Correlation & Forecasting

Learning Outcomes In this lecture, you'll: Understand the concept of Linear Correlation and degree of correlation through Pearson’s Correlation Coefficient & its calculations. Understand trendlines and how they can help you in sales forecasting. Use sample sales data to create a trendline in a chart and to forecast sales. Find the right trendline for your data. Name a trendline.

Correlation & Scatter Graph Correlation exists between two variables when one of them is related to the other in some way A scatter graph is a graph in which the paired (x,y) sample data are plotted with a horizontal x axis and a vertical y axis. Each individual (x,y) pair is plotted as a single point on the graph. page 509 of text

Correlation with Scatter Graph Scatter graph OR scattergram is a method of representing tabular data on a graphical figures. Examine following table:

Now the corresponding scattergram:

Notes: Note that each point represents a ‘pair’ of corresponding value & that 2 scales relate to the 2 variables under discussion. Term Scatter diagram comes from its appearance. From the illustration of scattergram, one could see that ‘taller men’ are ‘heavier’ than ‘shorter men’ This also shows that there is relation between the heights & weights of men!! But this does NOT mean larger ‘Height’ causes larger ‘Weight’. This is expressed in statistical terms as; The two variables, height & weight are CORRELATED.

Degrees of Correlation Perfectly Correlated Both variables increasing in linear fashion. All points lie exactly on the same straight line.

Degrees of Correlation Partly Correlated Points do not lie exactly on straight line. But do suggest a relation being clustered around a straight line. 8

Degrees of Correlation Un-correlated Random points on the graph. No suggestions of any relations. 9

Degrees of Correlation Positive Or Direct Correlation High values of ‘x’ related with High values of ‘y’. Negative Or Inverse Correlation High values of ‘x’ associated with Low values of ‘y’ & vice versa. 10

Pearson’s Correlation Coefficient- Formula Pearson developed a measure of the amount (degree) of linear correlation present in a set of pairs of data (x & y). Denoted by ‘r’; given as: Where ‘n’ is the number of data points; And; ‘x’ & ‘y’ are the correlated variables

Pearson’s Correlation Coefficient- Interpretation This measure has the property of always lying in the range of -1 to +1, where: r = +1 denotes perfect positive linear correlation (data points lie exactly on a straight line with positive gradient) r = -1 denotes perfect negative linear correlation (data points lie on straight line but with negative gradient) r = 0 denotes no linear correlation. Strength of correlation is judged by the proximity of the values towards +1 and -1. 12

Types of Correlation Linear Correlation When relationship of two variables can be represented by a straight line; eg: car ownership and family income. Non – Linear Correlation Relation between two variables cannot be illustrated with a straight line; eg; rainfall and crops; too high rainfall may mean lower crops, but good rainfall would still mean higher crops. 13

Nature of Regression lines Regression line: Straight line drawn through the scatter diagram that lies somewhere in the midst of the collection of points & slope in the direction suggested by the points Often referred to as ‘Line of Best Fit’, as it is drawn in order to represent the best possible linear relation between scatter points on the graph. E.g.: Monthly output of a factory; -y = total monthly costs -scatter diagram with a line that we think best fits the situation

Regression & connection with Correlation Interpolation & Extrapolation: Regression lines can be utilised to calculate values of dependent variable NOT observed in the data set. When estimated value of dependent variable lie within the observed range; it is called Interpolation; while forecast of values outside the observed range through regression is called Extrapolation; Assumption is made that there is always a linear relation between the two variables. If degree of correlation between 2 variables is high, then the estimates made through regression equations would be reasonably accurate. Care should be taken as correlation coefficient & regression equations are drawn from same data set; assumption of constant correlation Or extrapolation is better to be carried out on a larger sample.