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12. Bivariate Data Analysis

Check out the overview section for relevant importance of this section compared to other topics in this course. In 2020 exam, two questions appeared from this topic making up 6 marks (2020Q12, 2020Q36). This section covers the following parts of the syllabus:

– MS-S3 Further Statistical Analysis

– MS-S4 Bivariate Data Analysis

The infographic below shows all the past exam questions from 2010 to 2020 relevant to this topic sorted by difficulty level and further broken down into sub-topics. This will form the foundation of our study as we would like you to focus first on the easy questions and quickly develop skills to get those easy marks and then challenge yourself with the harder ones.

This section explores the relationship between two variables for which we are given a few data points and don’t know about the relationship that exists between them.

The questions that are of interest when exploring this relationship are:

- How do the data points look on a scatterplot and if there is a visible relationship from eyeballing? (Discussed under ‘Scatterplot’)
- If it seems there is a relationship what is the strength of that relationship in quantitative terms? (Discussed under ‘Correlation’)
- If there exists a line that best defines this relationship, what is that line and how it can be used to predict? (Discussed under ‘Best Fit Line’).

__Scatterplot:__

Scatterplot is a representation of two variables on a graph in the form of dots.

The following charts show different types of relationships:

Sometimes you will find a point on scatterplot which stands out from the otherwise explainable relationship. These points are called outliers.

*Example 1*

The following table shows the study hours versus marks for 20 students. Draw it on scatterplot and comment on the relationship and determine and explain any outliers.

__Correlation Coefficient:__

In the previous section, the relationship was determined by eyeballing and then categorized as strong, moderate or weak. Eyeballing may not work well sometimes when the two scatterplots look similar with little difference so there needs to be a standard way of quantifying this relationship which should make comparison easier.

This is called Pearson correlation coefficient (r).

*Example 2*

Which one of the following charts is most likely to be associated with these correlation coefficients; i) 1 ii) -0.3 iii) 0.6 iv) -0.75?

__Best Fit Line:__

Knowing that relationship exists between variables and knowing its strength leads to the next question as to if there is a way to define this relationship.

Best fit line is a line that, as the name suggests, fits these dots in the best possible way. By best possible way it means that the overall difference between the points and the line is minimized.

For example, it may look something like following where it can’t have all points on scatterplot on the line but overall the distance between line and each point is the least possible among all the ways line can be drawn.

Once this line is determined, it can then be used to determine values that are not in the existing dataset and can be used as a predictor for the dependent variable.

*Example 3*

Best fit line for the following data has equation Weight = 1.3421 x Height – 157.82. Draw the scatterplot along with this line and determine what is the weight of person with height of 152cm as per the line? Would it be reasonable to use the equation to determine weight of person with height of 190cm?

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You might find the following videos helpful related to this section:

The following are the types of questions you can expect in exam:

Study notes of this section and other resources can be accessed here:

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