Statistics 2 – (Two Variable)

IB Mathematical Studies SL

Syllabus reference

Content Detail

Inferential Statistics

Two variable statistics involves discovering if two variables are related or linked to each other in some way. e.g. - Does IQ determine income?

- Is there a link between foot size and the height of a person?

One variable is independent (x-axis) whilst the other is dependent (y-axis)

This section of statistics involves conclusions that can be made about data that has not been collected using data that has been collected.

Hence we can infer or predict certain points based on the data collected. Often this involves sampling as analysing an entire population can be difficult.

Methods

A scatter plot is necessary to quickly determine whether the variables are related, however, more formally we may need to measure:

(1) Correlation – initially it may be necessary to determine if a relationship exists between two or more variables(Pearson’s product moment correlation coefficient)

(2) Regression analysis – if a relationship appears to exist we can then conduct further analysis to determine the type and strength of the relationship(Linear Regression or Least Squares Regression)

Correlation

Correlation refers to the relationship or association between two variable.

They are classified qualitatively in three ways:› Direction – positive, negative, none› Strength – weak, moderate, strong› Type – linear or non-linear

They are classified quantitatively by Pearson’s product-moment correlation coefficient

Outliers must also be considered and usually appear as isolated points away from the main body (group) of data.

Exam hint - use this language!

Correlation Scatter graphsPositive linear correlations

Negative linear correlations

CAUTION! - Causation

Be careful not to jump to conclusions when you determine a strong correlation between two variables – why?› It does not mean that a causal relationship

exists, i.e. one variable does not necessarily cause the other.

› e.g – there is a strong correlation between arm length and running speed, does that mean that short arms cause a reduction in running speed?

A causal relationship only exists when they are directly correlated such that if one variable is changed the other changes as well.

Pearson’s product moment correlation coefficient (r)

The “r” value that your GDC gives in statistical calc mode is a measure of the strength of the correlation

It lies between -1 and 1› The closer to 1 the r-value is, the stronger

the (positive) correlation› The closer to 0 the r-value is, the weaker

the correlation› The closer to -1 the r-value is, the stronger

the (negative) correlation

Linking “r” to terminologyCorrelation coefficient value Description of strength &

direction

1 Perfect positive

0.8 to 1 Strong positive

0.6 to 0.8 Moderate positive

0.4 to 0.6 Weak positive

0 to 0.4 No correlation

-0.4 to 0 No correlation

-0.6 to -0.4 Weak negative

-0.8 to -0.6 Moderate negative

-1 to -0.8 Strong negative

-1 Perfect negative

Note: These are only guideline values, there is no specific division points where the description has to change from strong to

moderate etc.

Correlation Scatter graphs with “r” values

Formula for “r”

There are several formulae for calculating “r” but the one given and used in the IB course is:

› sxy is the covariance(It will always be given if required).

› Sx is the standard deviation of x data values

› Sy is the standard deviation of y data values

r =sxy

sxsy

Exam hint – make sure you know

that sx is σx and sy is σy on your

GDC!

Example – correlation coefficient

Use the data in the table below to calculate the r –value, given sxy=7.92

› Calculate the standard deviation of x› Calculate the standard deviation of y› Evaluate “r” using the IB formula and

compare it to your calculator.

# 1 2 3 4 5 6 7 8 9 10

x 72 85 94 92 73 81 86 95 78 72

y 6 5 6 7 4 4 6 7 5 3

Line of Best Fit & Linear Regression

The line of best fit is the “quick and easy” way of finding the trend of the data› By eye it should have approximately the same

number of data points above the line as below› A more accurate method is to calculate the mean of

the x data and y data and ensure the trend line passes through this point called the mean point

Linear regression is the most accurate process for determining the trend line, as the process takes every data point in to account via a formula.

x, y( )

Syllabus reference

Content Detail

Example – line of best fit A statistician wants to know if there is a correlation

between HSC maths scores and the Math Studies IB exam scores. She collected the following data from 10 randomly selected students.

› Is there a correlation ? If so, what kind?› Draw the scatter plot of IB vs HSC› Draw the line of best fit by finding the mean of each

variable.› If an HSC score is 77, predict the corresponding IB score.› If an IB score was a 2, predict the corresponding HSC score.

# 1 2 3 4 5 6 7 8 9 10

HSC 72 85 94 92 73 81 86 95 78 72

IB 6 5 6 7 4 4 6 7 5 3

Formula for Linear Regression

The line of best fit by the process of linear regression can be found using the given IB formula:

› sxy is the covariance(It will always be given if required).

› Sx is the standard deviation of x data values

› sx2 = (sx)2 i.e the std dev of x, then squared

y−y=sxy

sx2 x−x( )

Example – regression line

Find the equation of the line of best fit in y=ax+b form using the linear regression formula, if:sxy = 9.23sx = 3.46

= (14.4, 35.2)x, y( )

Extrapolation vs Interpolation

Once you have a line of best fit you can use that equation to infer or predict what would happen to one variable if the other changes.

If you are predicting values within the range of your current data then you are said to be “interpolating”.› The accuracy of interpolation depends on the accuracy of

your line of best fit and your r-value If you are predicting values outside the range of

your data then you are “extrapolating”.› The accuracy of extrapolation not only on the accuracy

of your line of best fit but also whether it is reasonable to assume that the same trend will continue outside your range of data.

Graph of extrapolation and interpolation ranges