EDUCATION: INTRODUCTION TO STATISTICS

Introduction to Statistics

III - Time Series

1. Definition: data for a variable at different points in time.
   
   
2. Simple Example of relating two time series:
   

   		Time 1	Time 2	Time 3	Time 4	Time 5
   Time Series (TS) I:		7	8	5	4	6
   Time Series (TS) II:		2	3	0	-1	1

   
   

   

   TS I: Mean = 6
   TS II: Mean = 1

   TS I Standard Deviation:

   S = Square root of ( 1/(N-1) ) [Sum of (each data set - mean)2]

   S = Square root of 1/4 (7-6)² + (8-6)² + (5-6)² + (4-6)² + (6-6)²

   S = Square root of 1/4 (1 + 4 + 1 + 4 + 0)

   S = Square root of 1/4 (10)

   S = Square root of 10/4

   S = Square root of 2.5

   S ~ 1.58

   TS II Standard Deviation:

   S = Square root of 1/4 (1 + 4 + 1 + 4 + 0)

   S = Square root of 10/4

   S = Square root of 2.5

   S ~ 1.58

Visually, it's obvious that these two time series are related. We can make this even more obvious by plotting one time series versus the other in a "scatter plot":

The two time series are linearly related. For example all the points in the scatter plot lie in a nice straight line. How do you express this relationship with numbers?

IV - Linear Correlation Coefficient

The statistical definition of "relatedness" of two time series is called correlation. We can calculate a "correlation coefficient" r that is a measure of how two time series are related. If r = 1, the two series are perfectly positively correlated, which means that as one variable gets larger, the other one does too. If r = -1, the two time series are perfectly negatively correlated, which means that as one variable gets larger the other one gets smaller. If r = 0, then the two variables are not related.

How do you calculate a correlation coefficient?

r = sum ( each time period of (It - Imean) (IIt - IImean) )
                     (N-1)*(SI)*(SII)

Where I_t is the value of Time Series I at time equals t and I_mean is the mean of Time Series I.

Example using TS I and TS II:

r = ((7-6) (2-1) + (8-6) (3-1) + (5-6) (0-1) + (4-6) (-1- 1) + 0)
                              (5-1) (1.58) (1.58)

r = 1 + 4 + 1 + 4 + 0
      (4) (1.58) (1.58)

r = 10/10

r = 1

Therefore, the two time series are perfectly positively correlated. But in real life, r is almost never 1, -1, or 0. In the next section we learn how to interpret the significance of r in a real-life situation.