Quantitative DataEssential Statistics

Quantitative DataO Review

O Quantitative data is any data that produces a measurement or amount of something

O Examples: Age, distance traveled, height, weightO Utilizes a variety of graphs

O Dot plotsO Stem plotsO Back to Back StemplotsO Line graphsO ScatterplotsO HistogramsO Boxplots

Numerical variables

Oor quantitative Oobservations or measurements

take on numerical valuesOmakes sense to average these

valuesOtwo types - discrete &

continuous

Discrete (numerical)

Olistable set of values

Ousually counts of items

Continuous (numerical)

Odata can take on any values in the domain of the variable

Ousually measurements of something

Classifying variables by the number of variables in a data set

Suppose that the PE coach records the height of each student in his class.

Univariate - data that describes a single characteristic of the population

This is an example of a univariate data

Classifying variables by the number of variables in a data set

Suppose that the PE coach records the height and weight of each student in his class.

Bivariate - data that describes two characteristics of the population

This is an example of a bivariate data

Classifying variables by the number of variables in a data set

Suppose that the PE coach records the height, weight, number of sit-ups, and number of push-ups for each student in his class.

Multivariate - data that describes more than two characteristics (beyond the scope of this course)

This is an example of a multivariate data

Identify the following variables:

1. the appraised value of homes in Plano

2. the color of cars in the teacher’s lot

3. the number of calculators owned by students at your school

4. the zip code of an individual

5. the amount of time it takes students to drive to school

Discrete numerical

Discrete numerical

Continuous numerical

Categorical

Categorical

Is money a measurement or a count?

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 GraphA 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 graphO 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.

O 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)

O Create a line graph for this data.

Our Line GraphCreate a table Create the graphYear # in

College

2004 106

2005 108

2006 99

2007 126

2008 117

2009 138

2010 139

2011 141

2012 138

2013 147 2004

2006

2008

2010

2012

020406080

100120140160

# of Students in college

# of Stu-dents in col-lege

ScatterplotsOStart 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

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?

Ht 74 65 77 72 68 60 62 73 61 64

Wt 256 124 320 185 158 129 103 196 110 130

Is it positive or negative? Weak or strong?

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.

1. Heights of mothers & heights of their adult daughters

+2. Age of a car in years and its current value

3. Weight of a person and calories consumed

4. Height of a person and the person’s birth month

5. Number of hours spent in safety training and the number of accidents that occur

-+NO

-

Correlation Coefficient (r)-

O A quantitative assessment of the strength & direction of the linear relationship between bivariate, quantitative data

O Pearson’s sample correlation is used most

O parameter - r (rho)O statistic – rO How do I determine strength?

Moderate CorrelationStrong correlation

Properties of r(correlation coefficient)

Olegitimate values of r is [-1,1]

0 .5 .8 1-1 -.8 -.5

No 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

14

20

20

19

22

23

19

21

22

25

30

22

15

18

16

18

19

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 TVHours doing homework

22.5

40.5

3.50.5

22

1.53

2.52

31

50

12

0.53

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

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

transformationsx (in mm) 12 15 21 32 26 19 24y 4 7 10 14 9 8 12

Find r.Change to cm & find r.

The correlations are the same.

Do the following transformations & calculate r

1) 5(x + 14)2) (y + 30) ÷ 4

• value of r does not depend on which of the two variables is labeled x

Switch x & y & find r.

The correlations are the same.

Type: LinReg L2, L1

•value of r is non-resistant

x 12 15 21 32 26 1924

y 4 7 10 14 9 822

Find r.Outliers affect the correlation

coefficient

• 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 9Y 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

Correlation does not imply causation

Correlation does not imply causation