STATISTICAL ANALYSIS

Statistical Analysis



Statistical Analysis

It is a common scenario: A practitioner has sales data for the past several months and wants to forecast next month's sales volume. This type of forecasting can help manufacturers and distributors ensure they have enough product to meet customer demands. But how is this forecasting done? Statistical analysis software offers two ways to plot the data in order to make a forecast: 1) a linear trend model or 2) a quadratic trend model. It is important for practitioners to understand both methods, as each can be beneficial, depending on the type of process being analyzed.

Part 1 - Linear Trend

The following explanations use the sample data shown in Table 1.

Table 1: Sample Volume Data

Month

Volume

Jan. 2009

98

Feb. 2009

105

March 2009

116

April 2009

119

May 2009

135

June 2009

156

July 2009

177

Aug. 2009

208

To begin, use statistical analysis software to create a time series plot with a linear trend analysis (Figure 1).

Figure 1: Trend Analysis Plot for Volume - Linear Trend Model

The software will generate a fitted line using the equation Yt = 71.43 + (15.1 x t). The t represents the time period during which each data point was collected - i.e., the first time period is 1, the second is 2 and so on. Hence, if someone wants to know the fitted value for January 2009, it is 71.43 +15.1*(1) = 86.53.

Table 2: Fitted Values for Past Months

Month

Volume

Fitted Value

t

Jan. 2009

98

86.53

1

Feb. 2009

105

101.63

2

March 2009

116

116.73

3

April 2009

119

131.83

4

May 2009

135

146.93

5

June 2009

156

162.03

6

July 2009

177

177.13

7

Aug. 2009

208

192.23

8

To forecast for September 2009, the practitioner would get 207.33 (71.43 + (15.1 x 9)).

But how does the software get the equation Yt = 71.43 + (15.1 x t)? It is nothing but linear regression. If practitioners used the linear regression function in their statistical analysis software instead, using volume for Y and the t (1, 2, 3, 4, etc.) for X they would get the same equation:

Regression Analysis: Volume versus t The regression equation isVolume = 71.4 + 15.1 t

Predictor    Coef  SE Coef     T      PConstant   71.429    8.626  8.28  0.000t                  15.071    1.708  8.82  0.000

S = 11.0701   R-Sq = 92.8%   R-Sq(adj) = 91.7%

Analysis of Variance

Source             DF        SS         MS       F         PRegression       1     9540.2  9540.2  77.85  0.000Residual Error   6      735.3    122.5Total                   7  10275.5

Another potentially confusing element of the linear trend plot is the forecast accuracy measures: MAD, MAPE and MSD. These are used to determine how well the trend will accurately predict the future volume.

MAD

MAD stands for mean absolute deviation, which is the average of the absolute deviations. An absolute deviation is the absolute value of the actual data minus the fitted value (Table 3).

Table 3: Sample Data Including Absolute Deviation

Month

Volume

Fitted Value

t

Absolute Deviation

Jan. 2009

98

86.53

1

11.47

Feb. 2009

105

101.63

2

3.37

March 2009

116

116.73

3

0.73

April 2009

119

131.83

4

12.83

May 2009

135

146.93

5

11.93

June 2009

156

162.03

6

6.03

July 2009

177

177.13

7

0.13

Aug. 2009

208

192.23

8

15.77

Sum

62.26

n

8

MAD

7.7825

Table 4: Forecast Using Last Month's Volume

Month

Volume

Fitted Value

t

Absolute Deviation

Jan. 2009

98

80

1

18

Feb. 2009

105

98

2

7

March 2009

116

105

3

11

April 2009

119

116

4

3

May 2009

135

119

5

16

June 2009

156

135

6

21

July 2009

177

156

7

21

Aug. 2009

208

177

8

31

 

 

 

Sum

128

 

 

 

n

8

 

 

 

MAD

16

The MAD value allows the practitioner to conclude that the model generated by linear regression is better than the model generated by last month's volume.

MSD

The linear trend plot also uses the accuracy measure MSD, which stands for mean square deviation. It is very similar to MAD, but instead of summing the absolute deviations, this measure sums up the squared deviations (Table ...