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ETC1000 / ETC9000 Business and Economic Statistics

Semester 1, 2010

Demonstration Lecture Week 3: Correlation, Covariance and Scatter Plots

This lecture provides examples of the material taught in this weeks lectures, to help

you see its potential for real world application, and to reinforce the ideas being

communicated.

Case Study: Attendance at lectures and your final mark!

There is a wide range of views among lecturers and students about the benefits of

attending lectures. Some say that with good quality, complete lecture notes and the

availability of audio lectures, there is little need to actually physically attend. Others

argue that a face-to-face lecture is a much better way to learn than simply reading or

listening on-line. Mostly it is lecturers who take this view after all, who wants tostand up and talk to a half-empty lecture theatre?

Is there evidence to support either view?

In 2009 one of the lecturers in Business and Economics decided to get some data to

try and address the question. She hoped that it would show that attendance at lectures

helps a lot, and that she could use this evidence to persuade students to attend lectures

more regularly.

Lets see whether the results worked out as she hoped.

Students were asked to record attendance at lectures throughout the semester, and

these will be compared with their final mark in the unit. Data is available for 100

students. The maximum number of lectures they could attend is 24 (12 weeks x 2

lectures per week).

Look first at some descriptive statistics on number of lectures attended.

Number of Lectures attended

Mean 15.62Standard Error 0.481743

Median 16

Mode 18

Standard Deviation 4.817435

Sample Variance 23.20768

Kurtosis -0.736

Skewness -0.08733

Range 18

Minimum 6

Maximum 24

Sum 1562

Count 100

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Now lets move to the potential relationship between attendance at class and the final

mark a student achieves. We look at this first with descriptive statistics for those who

attended 15 or less lectures, and compare with those who attended more than 15

lectures.

Attended 15 or less Attended 16 or more

Final Mark Final Mark

Mean 58.1087 Mean 68.11111

Standard Error 1.601397 Standard Error 1.32325

Median 59.5 Median 68

Mode 59 Mode 64StandardDeviation 10.8612

StandardDeviation 9.72386

SampleVariance 117.9657

SampleVariance 94.55346

Kurtosis 3.820277 Kurtosis 1.105246

Skewness -1.37546 Skewness -0.36536

Range 62 Range 53

Minimum 16 Minimum 39

Maximum 78 Maximum 92

Sum 2673 Sum 3678

Count 46 Count 54

Note the much higher mean and median for those who attend the higher number of

lectures.

Now consider a scatter plot.

Final Mark

0

10

20

30

40

5060

70

80

90

100

0 5 10 15 20 25 30

What do we learn from this scatter plot?

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There does appear to be a relationship (the dots are not totally random), it is positive

(higher attendance is associated with a higher mark), and a linear model would do a

reasonable job of summarising the relationship.

The message is that attending more lectures is associated with a higher mark in the

unit.

To get a handle on the strength of the relationship, lets calculate the correlation:

Number

of

Lectures

attended

Number of Lecturesattended 1

0.612081 1

A correlation of 0.612 indicates a reasonably clear relationship, but not a really strong

one. There are obviously a number of other factors that contribute to a students final

mark other than attending lectures. Some of these might be measurable things like

mathematical ability, VCE ENTER score, who the tutor is, etc. Others might be

individual-specific unmeasurable things like basic intelligence / ability, motivation,

etc.

Theres a couple of provisos

1. Correlation is not Causality

All we have shown is that attendance at lectures is associated with better final marks.

But we cannot prove with this analysis that attending more lectures will lead to an

improved mark. Maybe this correlation between these two factors is due to some

other common causal factor.

Eg. One scenario: students who attend lectures are generally more diligent and

organised, and hence work harder on their assignments and exam preparation, thus

performing well in their final mark. So it is not attending lectures per se that

improves your final mark, it is your diligence and organisational skills.

Determining causality is difficult!

2. Outliers

What if we had one freak student who was brilliant and hard working, but never

attended lectures they listened online instead. They attend just one lecture, but

scored 98%.

Lets change person number one in this way, and see what happens.

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Final Mark

0

20

40

60

80

100

120

0 5 10 15 20 25 30

Number of Lectures

attended

Final

Mark

Number of Lecturesattended 1

Final Mark 0.483083 1

Now we have a correlation of only 48%. Even though we have 100 students in thesample, this one outlier person can have quite a big impact on the correlation value.