? If so, this line is given by the equation $y=mx+b$. Where $m$ is the slope.

  An increasing line of best fit has a positive slope and thus a positive correlation.

  If the data isn't grouped at all there is no correlation.

  If the data doesn't change depending on $x$, that means a line of best fits parallel to the $x$-axis, there is no correlation.

  Solution

  Let's consider the diagrams from left to the right:

  1. When the date is spread out with no pattern we can conclude that there is no correlation and no trend.
  2. But even if data is grouped together we can't conclude a correlation. If the line of best fit is horizontal we have then measure on the $x$-axis has no influence on what we're measuring on the $y$-axis. Therefore, no correlation exists.
  3. If the line of best fit is a vertical line, the slope is undefined. Thus, we have no correlation and no trend.
  4. An increase in grouped data from left to right represents a positive correlation
  5. A decrease in grouped data from left to right represents a negative correlation.

  Note: correlation does not mean causation.

• Draw a scatter plot.

  Hints

  Pay attention to the labelling of the $x$- as well as $y$-axis.

  If you want to draw the point $(220,190)$, draw a line parallel to the $x$-axis passing $y=190$, and one parallel to the $y$-axis passing $x=220$. The intersection of those lines is the wanted point.

  The age is represented by $x$, while the number of friends is represented by $y$.

  Solution

  Here you can see the resulting scatter plot. To each age ($x$), there is a number of friends ($y$) given. So we can conclude, in total seven ordered pairs, which you can see in this diagram from left to the right:

  • $(220,190)$
  • $(230,170)$
  • $(250,160)$
  • $(280,140)$
  • $(320,140)$
  • $(350,130)$
  • $(380,120)$

  How can you draw a given ordered pair in a coordinate plane?

  Let's have a look at $(280,140)$:

  • Draw a line parallel to the $x$-axis passing $y=140$.
  • Draw a line parallel to the $y$-axis passing $x=280$.
  • The intersection of those lines is the wanted point.

• Interpret the given scatter plot.

  Hints

  An increasing line of best fit stands for a positive correlation.

  The $x$-axis represents the amount of effort needed, and the $y$-axis represents the amount of fun had.

  Solution

  Let's pick some pets:

  With turtles, the effort they take isn't so much... however, the resulting amount of fun isn't too high either.

  With cats and dogs, perhaps the most beloved pets, the effort for a cat is a little bit less than the effort for a dog. According to this diagram, the fun is also a little bit less for a cat than for a dog. But perhaps cat lovers wouldn't agree.

  The pets which take the most effort are the horses, and they are also the animal which are the most fun.

  We can conclude that the data seems to be grouped, and that the line of best fit is increasing. So we have a positive correlation. So, the higher the effort the higher the fun and vice versa.

• Determine the slope-intercept form of the line of best fit.

  Hints

  Use this formula to find the slope.

  Use the slope-intercept form of a line ($y=mx+b$) to find the $b$ term by plugging in either point as $x$ and $y$.

  "DJ with a 50% popularity rating has 200 guests in attendance" can be represented by the ordered pair $(50, 200)$.

  "DJ with 80% popularity leads to 350 guests" can be represented with the ordered pair $(80, 350)$.

  $(50, 200)$ this point gives us $x_1 = 50$ and $y_1 = 200$.

  $(80, 350)$ this point gives us $x_2 = 80$ and $y_2 = 350$.

  Solution

  Any linear equation can be expressed in slope intercept form as $y=mx+b$.

  1. We first determine the slope $m$ by the formula:

  • $m=\frac{y_2-y_1}{x_2-x_1}$.
  • So we need two points. Those are given by the information of the impact of 50% (80%) popularity rating on the number of guests 200 (350).
  • So we have two points $(50,200)$ and $(80,350)$. Now we put the coordinates of those points in the formula above to get
  • $m=\frac{350-200}{80-50}=\frac{150}{30}=5$.

  2. This gives us $y=5x+b$ with an unknown y-intercept. Last we put the coordinates of one point into this equation. We picked to use the point $(50, 200)$ and it looks like:

  • $200=5(50)+b$.
  • Subtracting $250$ results in the y-intercept $b=200-250=-50$.

  3. So, the linear equation is $y=5x-50$.

• Explain what kind of data you can represent in a scatter plot.

  Hints

  Here you see an example of a bar graph.

  An ordinal data set is one where each data point is assigned a numerical quantity, which establishes an ordering on the entire set of data.

  A nominal data set is one where each data point is assigned to a distinct category, which does not provide a measurement or order on the set of data.

  The bar graph represents a nominal data set.

  Let's have a look at an example: if three people lived in house number $1$, four people lived in house number $2$, five people lived in house number $3$, and so on, then we couldn't conclude that the house number tells us anything about the number of people living in that house.

  Solution

  Scatter plots are used to show relations between variables to recognize trends. The data use must be ordinal in order to make a scatter plot, as there must be a way to order the points so that they can be compared.

  You can use scatter plots to try to find correlations. However, a positive (or negative) correlation doesn't have to imply a trend. For example, if three people lived in house number $1$, four people lived in house number $2$, five people lived in house number $3$, and so on, then we couldn't conclude that the house number tells us anything about the number of people living in that house.

  For ordinal data, bar or line graphs can also be used as well.

  Nominal data cannot be represented with a scatter plot, so bar graphs are usually used instead.