Quantitative Data Essential Statistics

Quantitative Data Review Utilizes a variety of graphs Quantitative data is any data that produces a measurement or amount of something Examples: Age, distance traveled, height, weight Utilizes a variety of graphs Dot plots Stem plots Back to Back Stemplots Line graphs Scatterplots Histograms Boxplots

Scatter Plots Time Plots Scatter Plots – Start by placing the explanatory variable on the x-axis, and the response variable on the y-axis. Then plot each point, and look for tendencies. Positive linear correlation, Negative quadratic correlation, ect. Time Plots – Place the time on the x-axis, and the amount of the y-axis. Plot each point and then connect them. We utilize these to analyze trends as well.

Line Graph A line graph: Plots each observation against the time at which it was measured. Always put time on the horizontal axis and the variable you are measuring on the vertical axis. Connect the points by lines to display the change over time.

Creating a line graph In 2014, an parent in Belton ISD claimed that the number of students going to college each year is not growing with our growing population. Use the follow data to display the changes over time. The following is the number of students that attended college each given year starting in 2004: 106 (2004), 108 (2005), 99 (2006), 126 (2007), 117 (2008), 138 (2009), 139 (2010), 141 (2011), 138 (2012), and 147 (2013) Create a line graph for this data.

Our Line Graph Create the graph Create a table Year # in College 2004 106 2005 108 2006 99 2007 126 2008 117 2009 138 2010 139 2011 141 2012 2013 147

Scatterplots Start by placing the explanatory variable on the x-axis, and the response variable on the y-axis. Then plot each point, and look for tendencies. Positive linear correlation, Negative quadratic correlation, ect

Suppose we found the age and weight of a sample of 10 adults. Create a scatterplot of the data below. Is there any relationship between the age and weight of these adults? Age 24 30 41 28 50 46 49 35 20 39 Wt 256 124 320 185 158 129 103 196 110 130

Create a scatterplot of the data below. Suppose we found the height and weight of a sample of 10 adults. Create a scatterplot of the data below. Is there any relationship between the height and weight of these adults? Is it positive or negative? Weak or strong? Ht 74 65 77 72 68 60 62 73 61 64 Wt 256 124 320 185 158 129 103 196 110 130

The farther away from a straight line – the weaker the relationship The closer the points in a scatterplot are to a straight line - the stronger the relationship. The farther away from a straight line – the weaker the relationship

Identify as having a positive association, a negative association, or no association. + Heights of mothers & heights of their adult daughters - Age of a car in years and its current value + Weight of a person and calories consumed Height of a person and the person’s birth month NO Number of hours spent in safety training and the number of accidents that occur -

Correlation Coefficient (r)- A quantitative assessment of the strength & direction of the linear relationship between bivariate, quantitative data Pearson’s sample correlation is used most parameter - r (rho) statistic – r How do I determine strength?

Properties of r (correlation coefficient) legitimate values of r is [-1,1] Strong correlation No Correlation Moderate Correlation Weak correlation

Plotting scatter graphs This table shows the temperature on 10 days and the number of ice creams a shop sold. Plot the scatter graph. Temperature (°C) Ice creams sold 14 10 16 20 19 22 23 21 25 30 15 18 Ask pupils to predict whether or not there will be any correlation shown by the data. Decide on labels and an appropriate scale for the axes by considering the range for each measurement and number the axes using the pen tool. Ask volunteers to come to the board and plot the points from the table on the diagram. Decide if there is any correlation between the values and use the pen tool (set to draw straight lines) to draw a line of best fit.

Plotting scatter graphs We can use scatter graphs to find out if there is any relationship or correlation between two set of data. Hours watching TV Hours doing homework 2 2.5 4 0.5 3.5 1.5 3 1 5 Ask pupils to predict whether or not there will be any correlation shown by the data. Decide on labels and an appropriate scale for the axes by considering the range for each measurement and number the axes using the pen tool. Ask volunteers to come to the board and plot the points from the table on the diagram. Decide if there is any correlation between the values. Use the pen tool (set to draw straight lines) to draw a line of best fit, if appropriate. Copy this slide and modify the table to produce more examples if required.

Calculate r. Interpret r in context. Speed Limit (mph) 55 50 45 40 30 20 Avg. # of accidents (weekly) 28 25 21 17 11 6 Calculate r. Interpret r in context. There is a strong, positive, linear relationship between speed limit and average number of accidents per week.

value of r is not changed by any linear transformations x (in mm) 12 15 21 32 26 19 24 y 4 7 10 14 9 8 12 Find r. Change to cm & find r. Do the following transformations & calculate r 1) 5(x + 14) 2) (y + 30) ÷ 4 The correlations are the same.

value of r does not depend on which of the two variables is labeled x Switch x & y & find r. Type: LinReg L2, L1 The correlations are the same.

value of r is non-resistant x 12 15 21 32 26 19 24 y 4 7 10 14 9 8 22 Find r. Outliers affect the correlation coefficient

r = 0, but has a definite relationship! value of r is a measure of the extent to which x & y are linearly related Find the correlation for these points: x -3 -1 1 3 5 7 9 Y 40 20 8 4 8 20 40 What does this correlation mean? Sketch the scatterplot r = 0, but has a definite relationship!

Correlation does not imply causation