Allama Iqbal Open University

Subject Title:Subject Title:StatisticsStatistics

Presented By:Presented By:Adeel AhmadAdeel Ahmad

Roll No:Roll No:AD-514968AD-514968

Topic

• Measures of trend and seasonal variation

Area of discussion

• Time Series• Components of time series• SECULAR TREND OR TREND• Reasons for studying trends• Seasonal Variations• Reasons for studying seasonal variations

Time Series

• A time series is a set of observations taken at specified times usually at equal intervals.

• It is a set of data depending upon time.• A series of values over a period of time.• In time series, Time act as an independent

variable to estimate dependent variable. • Example: Monthly sales of a company for

the last year. 

Time Series

• The Time Series given as a variable Y, is usually given as a function of time t.

• Yt denotes the value of Y at time t. • According to additive model, the time series can be

expressed as: • Y=T+S+C+I , where • Y= Value of original time series. • T= Trend value. • S= Seasonal variation. • C= Cyclical variation. • I= Irregular variation.

Causes in variations in time series data

• Social Customs, Festivals• Seasons• The four phase of business:

1. Prosperity, decline ,depression ,recovery• Natural Climate:

1. Earthquake ,food etc• Political Movement/Changes, War etc

Importance of Time series Analysis

• A very popular tool for business forecasting

• Basis for understanding past behavior• Can forecast future activities/Planning for

future operation

Components of time series

• SECULAR TREND OR TREND• SEASONAL VARIATION/FLUCTUATIONS• CYCLICAL VARIATION/FLUCTUATIONS• IRREGULAR VARIATION AND MOVEMENT

Variation in time Series

Type of Variation

Long term Short term

SECULAR CYCLICAL SEASONAL IRREGULAR

SECULAR TREND OR TREND

• SECULAR TREND (or) TREND Secular trend is the smooth, regular and long term movement of series showing a continuous growth or decline over a long period of time.

• The general tendency of the data to grow or decline over a long period of time.

• Examples: • Upward trend in economic growth due to

increasing population, price, etc. Downward trends in a time series is relating to death and birth rates, etc. 

Reasons for studying trends

• The study of secular trends allows us to describe a historical pattern

• Studying secular trends permits us to project past patterns or trends in to the future.

• Two methods: • Freehand (or) graphic method. • Least square method.

FITTING THE TREND BY LEAST SQUARE METHOD

• FITTING THE TREND BY LEAST SQUARE METHOD Principle: The sum of squares of the deviations of the actual and computed values is least for the line of the fit. To fit a straight line,

• Y=a+bx, • where a= ΣY/n• b= ΣXY/ΣX2 • n: number of observations

Fit a straight line trends by the method of least squares to the following data:

Year 2003 2004 2005 2006 2007 2008

Production

Tones

24 25 29 26 22 24

Year Production

X

X-x

Xy x² Yc Fluctuation

1991 24 -2.5 -60 6.25 25.858 1.8575

1992 25 -1.5 -37.5 2.25 25.514 0.514

1993 29 -0.5 -14.5 0.25 25.172 -3.828

1994 26 0.5 13 0.25 24.828 -1.172

1995 22 1.5 33 2.25 24.486 2.486

1996 24 2.5 60 6.25 24.142 0.142

Total 150 0 -6 17.5

X = 1993.5 n= 6 a= Y = 25 b= (ΣXY/ ΣX2) = -0.343

Y=25 - 0.343X

Seasonal Variations

• SEASONAL VARIATIONS Involves patterns of change within a year that tend to be repeated from year to year. Short term periodic movement.

• Example: Increase in the number of flu and viral fever cases in winter every year.

SEASONAL VARIATION

• SEASONAL VARIATION For detecting seasonal variations, time intervals must be measured in small units, such as days, weeks, months or quarters.

• Reasons for studying seasonal variations: To describe a historical pattern.

• To project past patterns into future.

• Two methods: • Method of Averages. • Ratio-to-Moving-Average Method.

SEASONAL VARIATIONS METHOD OF AVERAGES Steps

• Find the quarterly averages for the 4 quarters of the given years: X1,X2,X3,X4.

• Calculate the Grand Average, G. • G= X1 + X2 + X3 + X4

4 • Find the seasonal indices for each quarter. • Seasonal Index for ith quarter = Xi * 100

G

The Quarterly sales for a graphics software company are given below. Determine the seasonal components.

Year 2003 2004 2005 2006 2007 2008 Total Average

Q1 500 450 350 550 550 750 3150 525

Q2 350 350 200 350 400 500 2150 358.33

Q3 250 200 150 250 350 400 1600 266.67

Q4 400 300 400 550 600 650 2900 483.33

G=(525+358.33+266.67+483.33) 4 = 408.3325

Seasonal IndexQUARTER SEASONAL INDEX

Q1 128.575

Q2 87.454

Q3 65.307

Q4 118.367

SEASONAL VARIATIONS Ratio-to-Moving-Average Method

• Steps: Calculate the 4-quarter moving total.

• Compute the 4-quarter moving average. • Center the 4-quarter moving average. • Calculate the percentage of actual value to

the moving average value. • Calculate the modified mean. Adjust the

modified mean.